Self-Organized Temporal Criticality: Bottom-Up Resilience...
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Research Article Self-Organized Temporal Criticality: Bottom-Up Resilience versus Top-Down Vulnerability Korosh Mahmoodi , 1 Bruce J. West, 2 and Paolo Grigolini 1 1 Center for Nonlinear Science, University of North Texas, Denton, TX, USA 2 Information Science Directorate, Army Research Office, Research Triangle Park, NC, USA Correspondence should be addressed to Korosh Mahmoodi; [email protected] Received 28 October 2017; Revised 11 February 2018; Accepted 21 February 2018; Published 26 March 2018 Academic Editor: Anand Nair Copyright © 2018 Korosh Mahmoodi et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We propose a social model of spontaneous self-organization generating criticality and resilience, called Self-Organized Temporal Criticality (SOTC). e criticality-induced long-range correlation favors the societal benefit and can be interpreted as the social system becoming cognizant of the fact that altruism generates societal benefit. We show that when the spontaneous bottom-up emergence of altruism is replaced by a top-down process, mimicking the leadership of an elite, the crucial events favoring the system’s resilience are turned into collapses, corresponding to the falls of the leading elites. We also show with numerical simulation that the top-down SOTC lacks the resilience of the bottom-up SOTC. We propose this theoretical model to contribute to the mathematical foundation of theoretical sociology illustrated in 1901 by Pareto to explain the rise and fall of elites. 1. Introduction e recent book of Haidt [1] aims at explaining the psy- chological reasons for the conflicts between parties with arguments ranging from psychology to evolutionary biology and from religion to theoretical sociology. ere exists a connection between these conflicts and the societal resilience that is supposed to be sufficiently robust as to prevent either societal collapses or rapid social changes. ese important sociological issues were addressed in 1901 by Pareto [2], who discussed the capability that elites should develop in order to adapt themselves to changing circumstances. e main goal of the present paper is to contribute to the discussion on the resilience issue, with a simplified model that was recently pro- posed by our group to resolve the altruism paradox, namely, the emergence of cooperation from the social interaction of individuals who make their choice between cooperation, , and defection, , on the basis of their self-interest [3]. e new model of spontaneous organization [3] is based on the conjecture that there are close connections between resilience and information transport, resilience and con- sciousness, and between consciousness and criticality. 1.1. Criticality and Temporal Complexity. Phase transitions and critical phenomena occur frequently in nature and have been widely studied by physicists; see, for instance, [4]. e Ising model [5] originally introduced to explain ferromag- netic phase transition is well known, and the exact solution found by Onsager [6] for the occurrence of phase transition in the two-dimensional case is widely recognized as an example of outstanding theoretical achievement. In the last few years some scientists have used the Ising model to shed light on biological and neurophysiological collective processes [7–10]. More precisely, the authors of [7] used the Ising model to explain the collective behavior of biological networks and the authors of [8–10] adopted the Ising model for the purpose of supporting their hypothesis that the brain works at criticality, but without establishing a clear distinction between phase transition and self-organized criticality [11]. Finally, we have to mention that the Ising model is frequently used, see, for instance, [12, 13], to model neurophysiological data subject to the constraint of maximal entropy. e term criticality is used to denote the physical condition corresponding to the onset of a phase transition, generated by the adoption of a suitable value of the control parameter . Hindawi Complexity Volume 2018, Article ID 8139058, 10 pages https://doi.org/10.1155/2018/8139058
Transcript of Self-Organized Temporal Criticality: Bottom-Up Resilience...
Research ArticleSelf-Organized Temporal Criticality Bottom-Up Resilienceversus Top-Down Vulnerability
Korosh Mahmoodi 1 Bruce J West2 and Paolo Grigolini1
1Center for Nonlinear Science University of North Texas Denton TX USA2Information Science Directorate Army Research Office Research Triangle Park NC USA
Correspondence should be addressed to Korosh Mahmoodi free3yahoocom
Received 28 October 2017 Revised 11 February 2018 Accepted 21 February 2018 Published 26 March 2018
Academic Editor Anand Nair
Copyright copy 2018 Korosh Mahmoodi et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We propose a social model of spontaneous self-organization generating criticality and resilience called Self-Organized TemporalCriticality (SOTC) The criticality-induced long-range correlation favors the societal benefit and can be interpreted as the socialsystem becoming cognizant of the fact that altruism generates societal benefit We show that when the spontaneous bottom-upemergence of altruism is replaced by a top-down process mimicking the leadership of an elite the crucial events favoring thesystemrsquos resilience are turned into collapses corresponding to the falls of the leading elitesWe also showwith numerical simulationthat the top-down SOTC lacks the resilience of the bottom-up SOTC We propose this theoretical model to contribute to themathematical foundation of theoretical sociology illustrated in 1901 by Pareto to explain the rise and fall of elites
1 Introduction
The recent book of Haidt [1] aims at explaining the psy-chological reasons for the conflicts between parties witharguments ranging from psychology to evolutionary biologyand from religion to theoretical sociology There exists aconnection between these conflicts and the societal resiliencethat is supposed to be sufficiently robust as to prevent eithersocietal collapses or rapid social changes These importantsociological issues were addressed in 1901 by Pareto [2] whodiscussed the capability that elites should develop in order toadapt themselves to changing circumstances The main goalof the present paper is to contribute to the discussion on theresilience issue with a simplifiedmodel that was recently pro-posed by our group to resolve the altruism paradox namelythe emergence of cooperation from the social interaction ofindividuals who make their choice between cooperation 119862and defection119863 on the basis of their self-interest [3]
The new model of spontaneous organization [3] is basedon the conjecture that there are close connections betweenresilience and information transport resilience and con-sciousness and between consciousness and criticality
11 Criticality and Temporal Complexity Phase transitionsand critical phenomena occur frequently in nature and havebeen widely studied by physicists see for instance [4] TheIsing model [5] originally introduced to explain ferromag-netic phase transition is well known and the exact solutionfound byOnsager [6] for the occurrence of phase transition inthe two-dimensional case is widely recognized as an exampleof outstanding theoretical achievement In the last few yearssome scientists have used the Ising model to shed light onbiological and neurophysiological collective processes [7ndash10]More precisely the authors of [7] used the Ising model toexplain the collective behavior of biological networks and theauthors of [8ndash10] adopted the Ising model for the purpose ofsupporting their hypothesis that the brain works at criticalitybut without establishing a clear distinction between phasetransition and self-organized criticality [11] Finally we haveto mention that the Ising model is frequently used see forinstance [12 13] to model neurophysiological data subject tothe constraint ofmaximal entropyThe term criticality is usedto denote the physical condition corresponding to the onsetof a phase transition generated by the adoption of a suitablevalue of the control parameter119870
HindawiComplexityVolume 2018 Article ID 8139058 10 pageshttpsdoiorg10115520188139058
2 Complexity
The Decision-Making Model (DMM) [14] which is usedin Sections 2 and 3 was proved [15] to generate phasetransition as a function of its control parameter 119870 identicalto that of the Ising model where the control parameter isthe temperature In other words the DMM belongs to Isinguniversality class [14]
At criticality namely when the dynamics of the systemare determined by the control parameter generating phasetransition the mean field 119909(119905) which in this paper is definedas the ratio of the difference between the number of coop-erator and the number of defectors to the total number ofunits fluctuates around the vanishing value The occurrenceof a vanishing value is a crucial event The crucial events aredefined as follows The time interval between consecutivecrucial events is described by the waiting-time probabilitydensity function (PDF) 120595(120591) that in the long-time limit 120591 rarrinfin has the inverse power law (IPL) structure
120595 (120591) prop 1120591120583 (1)
with 120583 lt 3 The crucial events are renewed thereby makingthe correlation function ⟨120591119894120591119895⟩ vanish if 119894 = 119895
In the case of the brain dynamics there is wide consensuson the connection between consciousness and criticality Seefor instance [9 16ndash18] and the recent review paper [19]The electroencephalogram (EEG) signals are characterizedby abrupt changes called rapid transition processes (RTP)which are proved [20 21] to be renewal non-Poisson eventswith 120583 asymp 2 This means that the brain in the awake state is agenerator of crucial events
The crucial events are responsible for the informationtransport from one system at criticality to another system atcriticality [22] Furthermore the emergence of crucial eventsrequires that the size of the complex system is finite In thispaper119872 is the total number of units within the system Theintensity of the fluctuations of the mean field 119909(119905) obeys thegeneral prescription
Δ120577 prop 1119872] (2)
where
Δ120577 = 119860 (119905) minus 119860 (3)
When working with DMM at criticality 119860 is the mean field119909 with 119909 = 0 ] = 025 [23] In the case of SOTC [3] with119860 = 119870 see Section 2 we find ] = 05 These criticality-induced fluctuations becoming visible for finite values of119872are referred to as an expression of temporal complexity
12 Swarm Intelligence and Resilience We may afford anintuitive interpretation of crucial (complex) events using theexample of a flock of birds flying in a given direction asan effect of self-organization A crucial (complex) event isequivalent to a complete rejuvenation of the flock that after anorganizational collapsemay freely select any new flying direc-tion An external fluctuation of even weak intensity can forcethe complex system to move in a given direction if it occursat the exact instant of the free will of the SOTCmodel system
It is important to stress that the organizational collapse isnot the fall of an elite which will be discussed subsequentlybecause the flock self-organization occurs spontaneouslyand does not rest on the action of a leader The choice ofa new flying direction is thus determined by an externalstimulus of even weak intensity occurring at the same timeas the collapse thereby implying the property of complexitymatching between the perturbed and the perturbing complexsystem [14]
As mentioned earlier the crucial events favor the trans-port of information fromone complex system to another [22]Crucial events are generated by criticality and consequentlythe transport of information becomes maximally efficient atcriticality [24]
However criticality may also be Achillesrsquo heel of acomplex system if criticality is generated by a fine tuningcontrol parameter In fact committedminorities acting whena crucial event occurs in the case of DMM can make thesystem jump from the state 119862 to the state 119863 [25] Herein weshow that this lack of resilience is not shared by the bottom-up approach to SOTC modeling in fact starting from thebottom generates a very resilient social organization
13 From Criticality Generated by the Fine Tuning of aControl Parameter to Self-Organization The model of [3]is a form of spontaneous transition to criticality revealedby the emergence of events with the temporal properties ofcrucial events thereby explaining the adoption of the nameSelf-Organized Temporal Criticality (SOTC) to define it Weshow that the bottom-up SOTC modeling is resilient andthat the top-down SOTC modeling is not We believe thatthe SOTC model may help to contribute to the discussionof the sociological issues of Haidt [1] with the tools ofComplexity Science In fact Haidt emphasized that thepolitical conflict between conservatives and liberals is dueto cultural and religious influences that have the effect ofcreating divisions We believe that the top-down SOTCapproachmay be used tomodel these cultural influencesThisis an extremely difficult problemmade evenmore difficult bythe philosophical controversies on definition ofmorality [26]According to the brilliant picture of Haidt the philosophyof Hume and Menciu may be compatible with the bottom-up origin of cooperation while the hypothesis that moralitytranscends human nature an interpretation moving fromPlato to Kant [1] may justify a top-down perspective Wemake the extremely simplified assumption that the top-downSOTC undermining social resilience explains the fall ofelites if they represent only limited groups a phenomenonthat may be explained by noticing that ldquoour minds weredesigned for groupish righteousnessrdquo [1]The source of socialconflict seems to be that cultural evolution differs from lifeevolution These culturally induced conflicts may overcomethe biological origin of cooperation
14 Bottom-Up versus Top-Down Approach to Morality Forclarity in Sections 2 and 3 we provide a review of the SOTCmodel [3] while stressing some properties of SOTC modelthat were not discussed in the original paper for instancethe behavior of single units with their frequent regression
Complexity 3
to the condition of independence of the other units for thebottom-up process and the Pareto cycles of the top-downversion of the modelThe original results of this earlier paperindicated a lack of resilience of the top-down SOTC modeland a robustness of the bottom-up SOTC model which areillustrated in Section 4 We devote Section 5 to balancing theresults of the present paper against the open problems that wepropose to study in future research
2 Bottom-Up Approach to Self-OrganizedTemporal Criticality
The decisions of single individuals in our model are made inaccordance with the criterion of bounded rationality [27 28]expanded by Kahneman [29] and more recently discussedfrom within the perspective of evolutionary game theory(EGT) [30 31] The nonrational component of the decision-making process is stressed also by the work of Gigerenzer[32 33] Herein individuals make decision using DMM Theindividuals of the social network aiming at increasing theirpayoffmake the control parameter119870119903 for individual 119903 evolvetowards criticality thereby creating an intelligent groupmind[34] perhaps connected to the intelligent unconscious ofGigerenzer [33] leading the individuals to fast but essentiallywise decisions As we shall see the time evolution of 119870119903 isslow because it depends on the payoffs of the individual atearlier time corresponding to the slow thinking mechanismdiscussed by Kahneman [29]
21 The Intuitive and Emotional Level In this paper we usethe DMM on a two-dimensional lattice of size 119871 with119872 =119871 times 119871 individuals and we set 119871 = 10 The DMM is basedon individuals imperfectly imitating the majority opinion oftheir four nearest neighbors thereby biasing the probabilityof making a transition from being a cooperator (119862) to beinga defector (119863)
119892(119903)119862119863 = 1198920 expminus119870119903 (119873(119903)119862 minus 119873(119903)119863 )119873 (4)
where 119873(119903)119862 is the number of nearest neighbors to individual119903 that are cooperators 119873(119903)119863 is the number of defectors andeach individual on the simple lattice has 119873 = 4 nearestneighbors In the same way the transition rate from defectorsto cooperators 119892(119903)119863119862 is
119892(119903)119863119862 = 1198920 exp119870119903 (119873(119903)119862 minus 119873(119903)119863 )119873 (5)
The unbiased transition rate is 1198920 = 001 throughout thecalculations and 11198920 defines the time scale for the processThe DMM has been shown [14] to undergo critical phasetransitions and to be a member of the Ising universalityclass in which all the members of the network can actcooperatively depending on themagnitude of the interactionstrength119870 [14] However this important result is obtained byassigning to all the individuals the same degree of attentionto the opinions of their nearest neighbors called 119870 Herein
Table 1 The payoffs of Prisonerrsquos dilemma game The first value ofeach pairs is the payoff of player119883 and the second value is the payoffof the player 119884
Player 119884119862 119863Player119883119862 (1 1) (0 19)119863 (19 0) (0 0)
each individual may have a different degree of attention andthis degree of attention does not fit the reciprocity principleThe degree of attention that the individual 119903 devotes to theindividual 1199031015840 may differ from the degree of attention that theindividual 1199031015840 devotes to the individual 119903 To explain how theindividual 119903 is influenced by her nearest neighbors let usconsider for instance (4) The individual we are consideringis a cooperator and (4) establishes the rate of her transition tothe defection state If 119873(119903)119862 gt 119873(119903)119863 the rate decreases and willvanish in the extreme limit119870119903 = infin Of course this will havethe effect of favoring the cooperation state
22 The Rational Level This decision-making process is fastand emotional and does not involve any direct reasoningabout the payoff The connection with the self-interestaccording to the slow thinking mechanism discussed byKanheman [29] is established over a more extended timescale where the single individual exerts an influence on theprocess aiming atmaximizing her payoff To define the payoffwe adopt the prisonerrsquos dilemma game (PDG) [35] Twoplayers interact and receive a payoff from their interactionadopting either the defection or the cooperation strategy Ifboth players select the cooperation strategies each of themreceives the payoff 119877 and their society receives the payoff2119877 The player choosing the defection strategy receives thepayoff 119879 The temptation to cheat is established by setting thecondition 119879 gt 119877 However this larger payoff is assigned tothe defector only if the other player selects cooperation Theplayer selecting cooperation receives the payoff 119878 which issmaller than 119877 If the other player also selects defection thepayoff for both players is119875 which is smaller than119877The PDGis based on the crucial payoffs 119879 gt 119877 gt 119875 gt 119878 and 119878+119879 lt 2119877
We adopt the choice of parameter values made by Gintis[35] and set 119877 = 1 119875 = 0 and 119878 = 0 The maximalpossible value of 119879 is 2 and we select the value 119879 = 19which is a very strong incentive to cheat These choices aresummarized in Table 1 We evaluate the social benefit for thesingle individual as well as for the community as a wholeas follows We define the payoff 119875119903 for individual 119903 as theaverage over the payoffs from the interactions with its fournearest neighbors If both players of a pair are cooperatorsthe contribution to the payoff of the individual 119903 is 119861119903 = 1If one of the two playing individuals is a cooperator and theother is a defector the contribution to the payoffof 119903 is119861119903 = 119879If both players are defectors the contribution to the payoff of119903 is 119861119903 = 0 The payoff 119875119903 to individual 119903 is the sum over thefour 119861119903rsquos
4 Complexity
Each individual receives a total payoff from the gamewiththe four nearest neighbors and adjusts her imitation strengthas follows
119870119903 (119905) = 119870119903 (119905 minus Δ119905) + 120594119875119903 (119905 minus Δ119905) minus 119875119903 (119905 minus 2Δ119905)119875119903 (119905 minus Δ119905) + 119875119903 (119905 minus 2Δ119905) (6)
where the parameter 120594 determines the intensity of interestof the individuals to the fractional change in their payoffs intimeThe key equation (6) is based on the assumption that theintuitive decision-making process is so fast that at both times119905 minus Δ119905 and 119905 minus 2Δ119905 it is possible to evaluate the correspondingpayoffs on the basis of fast decisions made by each individualand by her 4 nearest neighbors The decision of adjusting thesocial sensitivity 119870119903(119905) requires the time interval 2Δ119905 whilethe intuitive decision is virtually instantaneous
The second term on the right-hand side of (6) is the ratiobetween two quantities that for special cases vanish In thesecases we set the condition
119870119903 (119905) = 119877119870119903 (119905 minus Δ119905) (7)
with 119877 lt 1 We selected 119877 = 0 but for other values we getthe same result When 119870119903(119905) goes to negative values we set itequal to zero
Note that in the limit of vanishing time intervals (6)relates the time rate of change of an individualrsquos imitationstrength to the time rate of change of the logarithm of thelocal payoff to that individual On the global scale the meanbenefit to society of all the individuals is given by the averageover all the 119875119903rsquos
Π (119905) = 1119873119873sum119903=1
119875119903 (119905) (8)
whereas the mean imitation strength is given by the averageover all the119870119903(119905)
119870 (119905) = 1119873119873sum119903=1
119870119903 (119905) (9)
For the bottom-up case discussed in this section thecalculation is done with the parameters119872 = 100 1198920 = 001119879 = 19 and 120594 = 1 with the social benefit imitation strengthand mean field starting from zero
The results of Figure 1 are used to establish the bottom-up origin of altruism rather than interpreting it as it is fre-quently done to be the result of a religion-induced top-downprocess The calculations show that the top-down processgenerating altruism weakens the systemrsquos resilience whereasthe genuinely bottom-up approach makes the emergence ofaltruism robust against external perturbation Figure 1 showsthat the time evolution of the individual social sensitivity 119870119903is characterized by abrupt jumps that from time to time mayalso bring the single individual back to a behavior totallyindependent of the choices made by her nearest neighborsThis is a healthy social condition that has the effect of makingthe global properties 119909(119905) 119870(119905) and Π(119905) host crucial eventsfavoring the transmission of information between differentsocial systems either countries or parties
x(t)
K(t)
Kr(t)
4
3
2
1
0
minus10 2000 4000
t6000 8000 10000
Π(t)
Figure 1 For the bottom-up SOCT time evolution of the averagesocial benefit Π(119905) the average imitation strength 119870(119905) the meanfield 119909(119905) and the imitation strength of one of the units119870119903(119905) plottedversus time The mean field fluctuates around 08 which meansabout 90 percent of individuals are cooperators and 10 percentare defectors Mean field of 1 has all individuals in the state ofcooperation and the maximum average social benefit has the value4 because this is the number of nearest neighbors on the two-dimensional lattice
To stress the occurrence of crucial events in a socialsystem resting on the bottom-up emergence of altruism wehave to extend the method used for criticality generated bythe fine tuning of the control parameter 119870 In that caseat criticality the mean field fluctuates around the vanishingvalue and the crucial events correspond to the occurrenceof this vanishing value [15 23] We follow [36] and evaluatethe fluctuations around the proper nonvanishing mean valueof 119870 = 15 To explain this choice notice that in theconventional case of criticality generated by the choice of aproper control parameter 119870 with 119872 = 100 119870 = 15 isthe value at which the onset of phase transition occurs Thisis the value making the mean field 119909(119905) of the conventionalDMM fluctuate around 119909 = 0 with complex fluctuationsand which generates criticality-induced intelligence [37 38]In the case of the SOTC model this condition of criticality-induced intelligence with fluctuations of 119870(119905) around 15 isspontaneously generated When the criticality condition isreached the complex fluctuations of 119909(119905) do not occur anylonger around 119909 = 0 but around a positive value of the orderof 08 The time intervals 120591 between consecutive crossings ofthe 15 level are monitored and the corresponding waiting-time PDF 120595(120591) is illustrated in Figure 2
Figure 2 similar to Figure 4 of [3] is compared inSection 4 with Figure 4 illustrating the perturbation byindependents We limit ourselves to noticing that temporalcomplexity shows up in the intermediate asymptotics regime[3] and it is characterized by the IPL index 120583 = 13 a propertyshared by other systems at criticality see for instance [22]
Complexity 5
cong 13
()
1E minus 6
1E minus 5
1E minus 4
0001
001
01
10 100 10001
Figure 2 For bottom-up SOTC the waiting-time PDF of thetime interval between two consecutive crossings of the averagevalue of the mean value of 119870(119905) which is asymp15 is graphed 120595(120591) isexponentially truncated and has an intermediate asymptotic regimewith an index of 120583 asymp 13 This figure is similar to Figure 4 of [3]
This is evidence that the spontaneous transition to criticalityalso generates the crucial events responsible for informationtransport Earlier work [3] confirms that the crucial eventsfacilitate the transport of information from one complexsocial system to another if the two systems are at criticalityas a result of the spontaneous process of self-organization
In spite of the exponential truncation that may lead to themisleading conclusion that the Poisson-like character of thelong-time regime quenches the manifestations of complexitythe transport of information is determined by the interme-diate asymptotics In the earlier work of [3] we studied theefficiency of transport of information from one bottom-upSOTC to another identical SOTC and we found that themaximal efficiency of the information transport correspondsto about 119872 = 100 which is the condition studied inthis paper The explanation of these interesting effects is thefollowing The intensity of fluctuations generating crucialevents decreases with the size increase according to thefollowing formula (see Eq (14) of [3])
Δ120577 prop 1119872] (10)
with ] = 05 Note that Δ120577 denotes the intensity of thefluctuations of the variables 119870Π and 119909(119905) around theirmean values Therefore the systems with 119872 gt 100 havecrucial fluctuations of smaller intensity thereby explainingthe reduction of the process of information transport Forvalues 119872 lt 100 the role of the exponential truncationbecomes more important and the time extension of thecomplex intermediate asymptotics is reduced and eventuallythe intermediate asymptotics regime vanishes turning thesystem into a Poisson system with no complexity This hasthe effect of significantly reducing the efficiency of the processof information transport As far as the resilience of thebottom-up SOTC is concerned the theory of this paper rests
on the connection between resilience and the efficiency ofinformation transport As a consequence the results on theresilience of the bottom-up SOTC for119872 = 100 automaticallycorrespond on the condition of maximal resilience [39]
3 Top-Down Approach to Self-OrganizedTemporal Criticality
The top-down approach to self-organization is done usingagain (4) and (5) The adoption of the top-down perspectiveis realized by replacing (6) with
119870 (119905) = 119870 (119905 minus Δ119905) + 120594Π (119905 minus Δ119905) minus Π (119905 minus 2Δ119905)Π (119905 minus Δ119905) + Π (119905 minus 2Δ119905) (11)
The top-down origin of this process is made evident by thefact that all individuals in the network are forced to adoptthe same time-dependent imitation strength Furthermorerational choice is made on the basis of the collective payoffΠ(119905) using PDGThe conceptual difference with the bottom-up approach of Section 2 is impressive In fact with (11) all theindividuals of this society must change their social sensitivityat the same time and the information about the increase ordecrease of the global payoff implies that all the individualsare given this information from a central source such as thegovernment suggesting that a form of organization alreadyexists and is not created by the interaction between theindividuals In [3] the assumptionwasmade that a benevolentdictator exists and leads such a process Using Paretorsquos socialtheory we make the assumption that this process implies theleadership of an elite [2]
The second term on the right-hand side of (11) is theratio between two quantities that similarly to the bottom-upmodel for special cases vanishes In these cases as done inSection 2 we set the condition
119870119903 (119905) = 119877119870119903 (119905 minus Δ119905) (12)
with 119877 lt 1 We selected 119877 = 0 When 119870119903(119905) goes to negativevalues we set it equal to zero
For the top-down case discussed in this section thecalculation is done with the parameters119872 = 100 1198920 = 001119879 = 19 and 120594 = 4 with the social benefit imitation strengthand mean field starting from zero 120594 is chosen to be largerthan in the bottom-up case because of the fact that transitionto criticality in the top-down case is much slower (see Figures1 and 3)
Under the leadership of an elite see Figure 3 the controlparameter 119870(119905) shows a behavior totally different from thatof Figure 1 In this section we focus on the behavior of 119870(119905)in the absence of perturbation and discuss the effects ofperturbation in Section 4 With no perturbation there is atransient from 119905 = 0 to 119905 asymp 40000 after which time a sequenceof rises and falls occurs The value of 119870 adjusts according to(11) from small values around 02 to a maximal value of 18which is known to correspond to a supercritical conditionin the case of the conventional DMM When using the finetuning control parameter approach we set119870 = 18 the socialsystem is far from the intelligence condition that accordingto a widely accepted opinion [9 16ndash19] requires criticality
6 Complexity
Time100000800006000040000200000
minus02
00
02
04
06
08
10
12
14
16
18
20
K(t)
Figure 3 Black line the time evolution 119870(119905) of (11) for the sameregular two-dimensional network of Figure 1 red line the timeevolution of119870(119905) of the self-organized system described by the blackunder the influence of the weak noisy perturbation described inSection 4
The mean field 119909(119905) has very fast fluctuations around a meanvalue close to 1 but these fluctuations are Poisson and theconventional DMM system loses its complexity [22]
The falls to the small values of119870 are interpreted as falls ofelites The subcritical condition as well as the supercriticalis characterized by a lack of intelligence We have to remarkalso that values of 119870 significantly smaller than 119870 asymp 15indicate that there are many units with119870119903 = 0 like the singleunit of Figure 1 at a time close to 119905 asymp 2000 In conclusionboth small and maximal values of 119870 are affected by a lack ofconsciousness and the transitions through 119870 asymp 15 are toofast for the social system to benefit from the intelligence of thecritical condition This lack of intelligence is responsible forthe lack of resilience The sojourn times in the supercriticalstate correspond to the time durations of elites We do nothave to confuse the fluctuations of 119870(119905) with those of themean field 119909(119905) that are not shown here The fluctuations of119909(119905) are always Poisson around mean values close to 1 when119870(119905) is close to 18 and around the vanishing mean valuewhen 119870(119905) drops
It is interesting to notice that also the time intervalbetween consecutive falls of elite is a complex dynamicalprocess characterized by an IPL with 120583 asymp 2 in this case asshown by Figure 4 However the system is not resilient Thesojourn in a supercritical state with a 119870 significantly largerthan119870 asymp 15 is characterized by fast Poisson fluctuations andan external perturbation can easily affect the time durationof this regime [40] In fact the big difference betweenPoisson events and crucial events is that the former eventsobey conventional linear response theory and any formsof perturbation can deeply affect their dynamics therebyundermining the social resilience as we show in the nextsection
1 10 100 1000
1
10
100
()
1000
10000
Figure 4 For bottom-up SOTC the waiting-time PDF of the timeintervals between two consecutive crossings of the horizontal linewith 119870 = 07 (black and red line) and 119870 = 17 (blue line)for the top-down SOTC The blue and the black lines describethe unperturbed case and the red line describes the perturbationdescribed in Section 4
4 Perturbing the Self-Organized Society
To substantiate the arguments of the earlier section with theresults of a numerical simulation we devote this section toillustrating some numerical experiments on the effects of aperturbation on the process of societal self-organization
First of all let us define two different sources of pertur-bation the independent and the committed minorities Weassume that a minority of independent individual exists Anindependent individual is a unit that is characterized by119870119903 =0 As a consequence this unit does not adopt (6) and iscompletely insensitive to the connection between individualand societal benefit that yields the emergence of cooperation[3] The perturbing nature of this independent individualis realized by the fact that while the independent keeping119870119903 = 0 is completely independent of the choices made bythe other units her nearest neighbors are influenced by thechoices of the independent through the DMM and throughthe evaluation of the payoff 119875119903(119905) of (6)
In the top-down SOTCmodel the independent influencesthe process through his vanishing contribution to 119870(119905) andthrough his contribution to the global payoff of (11) Theperturbation of independents ismademore devastatingwhenthe independents are allowed to move randomly through thesocial network
The other kind of perturbation produced by committedminorities has already been studied elsewhere [25] Theseare minorities that keep selecting the state119863 The committedminorities are also called zealots and have been the subject ofmany publications see [41] for a wide set of referencesThesepublications emphasize the dramatic consequences that thezealots have on their societies thereby implying that theirmodels of organization are not resilient The experiment onthe perturbation of zealots done herein shows that the top-down SOTC model shares the lack of resilience observed in
Complexity 7
time10000080000600004000020000
00
02
04
06
08
10
12
14
16
18
K(t)
Figure 5119870(119905) as a function of time for the bottom-up SOTC in theno perturbation (black line) and perturbed case (red line)
these earlier studies on the social influences of zealots Thebottom-up SOTC model seems to be the only fully resilientmodel
Let us discuss first the strongest source of perturbationthe randomly moving independents At any time step oneof the 119872 = 100 is randomly selected to play the role ofindependent namely we force her to adopt the value 120585 = 1or the value 120585 = minus1 with equal probability In the case of thebottom-up SOTC this perturbation does not have significanteffect on the time evolution of119870(119905) as shown by Figure 5
In the case of the top-down SOTC the effects of thisperturbation are impressive The black line of Figure 3 showsthe rise and the fall of an elite The weak noisy perturbationmakes 119870(119905) evolve as illustrated by the red line of the samefigure which shows that the sojourn times of unperturbedelites are filled with many falls that are a clear manifestationof the lack of societal resilience
Important information on the lack of resilience of thetop-down SOTC is afforded by Figure 4 showing 120595(119905) fordifferent values of the threshold used to find the statistics ofcrucial events When the threshold is 17 close to the topsupercritical region reached by the system the intermediateasymptotics has a power index 120583 asymp 145 larger than that ofthe bottom-up SOTCThe adoption of the threshold119870 = 07as an effect of the collapse of elites cancels the intermediateasymptotic temporal complexity and favors the birth of aPoisson shoulder The noisy perturbation of independentsmakes this behavior even more pronounced This strongexponential shoulder is a signature of the death of dynamicalcomplexity and of the transition fromnon-Poisson to Poissonbehavior [3]
The perturbing action of independents is weaker if theindependent individuals do not move This response to thisform of perturbation is illustrated in Figure 6 showing thateven in this case the bottom-up SOTC model is more robustthan the top-down
We establish the perturbation of independent individ-uals in a different way We assume that all the units are
02 0800 100604
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
Figure 6 Dependence of mean field (at time 106) on the ratio 120588 ofindependent units (which are fixed on the lattice) for the bottom-up SOTCmodel (black line) top-down SOTCmodel (red line) andordinary DMMwith tuned control parameter119870119888 = 15 (green line)Ensemble average is done over 10 experiments
independent for a fraction 120578 of their time The results aredepicted in Figure 7 It is clear from the figure that thebottom-up SOTCmodel ismore resilient than the two-downeven to this most violent form of disruption
Finally in Figure 8 we show the action of committedminorities We see that only the bottom-up SOTC model isresilient The top-down SOTC model shares the same lack ofresilience shown by ordinary DMM at criticality
5 Conclusions
It is remarkable that according to SOTC the crucial eventsmay be harmful as well as beneficial If the global parameter119870(119905) fluctuating around the long-range correlation generat-ingmean value returns to the vanishing value the temporarycollapse is turned into a societal disaster The collapse into119870 = 0 would correspond to a new initial condition andas shown by Figure 1 119870(119905) would start increasing againgenerating a new organization led by a new elite [2] Howeverthe genuinely bottom-up process leading the time evolutionillustrated by Figure 1 is expected to keep forever the socialsystem in the condition of weak fluctuations around119870 asymp 15In other words there is an impressive difference between thecrucial events hosted by the weak fluctuation around119870 asymp 15and the regressions of 119870(119905) to values of 119870 ≪ 15 generatedby the adoption of a top-down process led by an elite
The main conclusion of this paper concerning resilienceis that criticality is necessary for resilience but it is notsufficient The top-down SOTC model generates criticalitybut it is not resilient Therefore information transport fromone top-down SOTC model system to another top-downSOTC model system is expected to occur by means ofcomplexity matching in spite of the fact that the two systemsare not resilient and the information transport may be easilyquenched by stray perturbing noises
8 Complexity
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10times(106)
0005 0010 0015 00200000
(a)
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
0005 0010 0015 00200000
(b)
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
0005 0010 0015 00200000
(c)
Figure 7 Dependence of mean field (at time 106) on 120578 (the ratio of time in which units behave individually) for bottom-up SOTCmodel (a)top-down SOTC model (b) and ordinary DMMwith tuned parameter119870119888 = 15 (c) Each set of colored dots corresponds to one experimentand the black solid lines are the average of them
An attractive interpretation of the resilient nature of thebottom-up SOTC model is that the ideal condition of fulldemocracy is the most robust form of social organization
In this paper the social payoff is evaluated using thePDG [35] The prisonerrsquos dilemma game is frequently usedin the field of EGT [42 43] EGTs aim at solving thealtruism paradox using the concept of network reciprocity[43] A game is played many times on a network where eachindividual is surrounded by a set of nearest neighbors andadopts the strategy of the most successful nearest neighborSince the clusters of cooperators are richer than the clustersof defectors it is plausible that the most successful nearestneighbor is a cooperator However this attempt at mimickingthe action of a collective intelligence failed because the socialactivity of the units being subcritical disrupts the beneficialeffects of network reciprocity [44 45] We note that SOTCmodeling represents an attempt to amend the field of EGT
by the limitations preventing for instance the concept ofnetwork reciprocity from yielding a satisfactory resolution ofthe altruism paradox
The human inclination to cooperate is the result of bio-logical evolution and of the spontaneous evolution towardscriticality The time appears ripe to unify the models ofbiology and physics made necessary to reach the ambitiousgoal of achieving a rigorous scientific foundation of thisimportant human characteristic [46 47] The spontaneoustransition to criticality of SOTC contributes to bypassing thecurrent limitations of the field of EGT SOTC models asshown in this paper can be adapted to take into account thetop-downprocesses connectedwith the nonresilient action ofelites It is possible to supplement the nonrational decision-making process based on (4) and (5) with self-righteousbiases [1] taking into account the influence of religion or otherpolarizing influences We expect that such generalizations
Complexity 9
minus12
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
06 0804 1000 02
Figure 8 Dependence of mean field (at time 106) on the ratio 120588of fanatics (having fixed position on the lattice) for the bottom-upSOTC model (red line) top-down SOTC model (black line) andordinary DMMwith tuned control parameter119870119888 = 15 (green line)Ensemble average is done over 10 experiments
of SOTC theory will lead to a lack of societal resilienceHowever this is left as a subject for future research
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
Korosh Mahmoodi and Paolo Grigolini warmly thank AROandWelch for support throughGrants noW911NF-15-1-0245and no B-1577 respectively
References
[1] J HaidtThe Righteous Mind Why Good People Are Divided byPolitics and Religion Pantheon Books 2012
[2] V ParetoTheRise and fall of Elites Transaction Publishers NewBrunswick Canada 1991
[3] K Mahmoodi B J West and P Grigolini ldquoSelf-organizingcomplex networks Individual versus global rulesrdquo Frontiers inPhysiology vol 8 article no 478 2017
[4] H E Stanley Introduction to Phase Transition and CriticalPhenomena OxfordUniversity Press NewYork NY USA 1987
[5] E Ising ldquoBeitrag zur theorie des ferromagnetismusrdquo Zeitschriftfur Physik vol 31 no 1 pp 253ndash258 1925
[6] L Onsager ldquoCrystal statistics I A two-dimensional modelwith an order-disorder transitionrdquo Physical Review A AtomicMolecular and Optical Physics vol 65 no 3-4 pp 117ndash149 1944
[7] T Mora and W Bialek ldquoAre biological systems poised atcriticalityrdquo Journal of Statistical Physics vol 144 no 2 pp 268ndash302 2011
[8] D Fraiman P Balenzuela J Foss and D R Chialvo ldquoIsing-like dynamics in large-scale functional brain networksrdquoPhysical
Review E Statistical Nonlinear and Soft Matter Physics vol 79no 6 Article ID 061922 2009
[9] D R Chialvo ldquoEmergent complex neural dynamicsrdquo NaturePhysics vol 6 no 10 pp 744ndash750 2010
[10] M G Kitzbichler M L Smith S R Christensen and EBullmore ldquoBroadband criticality of human brain networksynchronizationrdquo PLoS Computational Biology vol 5 no 3Article ID e1000314 2009
[11] P Bak andM Paczuski ldquoComplexity contingency and critical-ityrdquo Proceedings of the National Academy of Sciences vol 92 pp6689ndash6696 1995
[12] W Truccolo L R Hochberg and J P Donoghue ldquoCollectivedynamics in human and monkey sensorimotor cortex Predict-ing single neuron spikesrdquoNature Neuroscience vol 13 no 1 pp105ndash111 2010
[13] E Schneidman M J Berry II R Segev and W Bialek ldquoWeakpairwise correlations imply strongly correlated network statesin a neural populationrdquoNature vol 440 no 7087 pp 1007ndash10122006
[14] B J West M Turalska and P Grigolini Networks of EchoesImitation Innovation and Invisible Leaders Springer 2014
[15] M Turalska B J West and P Grigolini ldquoTemporal complexityof the order parameter at the phase transitionrdquo Physical ReviewE vol 91 Article ID 012907 2015
[16] P Paradisi P Allegrini A Gemignani M Laurino DMenicucci and A Piarulli ldquoScaling and intermittency of brainevents as a manifestation of consciousnessrdquo AIP ConferenceProceedings vol 1510 pp 151ndash161 2013
[17] P Paradisi and P Allegrini ldquoScaling law of diffusivity generatedby a noisy telegraph signal with fractal intermittencyrdquo ChaosSolitons amp Fractals vol 81 part B pp 451ndash462 2015
[18] E Tagliazucchi D R ChialvoM Siniatchkin et al ldquoLarge-scalesignatures of unconsciousness are consistent with a departurefrom critical dynamicsrdquo Journal of the Royal Society Interfacevol 13 no 114 Article ID 20151027 2016
[19] L Cocchi L L Gollo A Zalesky and M Breakspear ldquoCrit-icality in the brain A synthesis of neurobiology models andcognitionrdquo Progress in Neurobiology 2017
[20] P Allegrini P Paradisi D Menicucci and A GemignanildquoFractal complexity in spontaneous EEGmetastable-state tran-sitions new vistas on integrated neural dynamicsrdquo Frontiers inPhysiology vol 1 2010
[21] P Allegrini P Paradisi D Menicucci and A GemignanildquoRenewal Processes in the Critical Brainrdquo Studies of NonlinearPhenomena in Life Science vol 15 pp 161ndash179 2010
[22] M Lukovic F Vanni A Svenkeson and P Grigolini ldquoTrans-mission of information at criticalityrdquo Physica A StatisticalMechanics and its Applications vol 416 pp 430ndash438 2014
[23] M T Beig A Svenkeson M Bologna B J West and PGrigolini ldquoCritical slowing down in networks generating tem-poral complexityrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 91 no 1 Article ID 012907 2015
[24] J U Mafahim D Lambert M Zare and P Grigolini ldquoCom-plexity matching in neural networksrdquo New Journal of Physics vol 17 Article ID 015003 2015
[25] M Turalska B J West and P Grigolini ldquoRole of committedminorities in times of crisisrdquo Scientific Reports vol 3 article no1371 2013
[26] B Gert Morality a New Justification of Moral Rules OxfordUniversity Press New York NY USA 1988
10 Complexity
[27] H A Simon Models of Bounded Rationality vol 1 and 2 MITPress 1982
[28] J GMarch ldquoRationality foolishness and adaptive intelligencerdquoStrategic Management Journal vol 27 no 3 pp 201ndash214 2006
[29] D KahnemanThinking Fast and Slow Macmillan 2012[30] D G Rand and M A Nowak ldquoHuman cooperationrdquo Trends in
Cognitive Sciences vol 17 no 8 pp 413ndash425 2013[31] D G Rand ldquoCooperation Fast and Slow Meta-Analytic Evi-
dence for a Theory of Social Heuristics and Self-InterestedDeliberationrdquo Psychological Science vol 27 no 9 pp 1192ndash12062016
[32] G Gigerenzer Risk Savvy how to make good decisions PenguinBooks New York NY USA 2014
[33] G Gigerenzer Gut Feelings The Intelligence of the UnconsciousPenguin Books New York NY USA 2007
[34] P Grigolini N Piccinini A Svenkeson P Pramukkul D Lam-bert and B J West ldquoFrom Neural and Social Cooperation totheGlobal Emergence of Cognitionrdquo Frontiers in Bioengineeringand Biotechnology vol 3 article 78 2015
[35] HGintisThebounds of reasonmdashgame theory and the unificationof the behavioral sciences PrincetonUniversity Press PrincetonNJ USA 2014
[36] A Pease K Mahmoodi and B J West ldquoComplexity measuresof musicrdquo Chaos Solitons amp Fractals vol 108 pp 82ndash86 2018
[37] F VanniM Lukovic and PGrigolini ldquoCriticality and transmis-sion of information in a swarm of cooperative unitsrdquo PhysicalReview Letters vol 107 no 7 Article ID 078103 2011
[38] M Turalska E Geneston B J West P Allegrini and PGrigolini ldquoCooperation-induced topological complexity Apromising road to fault tolerance and Hebbian learningrdquoFrontiers in Physiology vol 3 article 52 2012
[39] KMahmoodi P Grigolini and B JWest ldquoOn Social Sensitivityto either Zealot or IndependentMinoritiesrdquo submitted to ChaosSolitons and Fractals
[40] P Allegrini M Bologna P Grigolini and B J WestldquoFluctuation-dissipation theorem for event-dominated pro-cessesrdquo Physical Review Letters vol 99 no 1 Article ID 0106032007
[41] N Khalil M San Miguel and R Toral ldquoZealots in themean-field noisy voter modelrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 97 no 1 Article ID012310 2018
[42] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[43] M A Nowak and R M May ldquoEvolutionary games and spatialchaosrdquo Nature vol 359 no 6398 pp 826ndash829 1992
[44] K Mahmoodi and P Grigolini ldquoEvolutionary game theory andcriticalityrdquo Journal of Physics A Mathematical and Theoreticalvol 50 no 1 Article ID 015101 2017
[45] C Gracia-Lazaro J A Cuesta A Sanchez and Y MorenoldquoHuman behavior in Prisonerrsquos Dilemma experiments sup-presses network reciprocityrdquo Scientific Reports vol 2 p 3252012
[46] W Bialek ldquoPerspectives on theory at the interface of physics andbiologyrdquo Reports on Progress in Physics vol 81 no 1 Article ID012601 2018
[47] M Munoz ldquoColloquium Criticality and dynamical scaling inliving systemsrdquo httpsarxivorgabs171204499
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2 Complexity
The Decision-Making Model (DMM) [14] which is usedin Sections 2 and 3 was proved [15] to generate phasetransition as a function of its control parameter 119870 identicalto that of the Ising model where the control parameter isthe temperature In other words the DMM belongs to Isinguniversality class [14]
At criticality namely when the dynamics of the systemare determined by the control parameter generating phasetransition the mean field 119909(119905) which in this paper is definedas the ratio of the difference between the number of coop-erator and the number of defectors to the total number ofunits fluctuates around the vanishing value The occurrenceof a vanishing value is a crucial event The crucial events aredefined as follows The time interval between consecutivecrucial events is described by the waiting-time probabilitydensity function (PDF) 120595(120591) that in the long-time limit 120591 rarrinfin has the inverse power law (IPL) structure
120595 (120591) prop 1120591120583 (1)
with 120583 lt 3 The crucial events are renewed thereby makingthe correlation function ⟨120591119894120591119895⟩ vanish if 119894 = 119895
In the case of the brain dynamics there is wide consensuson the connection between consciousness and criticality Seefor instance [9 16ndash18] and the recent review paper [19]The electroencephalogram (EEG) signals are characterizedby abrupt changes called rapid transition processes (RTP)which are proved [20 21] to be renewal non-Poisson eventswith 120583 asymp 2 This means that the brain in the awake state is agenerator of crucial events
The crucial events are responsible for the informationtransport from one system at criticality to another system atcriticality [22] Furthermore the emergence of crucial eventsrequires that the size of the complex system is finite In thispaper119872 is the total number of units within the system Theintensity of the fluctuations of the mean field 119909(119905) obeys thegeneral prescription
Δ120577 prop 1119872] (2)
where
Δ120577 = 119860 (119905) minus 119860 (3)
When working with DMM at criticality 119860 is the mean field119909 with 119909 = 0 ] = 025 [23] In the case of SOTC [3] with119860 = 119870 see Section 2 we find ] = 05 These criticality-induced fluctuations becoming visible for finite values of119872are referred to as an expression of temporal complexity
12 Swarm Intelligence and Resilience We may afford anintuitive interpretation of crucial (complex) events using theexample of a flock of birds flying in a given direction asan effect of self-organization A crucial (complex) event isequivalent to a complete rejuvenation of the flock that after anorganizational collapsemay freely select any new flying direc-tion An external fluctuation of even weak intensity can forcethe complex system to move in a given direction if it occursat the exact instant of the free will of the SOTCmodel system
It is important to stress that the organizational collapse isnot the fall of an elite which will be discussed subsequentlybecause the flock self-organization occurs spontaneouslyand does not rest on the action of a leader The choice ofa new flying direction is thus determined by an externalstimulus of even weak intensity occurring at the same timeas the collapse thereby implying the property of complexitymatching between the perturbed and the perturbing complexsystem [14]
As mentioned earlier the crucial events favor the trans-port of information fromone complex system to another [22]Crucial events are generated by criticality and consequentlythe transport of information becomes maximally efficient atcriticality [24]
However criticality may also be Achillesrsquo heel of acomplex system if criticality is generated by a fine tuningcontrol parameter In fact committedminorities acting whena crucial event occurs in the case of DMM can make thesystem jump from the state 119862 to the state 119863 [25] Herein weshow that this lack of resilience is not shared by the bottom-up approach to SOTC modeling in fact starting from thebottom generates a very resilient social organization
13 From Criticality Generated by the Fine Tuning of aControl Parameter to Self-Organization The model of [3]is a form of spontaneous transition to criticality revealedby the emergence of events with the temporal properties ofcrucial events thereby explaining the adoption of the nameSelf-Organized Temporal Criticality (SOTC) to define it Weshow that the bottom-up SOTC modeling is resilient andthat the top-down SOTC modeling is not We believe thatthe SOTC model may help to contribute to the discussionof the sociological issues of Haidt [1] with the tools ofComplexity Science In fact Haidt emphasized that thepolitical conflict between conservatives and liberals is dueto cultural and religious influences that have the effect ofcreating divisions We believe that the top-down SOTCapproachmay be used tomodel these cultural influencesThisis an extremely difficult problemmade evenmore difficult bythe philosophical controversies on definition ofmorality [26]According to the brilliant picture of Haidt the philosophyof Hume and Menciu may be compatible with the bottom-up origin of cooperation while the hypothesis that moralitytranscends human nature an interpretation moving fromPlato to Kant [1] may justify a top-down perspective Wemake the extremely simplified assumption that the top-downSOTC undermining social resilience explains the fall ofelites if they represent only limited groups a phenomenonthat may be explained by noticing that ldquoour minds weredesigned for groupish righteousnessrdquo [1]The source of socialconflict seems to be that cultural evolution differs from lifeevolution These culturally induced conflicts may overcomethe biological origin of cooperation
14 Bottom-Up versus Top-Down Approach to Morality Forclarity in Sections 2 and 3 we provide a review of the SOTCmodel [3] while stressing some properties of SOTC modelthat were not discussed in the original paper for instancethe behavior of single units with their frequent regression
Complexity 3
to the condition of independence of the other units for thebottom-up process and the Pareto cycles of the top-downversion of the modelThe original results of this earlier paperindicated a lack of resilience of the top-down SOTC modeland a robustness of the bottom-up SOTC model which areillustrated in Section 4 We devote Section 5 to balancing theresults of the present paper against the open problems that wepropose to study in future research
2 Bottom-Up Approach to Self-OrganizedTemporal Criticality
The decisions of single individuals in our model are made inaccordance with the criterion of bounded rationality [27 28]expanded by Kahneman [29] and more recently discussedfrom within the perspective of evolutionary game theory(EGT) [30 31] The nonrational component of the decision-making process is stressed also by the work of Gigerenzer[32 33] Herein individuals make decision using DMM Theindividuals of the social network aiming at increasing theirpayoffmake the control parameter119870119903 for individual 119903 evolvetowards criticality thereby creating an intelligent groupmind[34] perhaps connected to the intelligent unconscious ofGigerenzer [33] leading the individuals to fast but essentiallywise decisions As we shall see the time evolution of 119870119903 isslow because it depends on the payoffs of the individual atearlier time corresponding to the slow thinking mechanismdiscussed by Kahneman [29]
21 The Intuitive and Emotional Level In this paper we usethe DMM on a two-dimensional lattice of size 119871 with119872 =119871 times 119871 individuals and we set 119871 = 10 The DMM is basedon individuals imperfectly imitating the majority opinion oftheir four nearest neighbors thereby biasing the probabilityof making a transition from being a cooperator (119862) to beinga defector (119863)
119892(119903)119862119863 = 1198920 expminus119870119903 (119873(119903)119862 minus 119873(119903)119863 )119873 (4)
where 119873(119903)119862 is the number of nearest neighbors to individual119903 that are cooperators 119873(119903)119863 is the number of defectors andeach individual on the simple lattice has 119873 = 4 nearestneighbors In the same way the transition rate from defectorsto cooperators 119892(119903)119863119862 is
119892(119903)119863119862 = 1198920 exp119870119903 (119873(119903)119862 minus 119873(119903)119863 )119873 (5)
The unbiased transition rate is 1198920 = 001 throughout thecalculations and 11198920 defines the time scale for the processThe DMM has been shown [14] to undergo critical phasetransitions and to be a member of the Ising universalityclass in which all the members of the network can actcooperatively depending on themagnitude of the interactionstrength119870 [14] However this important result is obtained byassigning to all the individuals the same degree of attentionto the opinions of their nearest neighbors called 119870 Herein
Table 1 The payoffs of Prisonerrsquos dilemma game The first value ofeach pairs is the payoff of player119883 and the second value is the payoffof the player 119884
Player 119884119862 119863Player119883119862 (1 1) (0 19)119863 (19 0) (0 0)
each individual may have a different degree of attention andthis degree of attention does not fit the reciprocity principleThe degree of attention that the individual 119903 devotes to theindividual 1199031015840 may differ from the degree of attention that theindividual 1199031015840 devotes to the individual 119903 To explain how theindividual 119903 is influenced by her nearest neighbors let usconsider for instance (4) The individual we are consideringis a cooperator and (4) establishes the rate of her transition tothe defection state If 119873(119903)119862 gt 119873(119903)119863 the rate decreases and willvanish in the extreme limit119870119903 = infin Of course this will havethe effect of favoring the cooperation state
22 The Rational Level This decision-making process is fastand emotional and does not involve any direct reasoningabout the payoff The connection with the self-interestaccording to the slow thinking mechanism discussed byKanheman [29] is established over a more extended timescale where the single individual exerts an influence on theprocess aiming atmaximizing her payoff To define the payoffwe adopt the prisonerrsquos dilemma game (PDG) [35] Twoplayers interact and receive a payoff from their interactionadopting either the defection or the cooperation strategy Ifboth players select the cooperation strategies each of themreceives the payoff 119877 and their society receives the payoff2119877 The player choosing the defection strategy receives thepayoff 119879 The temptation to cheat is established by setting thecondition 119879 gt 119877 However this larger payoff is assigned tothe defector only if the other player selects cooperation Theplayer selecting cooperation receives the payoff 119878 which issmaller than 119877 If the other player also selects defection thepayoff for both players is119875 which is smaller than119877The PDGis based on the crucial payoffs 119879 gt 119877 gt 119875 gt 119878 and 119878+119879 lt 2119877
We adopt the choice of parameter values made by Gintis[35] and set 119877 = 1 119875 = 0 and 119878 = 0 The maximalpossible value of 119879 is 2 and we select the value 119879 = 19which is a very strong incentive to cheat These choices aresummarized in Table 1 We evaluate the social benefit for thesingle individual as well as for the community as a wholeas follows We define the payoff 119875119903 for individual 119903 as theaverage over the payoffs from the interactions with its fournearest neighbors If both players of a pair are cooperatorsthe contribution to the payoff of the individual 119903 is 119861119903 = 1If one of the two playing individuals is a cooperator and theother is a defector the contribution to the payoffof 119903 is119861119903 = 119879If both players are defectors the contribution to the payoff of119903 is 119861119903 = 0 The payoff 119875119903 to individual 119903 is the sum over thefour 119861119903rsquos
4 Complexity
Each individual receives a total payoff from the gamewiththe four nearest neighbors and adjusts her imitation strengthas follows
119870119903 (119905) = 119870119903 (119905 minus Δ119905) + 120594119875119903 (119905 minus Δ119905) minus 119875119903 (119905 minus 2Δ119905)119875119903 (119905 minus Δ119905) + 119875119903 (119905 minus 2Δ119905) (6)
where the parameter 120594 determines the intensity of interestof the individuals to the fractional change in their payoffs intimeThe key equation (6) is based on the assumption that theintuitive decision-making process is so fast that at both times119905 minus Δ119905 and 119905 minus 2Δ119905 it is possible to evaluate the correspondingpayoffs on the basis of fast decisions made by each individualand by her 4 nearest neighbors The decision of adjusting thesocial sensitivity 119870119903(119905) requires the time interval 2Δ119905 whilethe intuitive decision is virtually instantaneous
The second term on the right-hand side of (6) is the ratiobetween two quantities that for special cases vanish In thesecases we set the condition
119870119903 (119905) = 119877119870119903 (119905 minus Δ119905) (7)
with 119877 lt 1 We selected 119877 = 0 but for other values we getthe same result When 119870119903(119905) goes to negative values we set itequal to zero
Note that in the limit of vanishing time intervals (6)relates the time rate of change of an individualrsquos imitationstrength to the time rate of change of the logarithm of thelocal payoff to that individual On the global scale the meanbenefit to society of all the individuals is given by the averageover all the 119875119903rsquos
Π (119905) = 1119873119873sum119903=1
119875119903 (119905) (8)
whereas the mean imitation strength is given by the averageover all the119870119903(119905)
119870 (119905) = 1119873119873sum119903=1
119870119903 (119905) (9)
For the bottom-up case discussed in this section thecalculation is done with the parameters119872 = 100 1198920 = 001119879 = 19 and 120594 = 1 with the social benefit imitation strengthand mean field starting from zero
The results of Figure 1 are used to establish the bottom-up origin of altruism rather than interpreting it as it is fre-quently done to be the result of a religion-induced top-downprocess The calculations show that the top-down processgenerating altruism weakens the systemrsquos resilience whereasthe genuinely bottom-up approach makes the emergence ofaltruism robust against external perturbation Figure 1 showsthat the time evolution of the individual social sensitivity 119870119903is characterized by abrupt jumps that from time to time mayalso bring the single individual back to a behavior totallyindependent of the choices made by her nearest neighborsThis is a healthy social condition that has the effect of makingthe global properties 119909(119905) 119870(119905) and Π(119905) host crucial eventsfavoring the transmission of information between differentsocial systems either countries or parties
x(t)
K(t)
Kr(t)
4
3
2
1
0
minus10 2000 4000
t6000 8000 10000
Π(t)
Figure 1 For the bottom-up SOCT time evolution of the averagesocial benefit Π(119905) the average imitation strength 119870(119905) the meanfield 119909(119905) and the imitation strength of one of the units119870119903(119905) plottedversus time The mean field fluctuates around 08 which meansabout 90 percent of individuals are cooperators and 10 percentare defectors Mean field of 1 has all individuals in the state ofcooperation and the maximum average social benefit has the value4 because this is the number of nearest neighbors on the two-dimensional lattice
To stress the occurrence of crucial events in a socialsystem resting on the bottom-up emergence of altruism wehave to extend the method used for criticality generated bythe fine tuning of the control parameter 119870 In that caseat criticality the mean field fluctuates around the vanishingvalue and the crucial events correspond to the occurrenceof this vanishing value [15 23] We follow [36] and evaluatethe fluctuations around the proper nonvanishing mean valueof 119870 = 15 To explain this choice notice that in theconventional case of criticality generated by the choice of aproper control parameter 119870 with 119872 = 100 119870 = 15 isthe value at which the onset of phase transition occurs Thisis the value making the mean field 119909(119905) of the conventionalDMM fluctuate around 119909 = 0 with complex fluctuationsand which generates criticality-induced intelligence [37 38]In the case of the SOTC model this condition of criticality-induced intelligence with fluctuations of 119870(119905) around 15 isspontaneously generated When the criticality condition isreached the complex fluctuations of 119909(119905) do not occur anylonger around 119909 = 0 but around a positive value of the orderof 08 The time intervals 120591 between consecutive crossings ofthe 15 level are monitored and the corresponding waiting-time PDF 120595(120591) is illustrated in Figure 2
Figure 2 similar to Figure 4 of [3] is compared inSection 4 with Figure 4 illustrating the perturbation byindependents We limit ourselves to noticing that temporalcomplexity shows up in the intermediate asymptotics regime[3] and it is characterized by the IPL index 120583 = 13 a propertyshared by other systems at criticality see for instance [22]
Complexity 5
cong 13
()
1E minus 6
1E minus 5
1E minus 4
0001
001
01
10 100 10001
Figure 2 For bottom-up SOTC the waiting-time PDF of thetime interval between two consecutive crossings of the averagevalue of the mean value of 119870(119905) which is asymp15 is graphed 120595(120591) isexponentially truncated and has an intermediate asymptotic regimewith an index of 120583 asymp 13 This figure is similar to Figure 4 of [3]
This is evidence that the spontaneous transition to criticalityalso generates the crucial events responsible for informationtransport Earlier work [3] confirms that the crucial eventsfacilitate the transport of information from one complexsocial system to another if the two systems are at criticalityas a result of the spontaneous process of self-organization
In spite of the exponential truncation that may lead to themisleading conclusion that the Poisson-like character of thelong-time regime quenches the manifestations of complexitythe transport of information is determined by the interme-diate asymptotics In the earlier work of [3] we studied theefficiency of transport of information from one bottom-upSOTC to another identical SOTC and we found that themaximal efficiency of the information transport correspondsto about 119872 = 100 which is the condition studied inthis paper The explanation of these interesting effects is thefollowing The intensity of fluctuations generating crucialevents decreases with the size increase according to thefollowing formula (see Eq (14) of [3])
Δ120577 prop 1119872] (10)
with ] = 05 Note that Δ120577 denotes the intensity of thefluctuations of the variables 119870Π and 119909(119905) around theirmean values Therefore the systems with 119872 gt 100 havecrucial fluctuations of smaller intensity thereby explainingthe reduction of the process of information transport Forvalues 119872 lt 100 the role of the exponential truncationbecomes more important and the time extension of thecomplex intermediate asymptotics is reduced and eventuallythe intermediate asymptotics regime vanishes turning thesystem into a Poisson system with no complexity This hasthe effect of significantly reducing the efficiency of the processof information transport As far as the resilience of thebottom-up SOTC is concerned the theory of this paper rests
on the connection between resilience and the efficiency ofinformation transport As a consequence the results on theresilience of the bottom-up SOTC for119872 = 100 automaticallycorrespond on the condition of maximal resilience [39]
3 Top-Down Approach to Self-OrganizedTemporal Criticality
The top-down approach to self-organization is done usingagain (4) and (5) The adoption of the top-down perspectiveis realized by replacing (6) with
119870 (119905) = 119870 (119905 minus Δ119905) + 120594Π (119905 minus Δ119905) minus Π (119905 minus 2Δ119905)Π (119905 minus Δ119905) + Π (119905 minus 2Δ119905) (11)
The top-down origin of this process is made evident by thefact that all individuals in the network are forced to adoptthe same time-dependent imitation strength Furthermorerational choice is made on the basis of the collective payoffΠ(119905) using PDGThe conceptual difference with the bottom-up approach of Section 2 is impressive In fact with (11) all theindividuals of this society must change their social sensitivityat the same time and the information about the increase ordecrease of the global payoff implies that all the individualsare given this information from a central source such as thegovernment suggesting that a form of organization alreadyexists and is not created by the interaction between theindividuals In [3] the assumptionwasmade that a benevolentdictator exists and leads such a process Using Paretorsquos socialtheory we make the assumption that this process implies theleadership of an elite [2]
The second term on the right-hand side of (11) is theratio between two quantities that similarly to the bottom-upmodel for special cases vanishes In these cases as done inSection 2 we set the condition
119870119903 (119905) = 119877119870119903 (119905 minus Δ119905) (12)
with 119877 lt 1 We selected 119877 = 0 When 119870119903(119905) goes to negativevalues we set it equal to zero
For the top-down case discussed in this section thecalculation is done with the parameters119872 = 100 1198920 = 001119879 = 19 and 120594 = 4 with the social benefit imitation strengthand mean field starting from zero 120594 is chosen to be largerthan in the bottom-up case because of the fact that transitionto criticality in the top-down case is much slower (see Figures1 and 3)
Under the leadership of an elite see Figure 3 the controlparameter 119870(119905) shows a behavior totally different from thatof Figure 1 In this section we focus on the behavior of 119870(119905)in the absence of perturbation and discuss the effects ofperturbation in Section 4 With no perturbation there is atransient from 119905 = 0 to 119905 asymp 40000 after which time a sequenceof rises and falls occurs The value of 119870 adjusts according to(11) from small values around 02 to a maximal value of 18which is known to correspond to a supercritical conditionin the case of the conventional DMM When using the finetuning control parameter approach we set119870 = 18 the socialsystem is far from the intelligence condition that accordingto a widely accepted opinion [9 16ndash19] requires criticality
6 Complexity
Time100000800006000040000200000
minus02
00
02
04
06
08
10
12
14
16
18
20
K(t)
Figure 3 Black line the time evolution 119870(119905) of (11) for the sameregular two-dimensional network of Figure 1 red line the timeevolution of119870(119905) of the self-organized system described by the blackunder the influence of the weak noisy perturbation described inSection 4
The mean field 119909(119905) has very fast fluctuations around a meanvalue close to 1 but these fluctuations are Poisson and theconventional DMM system loses its complexity [22]
The falls to the small values of119870 are interpreted as falls ofelites The subcritical condition as well as the supercriticalis characterized by a lack of intelligence We have to remarkalso that values of 119870 significantly smaller than 119870 asymp 15indicate that there are many units with119870119903 = 0 like the singleunit of Figure 1 at a time close to 119905 asymp 2000 In conclusionboth small and maximal values of 119870 are affected by a lack ofconsciousness and the transitions through 119870 asymp 15 are toofast for the social system to benefit from the intelligence of thecritical condition This lack of intelligence is responsible forthe lack of resilience The sojourn times in the supercriticalstate correspond to the time durations of elites We do nothave to confuse the fluctuations of 119870(119905) with those of themean field 119909(119905) that are not shown here The fluctuations of119909(119905) are always Poisson around mean values close to 1 when119870(119905) is close to 18 and around the vanishing mean valuewhen 119870(119905) drops
It is interesting to notice that also the time intervalbetween consecutive falls of elite is a complex dynamicalprocess characterized by an IPL with 120583 asymp 2 in this case asshown by Figure 4 However the system is not resilient Thesojourn in a supercritical state with a 119870 significantly largerthan119870 asymp 15 is characterized by fast Poisson fluctuations andan external perturbation can easily affect the time durationof this regime [40] In fact the big difference betweenPoisson events and crucial events is that the former eventsobey conventional linear response theory and any formsof perturbation can deeply affect their dynamics therebyundermining the social resilience as we show in the nextsection
1 10 100 1000
1
10
100
()
1000
10000
Figure 4 For bottom-up SOTC the waiting-time PDF of the timeintervals between two consecutive crossings of the horizontal linewith 119870 = 07 (black and red line) and 119870 = 17 (blue line)for the top-down SOTC The blue and the black lines describethe unperturbed case and the red line describes the perturbationdescribed in Section 4
4 Perturbing the Self-Organized Society
To substantiate the arguments of the earlier section with theresults of a numerical simulation we devote this section toillustrating some numerical experiments on the effects of aperturbation on the process of societal self-organization
First of all let us define two different sources of pertur-bation the independent and the committed minorities Weassume that a minority of independent individual exists Anindependent individual is a unit that is characterized by119870119903 =0 As a consequence this unit does not adopt (6) and iscompletely insensitive to the connection between individualand societal benefit that yields the emergence of cooperation[3] The perturbing nature of this independent individualis realized by the fact that while the independent keeping119870119903 = 0 is completely independent of the choices made bythe other units her nearest neighbors are influenced by thechoices of the independent through the DMM and throughthe evaluation of the payoff 119875119903(119905) of (6)
In the top-down SOTCmodel the independent influencesthe process through his vanishing contribution to 119870(119905) andthrough his contribution to the global payoff of (11) Theperturbation of independents ismademore devastatingwhenthe independents are allowed to move randomly through thesocial network
The other kind of perturbation produced by committedminorities has already been studied elsewhere [25] Theseare minorities that keep selecting the state119863 The committedminorities are also called zealots and have been the subject ofmany publications see [41] for a wide set of referencesThesepublications emphasize the dramatic consequences that thezealots have on their societies thereby implying that theirmodels of organization are not resilient The experiment onthe perturbation of zealots done herein shows that the top-down SOTC model shares the lack of resilience observed in
Complexity 7
time10000080000600004000020000
00
02
04
06
08
10
12
14
16
18
K(t)
Figure 5119870(119905) as a function of time for the bottom-up SOTC in theno perturbation (black line) and perturbed case (red line)
these earlier studies on the social influences of zealots Thebottom-up SOTC model seems to be the only fully resilientmodel
Let us discuss first the strongest source of perturbationthe randomly moving independents At any time step oneof the 119872 = 100 is randomly selected to play the role ofindependent namely we force her to adopt the value 120585 = 1or the value 120585 = minus1 with equal probability In the case of thebottom-up SOTC this perturbation does not have significanteffect on the time evolution of119870(119905) as shown by Figure 5
In the case of the top-down SOTC the effects of thisperturbation are impressive The black line of Figure 3 showsthe rise and the fall of an elite The weak noisy perturbationmakes 119870(119905) evolve as illustrated by the red line of the samefigure which shows that the sojourn times of unperturbedelites are filled with many falls that are a clear manifestationof the lack of societal resilience
Important information on the lack of resilience of thetop-down SOTC is afforded by Figure 4 showing 120595(119905) fordifferent values of the threshold used to find the statistics ofcrucial events When the threshold is 17 close to the topsupercritical region reached by the system the intermediateasymptotics has a power index 120583 asymp 145 larger than that ofthe bottom-up SOTCThe adoption of the threshold119870 = 07as an effect of the collapse of elites cancels the intermediateasymptotic temporal complexity and favors the birth of aPoisson shoulder The noisy perturbation of independentsmakes this behavior even more pronounced This strongexponential shoulder is a signature of the death of dynamicalcomplexity and of the transition fromnon-Poisson to Poissonbehavior [3]
The perturbing action of independents is weaker if theindependent individuals do not move This response to thisform of perturbation is illustrated in Figure 6 showing thateven in this case the bottom-up SOTC model is more robustthan the top-down
We establish the perturbation of independent individ-uals in a different way We assume that all the units are
02 0800 100604
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
Figure 6 Dependence of mean field (at time 106) on the ratio 120588 ofindependent units (which are fixed on the lattice) for the bottom-up SOTCmodel (black line) top-down SOTCmodel (red line) andordinary DMMwith tuned control parameter119870119888 = 15 (green line)Ensemble average is done over 10 experiments
independent for a fraction 120578 of their time The results aredepicted in Figure 7 It is clear from the figure that thebottom-up SOTCmodel ismore resilient than the two-downeven to this most violent form of disruption
Finally in Figure 8 we show the action of committedminorities We see that only the bottom-up SOTC model isresilient The top-down SOTC model shares the same lack ofresilience shown by ordinary DMM at criticality
5 Conclusions
It is remarkable that according to SOTC the crucial eventsmay be harmful as well as beneficial If the global parameter119870(119905) fluctuating around the long-range correlation generat-ingmean value returns to the vanishing value the temporarycollapse is turned into a societal disaster The collapse into119870 = 0 would correspond to a new initial condition andas shown by Figure 1 119870(119905) would start increasing againgenerating a new organization led by a new elite [2] Howeverthe genuinely bottom-up process leading the time evolutionillustrated by Figure 1 is expected to keep forever the socialsystem in the condition of weak fluctuations around119870 asymp 15In other words there is an impressive difference between thecrucial events hosted by the weak fluctuation around119870 asymp 15and the regressions of 119870(119905) to values of 119870 ≪ 15 generatedby the adoption of a top-down process led by an elite
The main conclusion of this paper concerning resilienceis that criticality is necessary for resilience but it is notsufficient The top-down SOTC model generates criticalitybut it is not resilient Therefore information transport fromone top-down SOTC model system to another top-downSOTC model system is expected to occur by means ofcomplexity matching in spite of the fact that the two systemsare not resilient and the information transport may be easilyquenched by stray perturbing noises
8 Complexity
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10times(106)
0005 0010 0015 00200000
(a)
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
0005 0010 0015 00200000
(b)
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
0005 0010 0015 00200000
(c)
Figure 7 Dependence of mean field (at time 106) on 120578 (the ratio of time in which units behave individually) for bottom-up SOTCmodel (a)top-down SOTC model (b) and ordinary DMMwith tuned parameter119870119888 = 15 (c) Each set of colored dots corresponds to one experimentand the black solid lines are the average of them
An attractive interpretation of the resilient nature of thebottom-up SOTC model is that the ideal condition of fulldemocracy is the most robust form of social organization
In this paper the social payoff is evaluated using thePDG [35] The prisonerrsquos dilemma game is frequently usedin the field of EGT [42 43] EGTs aim at solving thealtruism paradox using the concept of network reciprocity[43] A game is played many times on a network where eachindividual is surrounded by a set of nearest neighbors andadopts the strategy of the most successful nearest neighborSince the clusters of cooperators are richer than the clustersof defectors it is plausible that the most successful nearestneighbor is a cooperator However this attempt at mimickingthe action of a collective intelligence failed because the socialactivity of the units being subcritical disrupts the beneficialeffects of network reciprocity [44 45] We note that SOTCmodeling represents an attempt to amend the field of EGT
by the limitations preventing for instance the concept ofnetwork reciprocity from yielding a satisfactory resolution ofthe altruism paradox
The human inclination to cooperate is the result of bio-logical evolution and of the spontaneous evolution towardscriticality The time appears ripe to unify the models ofbiology and physics made necessary to reach the ambitiousgoal of achieving a rigorous scientific foundation of thisimportant human characteristic [46 47] The spontaneoustransition to criticality of SOTC contributes to bypassing thecurrent limitations of the field of EGT SOTC models asshown in this paper can be adapted to take into account thetop-downprocesses connectedwith the nonresilient action ofelites It is possible to supplement the nonrational decision-making process based on (4) and (5) with self-righteousbiases [1] taking into account the influence of religion or otherpolarizing influences We expect that such generalizations
Complexity 9
minus12
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
06 0804 1000 02
Figure 8 Dependence of mean field (at time 106) on the ratio 120588of fanatics (having fixed position on the lattice) for the bottom-upSOTC model (red line) top-down SOTC model (black line) andordinary DMMwith tuned control parameter119870119888 = 15 (green line)Ensemble average is done over 10 experiments
of SOTC theory will lead to a lack of societal resilienceHowever this is left as a subject for future research
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
Korosh Mahmoodi and Paolo Grigolini warmly thank AROandWelch for support throughGrants noW911NF-15-1-0245and no B-1577 respectively
References
[1] J HaidtThe Righteous Mind Why Good People Are Divided byPolitics and Religion Pantheon Books 2012
[2] V ParetoTheRise and fall of Elites Transaction Publishers NewBrunswick Canada 1991
[3] K Mahmoodi B J West and P Grigolini ldquoSelf-organizingcomplex networks Individual versus global rulesrdquo Frontiers inPhysiology vol 8 article no 478 2017
[4] H E Stanley Introduction to Phase Transition and CriticalPhenomena OxfordUniversity Press NewYork NY USA 1987
[5] E Ising ldquoBeitrag zur theorie des ferromagnetismusrdquo Zeitschriftfur Physik vol 31 no 1 pp 253ndash258 1925
[6] L Onsager ldquoCrystal statistics I A two-dimensional modelwith an order-disorder transitionrdquo Physical Review A AtomicMolecular and Optical Physics vol 65 no 3-4 pp 117ndash149 1944
[7] T Mora and W Bialek ldquoAre biological systems poised atcriticalityrdquo Journal of Statistical Physics vol 144 no 2 pp 268ndash302 2011
[8] D Fraiman P Balenzuela J Foss and D R Chialvo ldquoIsing-like dynamics in large-scale functional brain networksrdquoPhysical
Review E Statistical Nonlinear and Soft Matter Physics vol 79no 6 Article ID 061922 2009
[9] D R Chialvo ldquoEmergent complex neural dynamicsrdquo NaturePhysics vol 6 no 10 pp 744ndash750 2010
[10] M G Kitzbichler M L Smith S R Christensen and EBullmore ldquoBroadband criticality of human brain networksynchronizationrdquo PLoS Computational Biology vol 5 no 3Article ID e1000314 2009
[11] P Bak andM Paczuski ldquoComplexity contingency and critical-ityrdquo Proceedings of the National Academy of Sciences vol 92 pp6689ndash6696 1995
[12] W Truccolo L R Hochberg and J P Donoghue ldquoCollectivedynamics in human and monkey sensorimotor cortex Predict-ing single neuron spikesrdquoNature Neuroscience vol 13 no 1 pp105ndash111 2010
[13] E Schneidman M J Berry II R Segev and W Bialek ldquoWeakpairwise correlations imply strongly correlated network statesin a neural populationrdquoNature vol 440 no 7087 pp 1007ndash10122006
[14] B J West M Turalska and P Grigolini Networks of EchoesImitation Innovation and Invisible Leaders Springer 2014
[15] M Turalska B J West and P Grigolini ldquoTemporal complexityof the order parameter at the phase transitionrdquo Physical ReviewE vol 91 Article ID 012907 2015
[16] P Paradisi P Allegrini A Gemignani M Laurino DMenicucci and A Piarulli ldquoScaling and intermittency of brainevents as a manifestation of consciousnessrdquo AIP ConferenceProceedings vol 1510 pp 151ndash161 2013
[17] P Paradisi and P Allegrini ldquoScaling law of diffusivity generatedby a noisy telegraph signal with fractal intermittencyrdquo ChaosSolitons amp Fractals vol 81 part B pp 451ndash462 2015
[18] E Tagliazucchi D R ChialvoM Siniatchkin et al ldquoLarge-scalesignatures of unconsciousness are consistent with a departurefrom critical dynamicsrdquo Journal of the Royal Society Interfacevol 13 no 114 Article ID 20151027 2016
[19] L Cocchi L L Gollo A Zalesky and M Breakspear ldquoCrit-icality in the brain A synthesis of neurobiology models andcognitionrdquo Progress in Neurobiology 2017
[20] P Allegrini P Paradisi D Menicucci and A GemignanildquoFractal complexity in spontaneous EEGmetastable-state tran-sitions new vistas on integrated neural dynamicsrdquo Frontiers inPhysiology vol 1 2010
[21] P Allegrini P Paradisi D Menicucci and A GemignanildquoRenewal Processes in the Critical Brainrdquo Studies of NonlinearPhenomena in Life Science vol 15 pp 161ndash179 2010
[22] M Lukovic F Vanni A Svenkeson and P Grigolini ldquoTrans-mission of information at criticalityrdquo Physica A StatisticalMechanics and its Applications vol 416 pp 430ndash438 2014
[23] M T Beig A Svenkeson M Bologna B J West and PGrigolini ldquoCritical slowing down in networks generating tem-poral complexityrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 91 no 1 Article ID 012907 2015
[24] J U Mafahim D Lambert M Zare and P Grigolini ldquoCom-plexity matching in neural networksrdquo New Journal of Physics vol 17 Article ID 015003 2015
[25] M Turalska B J West and P Grigolini ldquoRole of committedminorities in times of crisisrdquo Scientific Reports vol 3 article no1371 2013
[26] B Gert Morality a New Justification of Moral Rules OxfordUniversity Press New York NY USA 1988
10 Complexity
[27] H A Simon Models of Bounded Rationality vol 1 and 2 MITPress 1982
[28] J GMarch ldquoRationality foolishness and adaptive intelligencerdquoStrategic Management Journal vol 27 no 3 pp 201ndash214 2006
[29] D KahnemanThinking Fast and Slow Macmillan 2012[30] D G Rand and M A Nowak ldquoHuman cooperationrdquo Trends in
Cognitive Sciences vol 17 no 8 pp 413ndash425 2013[31] D G Rand ldquoCooperation Fast and Slow Meta-Analytic Evi-
dence for a Theory of Social Heuristics and Self-InterestedDeliberationrdquo Psychological Science vol 27 no 9 pp 1192ndash12062016
[32] G Gigerenzer Risk Savvy how to make good decisions PenguinBooks New York NY USA 2014
[33] G Gigerenzer Gut Feelings The Intelligence of the UnconsciousPenguin Books New York NY USA 2007
[34] P Grigolini N Piccinini A Svenkeson P Pramukkul D Lam-bert and B J West ldquoFrom Neural and Social Cooperation totheGlobal Emergence of Cognitionrdquo Frontiers in Bioengineeringand Biotechnology vol 3 article 78 2015
[35] HGintisThebounds of reasonmdashgame theory and the unificationof the behavioral sciences PrincetonUniversity Press PrincetonNJ USA 2014
[36] A Pease K Mahmoodi and B J West ldquoComplexity measuresof musicrdquo Chaos Solitons amp Fractals vol 108 pp 82ndash86 2018
[37] F VanniM Lukovic and PGrigolini ldquoCriticality and transmis-sion of information in a swarm of cooperative unitsrdquo PhysicalReview Letters vol 107 no 7 Article ID 078103 2011
[38] M Turalska E Geneston B J West P Allegrini and PGrigolini ldquoCooperation-induced topological complexity Apromising road to fault tolerance and Hebbian learningrdquoFrontiers in Physiology vol 3 article 52 2012
[39] KMahmoodi P Grigolini and B JWest ldquoOn Social Sensitivityto either Zealot or IndependentMinoritiesrdquo submitted to ChaosSolitons and Fractals
[40] P Allegrini M Bologna P Grigolini and B J WestldquoFluctuation-dissipation theorem for event-dominated pro-cessesrdquo Physical Review Letters vol 99 no 1 Article ID 0106032007
[41] N Khalil M San Miguel and R Toral ldquoZealots in themean-field noisy voter modelrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 97 no 1 Article ID012310 2018
[42] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[43] M A Nowak and R M May ldquoEvolutionary games and spatialchaosrdquo Nature vol 359 no 6398 pp 826ndash829 1992
[44] K Mahmoodi and P Grigolini ldquoEvolutionary game theory andcriticalityrdquo Journal of Physics A Mathematical and Theoreticalvol 50 no 1 Article ID 015101 2017
[45] C Gracia-Lazaro J A Cuesta A Sanchez and Y MorenoldquoHuman behavior in Prisonerrsquos Dilemma experiments sup-presses network reciprocityrdquo Scientific Reports vol 2 p 3252012
[46] W Bialek ldquoPerspectives on theory at the interface of physics andbiologyrdquo Reports on Progress in Physics vol 81 no 1 Article ID012601 2018
[47] M Munoz ldquoColloquium Criticality and dynamical scaling inliving systemsrdquo httpsarxivorgabs171204499
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Complexity 3
to the condition of independence of the other units for thebottom-up process and the Pareto cycles of the top-downversion of the modelThe original results of this earlier paperindicated a lack of resilience of the top-down SOTC modeland a robustness of the bottom-up SOTC model which areillustrated in Section 4 We devote Section 5 to balancing theresults of the present paper against the open problems that wepropose to study in future research
2 Bottom-Up Approach to Self-OrganizedTemporal Criticality
The decisions of single individuals in our model are made inaccordance with the criterion of bounded rationality [27 28]expanded by Kahneman [29] and more recently discussedfrom within the perspective of evolutionary game theory(EGT) [30 31] The nonrational component of the decision-making process is stressed also by the work of Gigerenzer[32 33] Herein individuals make decision using DMM Theindividuals of the social network aiming at increasing theirpayoffmake the control parameter119870119903 for individual 119903 evolvetowards criticality thereby creating an intelligent groupmind[34] perhaps connected to the intelligent unconscious ofGigerenzer [33] leading the individuals to fast but essentiallywise decisions As we shall see the time evolution of 119870119903 isslow because it depends on the payoffs of the individual atearlier time corresponding to the slow thinking mechanismdiscussed by Kahneman [29]
21 The Intuitive and Emotional Level In this paper we usethe DMM on a two-dimensional lattice of size 119871 with119872 =119871 times 119871 individuals and we set 119871 = 10 The DMM is basedon individuals imperfectly imitating the majority opinion oftheir four nearest neighbors thereby biasing the probabilityof making a transition from being a cooperator (119862) to beinga defector (119863)
119892(119903)119862119863 = 1198920 expminus119870119903 (119873(119903)119862 minus 119873(119903)119863 )119873 (4)
where 119873(119903)119862 is the number of nearest neighbors to individual119903 that are cooperators 119873(119903)119863 is the number of defectors andeach individual on the simple lattice has 119873 = 4 nearestneighbors In the same way the transition rate from defectorsto cooperators 119892(119903)119863119862 is
119892(119903)119863119862 = 1198920 exp119870119903 (119873(119903)119862 minus 119873(119903)119863 )119873 (5)
The unbiased transition rate is 1198920 = 001 throughout thecalculations and 11198920 defines the time scale for the processThe DMM has been shown [14] to undergo critical phasetransitions and to be a member of the Ising universalityclass in which all the members of the network can actcooperatively depending on themagnitude of the interactionstrength119870 [14] However this important result is obtained byassigning to all the individuals the same degree of attentionto the opinions of their nearest neighbors called 119870 Herein
Table 1 The payoffs of Prisonerrsquos dilemma game The first value ofeach pairs is the payoff of player119883 and the second value is the payoffof the player 119884
Player 119884119862 119863Player119883119862 (1 1) (0 19)119863 (19 0) (0 0)
each individual may have a different degree of attention andthis degree of attention does not fit the reciprocity principleThe degree of attention that the individual 119903 devotes to theindividual 1199031015840 may differ from the degree of attention that theindividual 1199031015840 devotes to the individual 119903 To explain how theindividual 119903 is influenced by her nearest neighbors let usconsider for instance (4) The individual we are consideringis a cooperator and (4) establishes the rate of her transition tothe defection state If 119873(119903)119862 gt 119873(119903)119863 the rate decreases and willvanish in the extreme limit119870119903 = infin Of course this will havethe effect of favoring the cooperation state
22 The Rational Level This decision-making process is fastand emotional and does not involve any direct reasoningabout the payoff The connection with the self-interestaccording to the slow thinking mechanism discussed byKanheman [29] is established over a more extended timescale where the single individual exerts an influence on theprocess aiming atmaximizing her payoff To define the payoffwe adopt the prisonerrsquos dilemma game (PDG) [35] Twoplayers interact and receive a payoff from their interactionadopting either the defection or the cooperation strategy Ifboth players select the cooperation strategies each of themreceives the payoff 119877 and their society receives the payoff2119877 The player choosing the defection strategy receives thepayoff 119879 The temptation to cheat is established by setting thecondition 119879 gt 119877 However this larger payoff is assigned tothe defector only if the other player selects cooperation Theplayer selecting cooperation receives the payoff 119878 which issmaller than 119877 If the other player also selects defection thepayoff for both players is119875 which is smaller than119877The PDGis based on the crucial payoffs 119879 gt 119877 gt 119875 gt 119878 and 119878+119879 lt 2119877
We adopt the choice of parameter values made by Gintis[35] and set 119877 = 1 119875 = 0 and 119878 = 0 The maximalpossible value of 119879 is 2 and we select the value 119879 = 19which is a very strong incentive to cheat These choices aresummarized in Table 1 We evaluate the social benefit for thesingle individual as well as for the community as a wholeas follows We define the payoff 119875119903 for individual 119903 as theaverage over the payoffs from the interactions with its fournearest neighbors If both players of a pair are cooperatorsthe contribution to the payoff of the individual 119903 is 119861119903 = 1If one of the two playing individuals is a cooperator and theother is a defector the contribution to the payoffof 119903 is119861119903 = 119879If both players are defectors the contribution to the payoff of119903 is 119861119903 = 0 The payoff 119875119903 to individual 119903 is the sum over thefour 119861119903rsquos
4 Complexity
Each individual receives a total payoff from the gamewiththe four nearest neighbors and adjusts her imitation strengthas follows
119870119903 (119905) = 119870119903 (119905 minus Δ119905) + 120594119875119903 (119905 minus Δ119905) minus 119875119903 (119905 minus 2Δ119905)119875119903 (119905 minus Δ119905) + 119875119903 (119905 minus 2Δ119905) (6)
where the parameter 120594 determines the intensity of interestof the individuals to the fractional change in their payoffs intimeThe key equation (6) is based on the assumption that theintuitive decision-making process is so fast that at both times119905 minus Δ119905 and 119905 minus 2Δ119905 it is possible to evaluate the correspondingpayoffs on the basis of fast decisions made by each individualand by her 4 nearest neighbors The decision of adjusting thesocial sensitivity 119870119903(119905) requires the time interval 2Δ119905 whilethe intuitive decision is virtually instantaneous
The second term on the right-hand side of (6) is the ratiobetween two quantities that for special cases vanish In thesecases we set the condition
119870119903 (119905) = 119877119870119903 (119905 minus Δ119905) (7)
with 119877 lt 1 We selected 119877 = 0 but for other values we getthe same result When 119870119903(119905) goes to negative values we set itequal to zero
Note that in the limit of vanishing time intervals (6)relates the time rate of change of an individualrsquos imitationstrength to the time rate of change of the logarithm of thelocal payoff to that individual On the global scale the meanbenefit to society of all the individuals is given by the averageover all the 119875119903rsquos
Π (119905) = 1119873119873sum119903=1
119875119903 (119905) (8)
whereas the mean imitation strength is given by the averageover all the119870119903(119905)
119870 (119905) = 1119873119873sum119903=1
119870119903 (119905) (9)
For the bottom-up case discussed in this section thecalculation is done with the parameters119872 = 100 1198920 = 001119879 = 19 and 120594 = 1 with the social benefit imitation strengthand mean field starting from zero
The results of Figure 1 are used to establish the bottom-up origin of altruism rather than interpreting it as it is fre-quently done to be the result of a religion-induced top-downprocess The calculations show that the top-down processgenerating altruism weakens the systemrsquos resilience whereasthe genuinely bottom-up approach makes the emergence ofaltruism robust against external perturbation Figure 1 showsthat the time evolution of the individual social sensitivity 119870119903is characterized by abrupt jumps that from time to time mayalso bring the single individual back to a behavior totallyindependent of the choices made by her nearest neighborsThis is a healthy social condition that has the effect of makingthe global properties 119909(119905) 119870(119905) and Π(119905) host crucial eventsfavoring the transmission of information between differentsocial systems either countries or parties
x(t)
K(t)
Kr(t)
4
3
2
1
0
minus10 2000 4000
t6000 8000 10000
Π(t)
Figure 1 For the bottom-up SOCT time evolution of the averagesocial benefit Π(119905) the average imitation strength 119870(119905) the meanfield 119909(119905) and the imitation strength of one of the units119870119903(119905) plottedversus time The mean field fluctuates around 08 which meansabout 90 percent of individuals are cooperators and 10 percentare defectors Mean field of 1 has all individuals in the state ofcooperation and the maximum average social benefit has the value4 because this is the number of nearest neighbors on the two-dimensional lattice
To stress the occurrence of crucial events in a socialsystem resting on the bottom-up emergence of altruism wehave to extend the method used for criticality generated bythe fine tuning of the control parameter 119870 In that caseat criticality the mean field fluctuates around the vanishingvalue and the crucial events correspond to the occurrenceof this vanishing value [15 23] We follow [36] and evaluatethe fluctuations around the proper nonvanishing mean valueof 119870 = 15 To explain this choice notice that in theconventional case of criticality generated by the choice of aproper control parameter 119870 with 119872 = 100 119870 = 15 isthe value at which the onset of phase transition occurs Thisis the value making the mean field 119909(119905) of the conventionalDMM fluctuate around 119909 = 0 with complex fluctuationsand which generates criticality-induced intelligence [37 38]In the case of the SOTC model this condition of criticality-induced intelligence with fluctuations of 119870(119905) around 15 isspontaneously generated When the criticality condition isreached the complex fluctuations of 119909(119905) do not occur anylonger around 119909 = 0 but around a positive value of the orderof 08 The time intervals 120591 between consecutive crossings ofthe 15 level are monitored and the corresponding waiting-time PDF 120595(120591) is illustrated in Figure 2
Figure 2 similar to Figure 4 of [3] is compared inSection 4 with Figure 4 illustrating the perturbation byindependents We limit ourselves to noticing that temporalcomplexity shows up in the intermediate asymptotics regime[3] and it is characterized by the IPL index 120583 = 13 a propertyshared by other systems at criticality see for instance [22]
Complexity 5
cong 13
()
1E minus 6
1E minus 5
1E minus 4
0001
001
01
10 100 10001
Figure 2 For bottom-up SOTC the waiting-time PDF of thetime interval between two consecutive crossings of the averagevalue of the mean value of 119870(119905) which is asymp15 is graphed 120595(120591) isexponentially truncated and has an intermediate asymptotic regimewith an index of 120583 asymp 13 This figure is similar to Figure 4 of [3]
This is evidence that the spontaneous transition to criticalityalso generates the crucial events responsible for informationtransport Earlier work [3] confirms that the crucial eventsfacilitate the transport of information from one complexsocial system to another if the two systems are at criticalityas a result of the spontaneous process of self-organization
In spite of the exponential truncation that may lead to themisleading conclusion that the Poisson-like character of thelong-time regime quenches the manifestations of complexitythe transport of information is determined by the interme-diate asymptotics In the earlier work of [3] we studied theefficiency of transport of information from one bottom-upSOTC to another identical SOTC and we found that themaximal efficiency of the information transport correspondsto about 119872 = 100 which is the condition studied inthis paper The explanation of these interesting effects is thefollowing The intensity of fluctuations generating crucialevents decreases with the size increase according to thefollowing formula (see Eq (14) of [3])
Δ120577 prop 1119872] (10)
with ] = 05 Note that Δ120577 denotes the intensity of thefluctuations of the variables 119870Π and 119909(119905) around theirmean values Therefore the systems with 119872 gt 100 havecrucial fluctuations of smaller intensity thereby explainingthe reduction of the process of information transport Forvalues 119872 lt 100 the role of the exponential truncationbecomes more important and the time extension of thecomplex intermediate asymptotics is reduced and eventuallythe intermediate asymptotics regime vanishes turning thesystem into a Poisson system with no complexity This hasthe effect of significantly reducing the efficiency of the processof information transport As far as the resilience of thebottom-up SOTC is concerned the theory of this paper rests
on the connection between resilience and the efficiency ofinformation transport As a consequence the results on theresilience of the bottom-up SOTC for119872 = 100 automaticallycorrespond on the condition of maximal resilience [39]
3 Top-Down Approach to Self-OrganizedTemporal Criticality
The top-down approach to self-organization is done usingagain (4) and (5) The adoption of the top-down perspectiveis realized by replacing (6) with
119870 (119905) = 119870 (119905 minus Δ119905) + 120594Π (119905 minus Δ119905) minus Π (119905 minus 2Δ119905)Π (119905 minus Δ119905) + Π (119905 minus 2Δ119905) (11)
The top-down origin of this process is made evident by thefact that all individuals in the network are forced to adoptthe same time-dependent imitation strength Furthermorerational choice is made on the basis of the collective payoffΠ(119905) using PDGThe conceptual difference with the bottom-up approach of Section 2 is impressive In fact with (11) all theindividuals of this society must change their social sensitivityat the same time and the information about the increase ordecrease of the global payoff implies that all the individualsare given this information from a central source such as thegovernment suggesting that a form of organization alreadyexists and is not created by the interaction between theindividuals In [3] the assumptionwasmade that a benevolentdictator exists and leads such a process Using Paretorsquos socialtheory we make the assumption that this process implies theleadership of an elite [2]
The second term on the right-hand side of (11) is theratio between two quantities that similarly to the bottom-upmodel for special cases vanishes In these cases as done inSection 2 we set the condition
119870119903 (119905) = 119877119870119903 (119905 minus Δ119905) (12)
with 119877 lt 1 We selected 119877 = 0 When 119870119903(119905) goes to negativevalues we set it equal to zero
For the top-down case discussed in this section thecalculation is done with the parameters119872 = 100 1198920 = 001119879 = 19 and 120594 = 4 with the social benefit imitation strengthand mean field starting from zero 120594 is chosen to be largerthan in the bottom-up case because of the fact that transitionto criticality in the top-down case is much slower (see Figures1 and 3)
Under the leadership of an elite see Figure 3 the controlparameter 119870(119905) shows a behavior totally different from thatof Figure 1 In this section we focus on the behavior of 119870(119905)in the absence of perturbation and discuss the effects ofperturbation in Section 4 With no perturbation there is atransient from 119905 = 0 to 119905 asymp 40000 after which time a sequenceof rises and falls occurs The value of 119870 adjusts according to(11) from small values around 02 to a maximal value of 18which is known to correspond to a supercritical conditionin the case of the conventional DMM When using the finetuning control parameter approach we set119870 = 18 the socialsystem is far from the intelligence condition that accordingto a widely accepted opinion [9 16ndash19] requires criticality
6 Complexity
Time100000800006000040000200000
minus02
00
02
04
06
08
10
12
14
16
18
20
K(t)
Figure 3 Black line the time evolution 119870(119905) of (11) for the sameregular two-dimensional network of Figure 1 red line the timeevolution of119870(119905) of the self-organized system described by the blackunder the influence of the weak noisy perturbation described inSection 4
The mean field 119909(119905) has very fast fluctuations around a meanvalue close to 1 but these fluctuations are Poisson and theconventional DMM system loses its complexity [22]
The falls to the small values of119870 are interpreted as falls ofelites The subcritical condition as well as the supercriticalis characterized by a lack of intelligence We have to remarkalso that values of 119870 significantly smaller than 119870 asymp 15indicate that there are many units with119870119903 = 0 like the singleunit of Figure 1 at a time close to 119905 asymp 2000 In conclusionboth small and maximal values of 119870 are affected by a lack ofconsciousness and the transitions through 119870 asymp 15 are toofast for the social system to benefit from the intelligence of thecritical condition This lack of intelligence is responsible forthe lack of resilience The sojourn times in the supercriticalstate correspond to the time durations of elites We do nothave to confuse the fluctuations of 119870(119905) with those of themean field 119909(119905) that are not shown here The fluctuations of119909(119905) are always Poisson around mean values close to 1 when119870(119905) is close to 18 and around the vanishing mean valuewhen 119870(119905) drops
It is interesting to notice that also the time intervalbetween consecutive falls of elite is a complex dynamicalprocess characterized by an IPL with 120583 asymp 2 in this case asshown by Figure 4 However the system is not resilient Thesojourn in a supercritical state with a 119870 significantly largerthan119870 asymp 15 is characterized by fast Poisson fluctuations andan external perturbation can easily affect the time durationof this regime [40] In fact the big difference betweenPoisson events and crucial events is that the former eventsobey conventional linear response theory and any formsof perturbation can deeply affect their dynamics therebyundermining the social resilience as we show in the nextsection
1 10 100 1000
1
10
100
()
1000
10000
Figure 4 For bottom-up SOTC the waiting-time PDF of the timeintervals between two consecutive crossings of the horizontal linewith 119870 = 07 (black and red line) and 119870 = 17 (blue line)for the top-down SOTC The blue and the black lines describethe unperturbed case and the red line describes the perturbationdescribed in Section 4
4 Perturbing the Self-Organized Society
To substantiate the arguments of the earlier section with theresults of a numerical simulation we devote this section toillustrating some numerical experiments on the effects of aperturbation on the process of societal self-organization
First of all let us define two different sources of pertur-bation the independent and the committed minorities Weassume that a minority of independent individual exists Anindependent individual is a unit that is characterized by119870119903 =0 As a consequence this unit does not adopt (6) and iscompletely insensitive to the connection between individualand societal benefit that yields the emergence of cooperation[3] The perturbing nature of this independent individualis realized by the fact that while the independent keeping119870119903 = 0 is completely independent of the choices made bythe other units her nearest neighbors are influenced by thechoices of the independent through the DMM and throughthe evaluation of the payoff 119875119903(119905) of (6)
In the top-down SOTCmodel the independent influencesthe process through his vanishing contribution to 119870(119905) andthrough his contribution to the global payoff of (11) Theperturbation of independents ismademore devastatingwhenthe independents are allowed to move randomly through thesocial network
The other kind of perturbation produced by committedminorities has already been studied elsewhere [25] Theseare minorities that keep selecting the state119863 The committedminorities are also called zealots and have been the subject ofmany publications see [41] for a wide set of referencesThesepublications emphasize the dramatic consequences that thezealots have on their societies thereby implying that theirmodels of organization are not resilient The experiment onthe perturbation of zealots done herein shows that the top-down SOTC model shares the lack of resilience observed in
Complexity 7
time10000080000600004000020000
00
02
04
06
08
10
12
14
16
18
K(t)
Figure 5119870(119905) as a function of time for the bottom-up SOTC in theno perturbation (black line) and perturbed case (red line)
these earlier studies on the social influences of zealots Thebottom-up SOTC model seems to be the only fully resilientmodel
Let us discuss first the strongest source of perturbationthe randomly moving independents At any time step oneof the 119872 = 100 is randomly selected to play the role ofindependent namely we force her to adopt the value 120585 = 1or the value 120585 = minus1 with equal probability In the case of thebottom-up SOTC this perturbation does not have significanteffect on the time evolution of119870(119905) as shown by Figure 5
In the case of the top-down SOTC the effects of thisperturbation are impressive The black line of Figure 3 showsthe rise and the fall of an elite The weak noisy perturbationmakes 119870(119905) evolve as illustrated by the red line of the samefigure which shows that the sojourn times of unperturbedelites are filled with many falls that are a clear manifestationof the lack of societal resilience
Important information on the lack of resilience of thetop-down SOTC is afforded by Figure 4 showing 120595(119905) fordifferent values of the threshold used to find the statistics ofcrucial events When the threshold is 17 close to the topsupercritical region reached by the system the intermediateasymptotics has a power index 120583 asymp 145 larger than that ofthe bottom-up SOTCThe adoption of the threshold119870 = 07as an effect of the collapse of elites cancels the intermediateasymptotic temporal complexity and favors the birth of aPoisson shoulder The noisy perturbation of independentsmakes this behavior even more pronounced This strongexponential shoulder is a signature of the death of dynamicalcomplexity and of the transition fromnon-Poisson to Poissonbehavior [3]
The perturbing action of independents is weaker if theindependent individuals do not move This response to thisform of perturbation is illustrated in Figure 6 showing thateven in this case the bottom-up SOTC model is more robustthan the top-down
We establish the perturbation of independent individ-uals in a different way We assume that all the units are
02 0800 100604
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
Figure 6 Dependence of mean field (at time 106) on the ratio 120588 ofindependent units (which are fixed on the lattice) for the bottom-up SOTCmodel (black line) top-down SOTCmodel (red line) andordinary DMMwith tuned control parameter119870119888 = 15 (green line)Ensemble average is done over 10 experiments
independent for a fraction 120578 of their time The results aredepicted in Figure 7 It is clear from the figure that thebottom-up SOTCmodel ismore resilient than the two-downeven to this most violent form of disruption
Finally in Figure 8 we show the action of committedminorities We see that only the bottom-up SOTC model isresilient The top-down SOTC model shares the same lack ofresilience shown by ordinary DMM at criticality
5 Conclusions
It is remarkable that according to SOTC the crucial eventsmay be harmful as well as beneficial If the global parameter119870(119905) fluctuating around the long-range correlation generat-ingmean value returns to the vanishing value the temporarycollapse is turned into a societal disaster The collapse into119870 = 0 would correspond to a new initial condition andas shown by Figure 1 119870(119905) would start increasing againgenerating a new organization led by a new elite [2] Howeverthe genuinely bottom-up process leading the time evolutionillustrated by Figure 1 is expected to keep forever the socialsystem in the condition of weak fluctuations around119870 asymp 15In other words there is an impressive difference between thecrucial events hosted by the weak fluctuation around119870 asymp 15and the regressions of 119870(119905) to values of 119870 ≪ 15 generatedby the adoption of a top-down process led by an elite
The main conclusion of this paper concerning resilienceis that criticality is necessary for resilience but it is notsufficient The top-down SOTC model generates criticalitybut it is not resilient Therefore information transport fromone top-down SOTC model system to another top-downSOTC model system is expected to occur by means ofcomplexity matching in spite of the fact that the two systemsare not resilient and the information transport may be easilyquenched by stray perturbing noises
8 Complexity
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10times(106)
0005 0010 0015 00200000
(a)
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
0005 0010 0015 00200000
(b)
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
0005 0010 0015 00200000
(c)
Figure 7 Dependence of mean field (at time 106) on 120578 (the ratio of time in which units behave individually) for bottom-up SOTCmodel (a)top-down SOTC model (b) and ordinary DMMwith tuned parameter119870119888 = 15 (c) Each set of colored dots corresponds to one experimentand the black solid lines are the average of them
An attractive interpretation of the resilient nature of thebottom-up SOTC model is that the ideal condition of fulldemocracy is the most robust form of social organization
In this paper the social payoff is evaluated using thePDG [35] The prisonerrsquos dilemma game is frequently usedin the field of EGT [42 43] EGTs aim at solving thealtruism paradox using the concept of network reciprocity[43] A game is played many times on a network where eachindividual is surrounded by a set of nearest neighbors andadopts the strategy of the most successful nearest neighborSince the clusters of cooperators are richer than the clustersof defectors it is plausible that the most successful nearestneighbor is a cooperator However this attempt at mimickingthe action of a collective intelligence failed because the socialactivity of the units being subcritical disrupts the beneficialeffects of network reciprocity [44 45] We note that SOTCmodeling represents an attempt to amend the field of EGT
by the limitations preventing for instance the concept ofnetwork reciprocity from yielding a satisfactory resolution ofthe altruism paradox
The human inclination to cooperate is the result of bio-logical evolution and of the spontaneous evolution towardscriticality The time appears ripe to unify the models ofbiology and physics made necessary to reach the ambitiousgoal of achieving a rigorous scientific foundation of thisimportant human characteristic [46 47] The spontaneoustransition to criticality of SOTC contributes to bypassing thecurrent limitations of the field of EGT SOTC models asshown in this paper can be adapted to take into account thetop-downprocesses connectedwith the nonresilient action ofelites It is possible to supplement the nonrational decision-making process based on (4) and (5) with self-righteousbiases [1] taking into account the influence of religion or otherpolarizing influences We expect that such generalizations
Complexity 9
minus12
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
06 0804 1000 02
Figure 8 Dependence of mean field (at time 106) on the ratio 120588of fanatics (having fixed position on the lattice) for the bottom-upSOTC model (red line) top-down SOTC model (black line) andordinary DMMwith tuned control parameter119870119888 = 15 (green line)Ensemble average is done over 10 experiments
of SOTC theory will lead to a lack of societal resilienceHowever this is left as a subject for future research
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
Korosh Mahmoodi and Paolo Grigolini warmly thank AROandWelch for support throughGrants noW911NF-15-1-0245and no B-1577 respectively
References
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[2] V ParetoTheRise and fall of Elites Transaction Publishers NewBrunswick Canada 1991
[3] K Mahmoodi B J West and P Grigolini ldquoSelf-organizingcomplex networks Individual versus global rulesrdquo Frontiers inPhysiology vol 8 article no 478 2017
[4] H E Stanley Introduction to Phase Transition and CriticalPhenomena OxfordUniversity Press NewYork NY USA 1987
[5] E Ising ldquoBeitrag zur theorie des ferromagnetismusrdquo Zeitschriftfur Physik vol 31 no 1 pp 253ndash258 1925
[6] L Onsager ldquoCrystal statistics I A two-dimensional modelwith an order-disorder transitionrdquo Physical Review A AtomicMolecular and Optical Physics vol 65 no 3-4 pp 117ndash149 1944
[7] T Mora and W Bialek ldquoAre biological systems poised atcriticalityrdquo Journal of Statistical Physics vol 144 no 2 pp 268ndash302 2011
[8] D Fraiman P Balenzuela J Foss and D R Chialvo ldquoIsing-like dynamics in large-scale functional brain networksrdquoPhysical
Review E Statistical Nonlinear and Soft Matter Physics vol 79no 6 Article ID 061922 2009
[9] D R Chialvo ldquoEmergent complex neural dynamicsrdquo NaturePhysics vol 6 no 10 pp 744ndash750 2010
[10] M G Kitzbichler M L Smith S R Christensen and EBullmore ldquoBroadband criticality of human brain networksynchronizationrdquo PLoS Computational Biology vol 5 no 3Article ID e1000314 2009
[11] P Bak andM Paczuski ldquoComplexity contingency and critical-ityrdquo Proceedings of the National Academy of Sciences vol 92 pp6689ndash6696 1995
[12] W Truccolo L R Hochberg and J P Donoghue ldquoCollectivedynamics in human and monkey sensorimotor cortex Predict-ing single neuron spikesrdquoNature Neuroscience vol 13 no 1 pp105ndash111 2010
[13] E Schneidman M J Berry II R Segev and W Bialek ldquoWeakpairwise correlations imply strongly correlated network statesin a neural populationrdquoNature vol 440 no 7087 pp 1007ndash10122006
[14] B J West M Turalska and P Grigolini Networks of EchoesImitation Innovation and Invisible Leaders Springer 2014
[15] M Turalska B J West and P Grigolini ldquoTemporal complexityof the order parameter at the phase transitionrdquo Physical ReviewE vol 91 Article ID 012907 2015
[16] P Paradisi P Allegrini A Gemignani M Laurino DMenicucci and A Piarulli ldquoScaling and intermittency of brainevents as a manifestation of consciousnessrdquo AIP ConferenceProceedings vol 1510 pp 151ndash161 2013
[17] P Paradisi and P Allegrini ldquoScaling law of diffusivity generatedby a noisy telegraph signal with fractal intermittencyrdquo ChaosSolitons amp Fractals vol 81 part B pp 451ndash462 2015
[18] E Tagliazucchi D R ChialvoM Siniatchkin et al ldquoLarge-scalesignatures of unconsciousness are consistent with a departurefrom critical dynamicsrdquo Journal of the Royal Society Interfacevol 13 no 114 Article ID 20151027 2016
[19] L Cocchi L L Gollo A Zalesky and M Breakspear ldquoCrit-icality in the brain A synthesis of neurobiology models andcognitionrdquo Progress in Neurobiology 2017
[20] P Allegrini P Paradisi D Menicucci and A GemignanildquoFractal complexity in spontaneous EEGmetastable-state tran-sitions new vistas on integrated neural dynamicsrdquo Frontiers inPhysiology vol 1 2010
[21] P Allegrini P Paradisi D Menicucci and A GemignanildquoRenewal Processes in the Critical Brainrdquo Studies of NonlinearPhenomena in Life Science vol 15 pp 161ndash179 2010
[22] M Lukovic F Vanni A Svenkeson and P Grigolini ldquoTrans-mission of information at criticalityrdquo Physica A StatisticalMechanics and its Applications vol 416 pp 430ndash438 2014
[23] M T Beig A Svenkeson M Bologna B J West and PGrigolini ldquoCritical slowing down in networks generating tem-poral complexityrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 91 no 1 Article ID 012907 2015
[24] J U Mafahim D Lambert M Zare and P Grigolini ldquoCom-plexity matching in neural networksrdquo New Journal of Physics vol 17 Article ID 015003 2015
[25] M Turalska B J West and P Grigolini ldquoRole of committedminorities in times of crisisrdquo Scientific Reports vol 3 article no1371 2013
[26] B Gert Morality a New Justification of Moral Rules OxfordUniversity Press New York NY USA 1988
10 Complexity
[27] H A Simon Models of Bounded Rationality vol 1 and 2 MITPress 1982
[28] J GMarch ldquoRationality foolishness and adaptive intelligencerdquoStrategic Management Journal vol 27 no 3 pp 201ndash214 2006
[29] D KahnemanThinking Fast and Slow Macmillan 2012[30] D G Rand and M A Nowak ldquoHuman cooperationrdquo Trends in
Cognitive Sciences vol 17 no 8 pp 413ndash425 2013[31] D G Rand ldquoCooperation Fast and Slow Meta-Analytic Evi-
dence for a Theory of Social Heuristics and Self-InterestedDeliberationrdquo Psychological Science vol 27 no 9 pp 1192ndash12062016
[32] G Gigerenzer Risk Savvy how to make good decisions PenguinBooks New York NY USA 2014
[33] G Gigerenzer Gut Feelings The Intelligence of the UnconsciousPenguin Books New York NY USA 2007
[34] P Grigolini N Piccinini A Svenkeson P Pramukkul D Lam-bert and B J West ldquoFrom Neural and Social Cooperation totheGlobal Emergence of Cognitionrdquo Frontiers in Bioengineeringand Biotechnology vol 3 article 78 2015
[35] HGintisThebounds of reasonmdashgame theory and the unificationof the behavioral sciences PrincetonUniversity Press PrincetonNJ USA 2014
[36] A Pease K Mahmoodi and B J West ldquoComplexity measuresof musicrdquo Chaos Solitons amp Fractals vol 108 pp 82ndash86 2018
[37] F VanniM Lukovic and PGrigolini ldquoCriticality and transmis-sion of information in a swarm of cooperative unitsrdquo PhysicalReview Letters vol 107 no 7 Article ID 078103 2011
[38] M Turalska E Geneston B J West P Allegrini and PGrigolini ldquoCooperation-induced topological complexity Apromising road to fault tolerance and Hebbian learningrdquoFrontiers in Physiology vol 3 article 52 2012
[39] KMahmoodi P Grigolini and B JWest ldquoOn Social Sensitivityto either Zealot or IndependentMinoritiesrdquo submitted to ChaosSolitons and Fractals
[40] P Allegrini M Bologna P Grigolini and B J WestldquoFluctuation-dissipation theorem for event-dominated pro-cessesrdquo Physical Review Letters vol 99 no 1 Article ID 0106032007
[41] N Khalil M San Miguel and R Toral ldquoZealots in themean-field noisy voter modelrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 97 no 1 Article ID012310 2018
[42] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[43] M A Nowak and R M May ldquoEvolutionary games and spatialchaosrdquo Nature vol 359 no 6398 pp 826ndash829 1992
[44] K Mahmoodi and P Grigolini ldquoEvolutionary game theory andcriticalityrdquo Journal of Physics A Mathematical and Theoreticalvol 50 no 1 Article ID 015101 2017
[45] C Gracia-Lazaro J A Cuesta A Sanchez and Y MorenoldquoHuman behavior in Prisonerrsquos Dilemma experiments sup-presses network reciprocityrdquo Scientific Reports vol 2 p 3252012
[46] W Bialek ldquoPerspectives on theory at the interface of physics andbiologyrdquo Reports on Progress in Physics vol 81 no 1 Article ID012601 2018
[47] M Munoz ldquoColloquium Criticality and dynamical scaling inliving systemsrdquo httpsarxivorgabs171204499
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4 Complexity
Each individual receives a total payoff from the gamewiththe four nearest neighbors and adjusts her imitation strengthas follows
119870119903 (119905) = 119870119903 (119905 minus Δ119905) + 120594119875119903 (119905 minus Δ119905) minus 119875119903 (119905 minus 2Δ119905)119875119903 (119905 minus Δ119905) + 119875119903 (119905 minus 2Δ119905) (6)
where the parameter 120594 determines the intensity of interestof the individuals to the fractional change in their payoffs intimeThe key equation (6) is based on the assumption that theintuitive decision-making process is so fast that at both times119905 minus Δ119905 and 119905 minus 2Δ119905 it is possible to evaluate the correspondingpayoffs on the basis of fast decisions made by each individualand by her 4 nearest neighbors The decision of adjusting thesocial sensitivity 119870119903(119905) requires the time interval 2Δ119905 whilethe intuitive decision is virtually instantaneous
The second term on the right-hand side of (6) is the ratiobetween two quantities that for special cases vanish In thesecases we set the condition
119870119903 (119905) = 119877119870119903 (119905 minus Δ119905) (7)
with 119877 lt 1 We selected 119877 = 0 but for other values we getthe same result When 119870119903(119905) goes to negative values we set itequal to zero
Note that in the limit of vanishing time intervals (6)relates the time rate of change of an individualrsquos imitationstrength to the time rate of change of the logarithm of thelocal payoff to that individual On the global scale the meanbenefit to society of all the individuals is given by the averageover all the 119875119903rsquos
Π (119905) = 1119873119873sum119903=1
119875119903 (119905) (8)
whereas the mean imitation strength is given by the averageover all the119870119903(119905)
119870 (119905) = 1119873119873sum119903=1
119870119903 (119905) (9)
For the bottom-up case discussed in this section thecalculation is done with the parameters119872 = 100 1198920 = 001119879 = 19 and 120594 = 1 with the social benefit imitation strengthand mean field starting from zero
The results of Figure 1 are used to establish the bottom-up origin of altruism rather than interpreting it as it is fre-quently done to be the result of a religion-induced top-downprocess The calculations show that the top-down processgenerating altruism weakens the systemrsquos resilience whereasthe genuinely bottom-up approach makes the emergence ofaltruism robust against external perturbation Figure 1 showsthat the time evolution of the individual social sensitivity 119870119903is characterized by abrupt jumps that from time to time mayalso bring the single individual back to a behavior totallyindependent of the choices made by her nearest neighborsThis is a healthy social condition that has the effect of makingthe global properties 119909(119905) 119870(119905) and Π(119905) host crucial eventsfavoring the transmission of information between differentsocial systems either countries or parties
x(t)
K(t)
Kr(t)
4
3
2
1
0
minus10 2000 4000
t6000 8000 10000
Π(t)
Figure 1 For the bottom-up SOCT time evolution of the averagesocial benefit Π(119905) the average imitation strength 119870(119905) the meanfield 119909(119905) and the imitation strength of one of the units119870119903(119905) plottedversus time The mean field fluctuates around 08 which meansabout 90 percent of individuals are cooperators and 10 percentare defectors Mean field of 1 has all individuals in the state ofcooperation and the maximum average social benefit has the value4 because this is the number of nearest neighbors on the two-dimensional lattice
To stress the occurrence of crucial events in a socialsystem resting on the bottom-up emergence of altruism wehave to extend the method used for criticality generated bythe fine tuning of the control parameter 119870 In that caseat criticality the mean field fluctuates around the vanishingvalue and the crucial events correspond to the occurrenceof this vanishing value [15 23] We follow [36] and evaluatethe fluctuations around the proper nonvanishing mean valueof 119870 = 15 To explain this choice notice that in theconventional case of criticality generated by the choice of aproper control parameter 119870 with 119872 = 100 119870 = 15 isthe value at which the onset of phase transition occurs Thisis the value making the mean field 119909(119905) of the conventionalDMM fluctuate around 119909 = 0 with complex fluctuationsand which generates criticality-induced intelligence [37 38]In the case of the SOTC model this condition of criticality-induced intelligence with fluctuations of 119870(119905) around 15 isspontaneously generated When the criticality condition isreached the complex fluctuations of 119909(119905) do not occur anylonger around 119909 = 0 but around a positive value of the orderof 08 The time intervals 120591 between consecutive crossings ofthe 15 level are monitored and the corresponding waiting-time PDF 120595(120591) is illustrated in Figure 2
Figure 2 similar to Figure 4 of [3] is compared inSection 4 with Figure 4 illustrating the perturbation byindependents We limit ourselves to noticing that temporalcomplexity shows up in the intermediate asymptotics regime[3] and it is characterized by the IPL index 120583 = 13 a propertyshared by other systems at criticality see for instance [22]
Complexity 5
cong 13
()
1E minus 6
1E minus 5
1E minus 4
0001
001
01
10 100 10001
Figure 2 For bottom-up SOTC the waiting-time PDF of thetime interval between two consecutive crossings of the averagevalue of the mean value of 119870(119905) which is asymp15 is graphed 120595(120591) isexponentially truncated and has an intermediate asymptotic regimewith an index of 120583 asymp 13 This figure is similar to Figure 4 of [3]
This is evidence that the spontaneous transition to criticalityalso generates the crucial events responsible for informationtransport Earlier work [3] confirms that the crucial eventsfacilitate the transport of information from one complexsocial system to another if the two systems are at criticalityas a result of the spontaneous process of self-organization
In spite of the exponential truncation that may lead to themisleading conclusion that the Poisson-like character of thelong-time regime quenches the manifestations of complexitythe transport of information is determined by the interme-diate asymptotics In the earlier work of [3] we studied theefficiency of transport of information from one bottom-upSOTC to another identical SOTC and we found that themaximal efficiency of the information transport correspondsto about 119872 = 100 which is the condition studied inthis paper The explanation of these interesting effects is thefollowing The intensity of fluctuations generating crucialevents decreases with the size increase according to thefollowing formula (see Eq (14) of [3])
Δ120577 prop 1119872] (10)
with ] = 05 Note that Δ120577 denotes the intensity of thefluctuations of the variables 119870Π and 119909(119905) around theirmean values Therefore the systems with 119872 gt 100 havecrucial fluctuations of smaller intensity thereby explainingthe reduction of the process of information transport Forvalues 119872 lt 100 the role of the exponential truncationbecomes more important and the time extension of thecomplex intermediate asymptotics is reduced and eventuallythe intermediate asymptotics regime vanishes turning thesystem into a Poisson system with no complexity This hasthe effect of significantly reducing the efficiency of the processof information transport As far as the resilience of thebottom-up SOTC is concerned the theory of this paper rests
on the connection between resilience and the efficiency ofinformation transport As a consequence the results on theresilience of the bottom-up SOTC for119872 = 100 automaticallycorrespond on the condition of maximal resilience [39]
3 Top-Down Approach to Self-OrganizedTemporal Criticality
The top-down approach to self-organization is done usingagain (4) and (5) The adoption of the top-down perspectiveis realized by replacing (6) with
119870 (119905) = 119870 (119905 minus Δ119905) + 120594Π (119905 minus Δ119905) minus Π (119905 minus 2Δ119905)Π (119905 minus Δ119905) + Π (119905 minus 2Δ119905) (11)
The top-down origin of this process is made evident by thefact that all individuals in the network are forced to adoptthe same time-dependent imitation strength Furthermorerational choice is made on the basis of the collective payoffΠ(119905) using PDGThe conceptual difference with the bottom-up approach of Section 2 is impressive In fact with (11) all theindividuals of this society must change their social sensitivityat the same time and the information about the increase ordecrease of the global payoff implies that all the individualsare given this information from a central source such as thegovernment suggesting that a form of organization alreadyexists and is not created by the interaction between theindividuals In [3] the assumptionwasmade that a benevolentdictator exists and leads such a process Using Paretorsquos socialtheory we make the assumption that this process implies theleadership of an elite [2]
The second term on the right-hand side of (11) is theratio between two quantities that similarly to the bottom-upmodel for special cases vanishes In these cases as done inSection 2 we set the condition
119870119903 (119905) = 119877119870119903 (119905 minus Δ119905) (12)
with 119877 lt 1 We selected 119877 = 0 When 119870119903(119905) goes to negativevalues we set it equal to zero
For the top-down case discussed in this section thecalculation is done with the parameters119872 = 100 1198920 = 001119879 = 19 and 120594 = 4 with the social benefit imitation strengthand mean field starting from zero 120594 is chosen to be largerthan in the bottom-up case because of the fact that transitionto criticality in the top-down case is much slower (see Figures1 and 3)
Under the leadership of an elite see Figure 3 the controlparameter 119870(119905) shows a behavior totally different from thatof Figure 1 In this section we focus on the behavior of 119870(119905)in the absence of perturbation and discuss the effects ofperturbation in Section 4 With no perturbation there is atransient from 119905 = 0 to 119905 asymp 40000 after which time a sequenceof rises and falls occurs The value of 119870 adjusts according to(11) from small values around 02 to a maximal value of 18which is known to correspond to a supercritical conditionin the case of the conventional DMM When using the finetuning control parameter approach we set119870 = 18 the socialsystem is far from the intelligence condition that accordingto a widely accepted opinion [9 16ndash19] requires criticality
6 Complexity
Time100000800006000040000200000
minus02
00
02
04
06
08
10
12
14
16
18
20
K(t)
Figure 3 Black line the time evolution 119870(119905) of (11) for the sameregular two-dimensional network of Figure 1 red line the timeevolution of119870(119905) of the self-organized system described by the blackunder the influence of the weak noisy perturbation described inSection 4
The mean field 119909(119905) has very fast fluctuations around a meanvalue close to 1 but these fluctuations are Poisson and theconventional DMM system loses its complexity [22]
The falls to the small values of119870 are interpreted as falls ofelites The subcritical condition as well as the supercriticalis characterized by a lack of intelligence We have to remarkalso that values of 119870 significantly smaller than 119870 asymp 15indicate that there are many units with119870119903 = 0 like the singleunit of Figure 1 at a time close to 119905 asymp 2000 In conclusionboth small and maximal values of 119870 are affected by a lack ofconsciousness and the transitions through 119870 asymp 15 are toofast for the social system to benefit from the intelligence of thecritical condition This lack of intelligence is responsible forthe lack of resilience The sojourn times in the supercriticalstate correspond to the time durations of elites We do nothave to confuse the fluctuations of 119870(119905) with those of themean field 119909(119905) that are not shown here The fluctuations of119909(119905) are always Poisson around mean values close to 1 when119870(119905) is close to 18 and around the vanishing mean valuewhen 119870(119905) drops
It is interesting to notice that also the time intervalbetween consecutive falls of elite is a complex dynamicalprocess characterized by an IPL with 120583 asymp 2 in this case asshown by Figure 4 However the system is not resilient Thesojourn in a supercritical state with a 119870 significantly largerthan119870 asymp 15 is characterized by fast Poisson fluctuations andan external perturbation can easily affect the time durationof this regime [40] In fact the big difference betweenPoisson events and crucial events is that the former eventsobey conventional linear response theory and any formsof perturbation can deeply affect their dynamics therebyundermining the social resilience as we show in the nextsection
1 10 100 1000
1
10
100
()
1000
10000
Figure 4 For bottom-up SOTC the waiting-time PDF of the timeintervals between two consecutive crossings of the horizontal linewith 119870 = 07 (black and red line) and 119870 = 17 (blue line)for the top-down SOTC The blue and the black lines describethe unperturbed case and the red line describes the perturbationdescribed in Section 4
4 Perturbing the Self-Organized Society
To substantiate the arguments of the earlier section with theresults of a numerical simulation we devote this section toillustrating some numerical experiments on the effects of aperturbation on the process of societal self-organization
First of all let us define two different sources of pertur-bation the independent and the committed minorities Weassume that a minority of independent individual exists Anindependent individual is a unit that is characterized by119870119903 =0 As a consequence this unit does not adopt (6) and iscompletely insensitive to the connection between individualand societal benefit that yields the emergence of cooperation[3] The perturbing nature of this independent individualis realized by the fact that while the independent keeping119870119903 = 0 is completely independent of the choices made bythe other units her nearest neighbors are influenced by thechoices of the independent through the DMM and throughthe evaluation of the payoff 119875119903(119905) of (6)
In the top-down SOTCmodel the independent influencesthe process through his vanishing contribution to 119870(119905) andthrough his contribution to the global payoff of (11) Theperturbation of independents ismademore devastatingwhenthe independents are allowed to move randomly through thesocial network
The other kind of perturbation produced by committedminorities has already been studied elsewhere [25] Theseare minorities that keep selecting the state119863 The committedminorities are also called zealots and have been the subject ofmany publications see [41] for a wide set of referencesThesepublications emphasize the dramatic consequences that thezealots have on their societies thereby implying that theirmodels of organization are not resilient The experiment onthe perturbation of zealots done herein shows that the top-down SOTC model shares the lack of resilience observed in
Complexity 7
time10000080000600004000020000
00
02
04
06
08
10
12
14
16
18
K(t)
Figure 5119870(119905) as a function of time for the bottom-up SOTC in theno perturbation (black line) and perturbed case (red line)
these earlier studies on the social influences of zealots Thebottom-up SOTC model seems to be the only fully resilientmodel
Let us discuss first the strongest source of perturbationthe randomly moving independents At any time step oneof the 119872 = 100 is randomly selected to play the role ofindependent namely we force her to adopt the value 120585 = 1or the value 120585 = minus1 with equal probability In the case of thebottom-up SOTC this perturbation does not have significanteffect on the time evolution of119870(119905) as shown by Figure 5
In the case of the top-down SOTC the effects of thisperturbation are impressive The black line of Figure 3 showsthe rise and the fall of an elite The weak noisy perturbationmakes 119870(119905) evolve as illustrated by the red line of the samefigure which shows that the sojourn times of unperturbedelites are filled with many falls that are a clear manifestationof the lack of societal resilience
Important information on the lack of resilience of thetop-down SOTC is afforded by Figure 4 showing 120595(119905) fordifferent values of the threshold used to find the statistics ofcrucial events When the threshold is 17 close to the topsupercritical region reached by the system the intermediateasymptotics has a power index 120583 asymp 145 larger than that ofthe bottom-up SOTCThe adoption of the threshold119870 = 07as an effect of the collapse of elites cancels the intermediateasymptotic temporal complexity and favors the birth of aPoisson shoulder The noisy perturbation of independentsmakes this behavior even more pronounced This strongexponential shoulder is a signature of the death of dynamicalcomplexity and of the transition fromnon-Poisson to Poissonbehavior [3]
The perturbing action of independents is weaker if theindependent individuals do not move This response to thisform of perturbation is illustrated in Figure 6 showing thateven in this case the bottom-up SOTC model is more robustthan the top-down
We establish the perturbation of independent individ-uals in a different way We assume that all the units are
02 0800 100604
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
Figure 6 Dependence of mean field (at time 106) on the ratio 120588 ofindependent units (which are fixed on the lattice) for the bottom-up SOTCmodel (black line) top-down SOTCmodel (red line) andordinary DMMwith tuned control parameter119870119888 = 15 (green line)Ensemble average is done over 10 experiments
independent for a fraction 120578 of their time The results aredepicted in Figure 7 It is clear from the figure that thebottom-up SOTCmodel ismore resilient than the two-downeven to this most violent form of disruption
Finally in Figure 8 we show the action of committedminorities We see that only the bottom-up SOTC model isresilient The top-down SOTC model shares the same lack ofresilience shown by ordinary DMM at criticality
5 Conclusions
It is remarkable that according to SOTC the crucial eventsmay be harmful as well as beneficial If the global parameter119870(119905) fluctuating around the long-range correlation generat-ingmean value returns to the vanishing value the temporarycollapse is turned into a societal disaster The collapse into119870 = 0 would correspond to a new initial condition andas shown by Figure 1 119870(119905) would start increasing againgenerating a new organization led by a new elite [2] Howeverthe genuinely bottom-up process leading the time evolutionillustrated by Figure 1 is expected to keep forever the socialsystem in the condition of weak fluctuations around119870 asymp 15In other words there is an impressive difference between thecrucial events hosted by the weak fluctuation around119870 asymp 15and the regressions of 119870(119905) to values of 119870 ≪ 15 generatedby the adoption of a top-down process led by an elite
The main conclusion of this paper concerning resilienceis that criticality is necessary for resilience but it is notsufficient The top-down SOTC model generates criticalitybut it is not resilient Therefore information transport fromone top-down SOTC model system to another top-downSOTC model system is expected to occur by means ofcomplexity matching in spite of the fact that the two systemsare not resilient and the information transport may be easilyquenched by stray perturbing noises
8 Complexity
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10times(106)
0005 0010 0015 00200000
(a)
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
0005 0010 0015 00200000
(b)
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
0005 0010 0015 00200000
(c)
Figure 7 Dependence of mean field (at time 106) on 120578 (the ratio of time in which units behave individually) for bottom-up SOTCmodel (a)top-down SOTC model (b) and ordinary DMMwith tuned parameter119870119888 = 15 (c) Each set of colored dots corresponds to one experimentand the black solid lines are the average of them
An attractive interpretation of the resilient nature of thebottom-up SOTC model is that the ideal condition of fulldemocracy is the most robust form of social organization
In this paper the social payoff is evaluated using thePDG [35] The prisonerrsquos dilemma game is frequently usedin the field of EGT [42 43] EGTs aim at solving thealtruism paradox using the concept of network reciprocity[43] A game is played many times on a network where eachindividual is surrounded by a set of nearest neighbors andadopts the strategy of the most successful nearest neighborSince the clusters of cooperators are richer than the clustersof defectors it is plausible that the most successful nearestneighbor is a cooperator However this attempt at mimickingthe action of a collective intelligence failed because the socialactivity of the units being subcritical disrupts the beneficialeffects of network reciprocity [44 45] We note that SOTCmodeling represents an attempt to amend the field of EGT
by the limitations preventing for instance the concept ofnetwork reciprocity from yielding a satisfactory resolution ofthe altruism paradox
The human inclination to cooperate is the result of bio-logical evolution and of the spontaneous evolution towardscriticality The time appears ripe to unify the models ofbiology and physics made necessary to reach the ambitiousgoal of achieving a rigorous scientific foundation of thisimportant human characteristic [46 47] The spontaneoustransition to criticality of SOTC contributes to bypassing thecurrent limitations of the field of EGT SOTC models asshown in this paper can be adapted to take into account thetop-downprocesses connectedwith the nonresilient action ofelites It is possible to supplement the nonrational decision-making process based on (4) and (5) with self-righteousbiases [1] taking into account the influence of religion or otherpolarizing influences We expect that such generalizations
Complexity 9
minus12
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
06 0804 1000 02
Figure 8 Dependence of mean field (at time 106) on the ratio 120588of fanatics (having fixed position on the lattice) for the bottom-upSOTC model (red line) top-down SOTC model (black line) andordinary DMMwith tuned control parameter119870119888 = 15 (green line)Ensemble average is done over 10 experiments
of SOTC theory will lead to a lack of societal resilienceHowever this is left as a subject for future research
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
Korosh Mahmoodi and Paolo Grigolini warmly thank AROandWelch for support throughGrants noW911NF-15-1-0245and no B-1577 respectively
References
[1] J HaidtThe Righteous Mind Why Good People Are Divided byPolitics and Religion Pantheon Books 2012
[2] V ParetoTheRise and fall of Elites Transaction Publishers NewBrunswick Canada 1991
[3] K Mahmoodi B J West and P Grigolini ldquoSelf-organizingcomplex networks Individual versus global rulesrdquo Frontiers inPhysiology vol 8 article no 478 2017
[4] H E Stanley Introduction to Phase Transition and CriticalPhenomena OxfordUniversity Press NewYork NY USA 1987
[5] E Ising ldquoBeitrag zur theorie des ferromagnetismusrdquo Zeitschriftfur Physik vol 31 no 1 pp 253ndash258 1925
[6] L Onsager ldquoCrystal statistics I A two-dimensional modelwith an order-disorder transitionrdquo Physical Review A AtomicMolecular and Optical Physics vol 65 no 3-4 pp 117ndash149 1944
[7] T Mora and W Bialek ldquoAre biological systems poised atcriticalityrdquo Journal of Statistical Physics vol 144 no 2 pp 268ndash302 2011
[8] D Fraiman P Balenzuela J Foss and D R Chialvo ldquoIsing-like dynamics in large-scale functional brain networksrdquoPhysical
Review E Statistical Nonlinear and Soft Matter Physics vol 79no 6 Article ID 061922 2009
[9] D R Chialvo ldquoEmergent complex neural dynamicsrdquo NaturePhysics vol 6 no 10 pp 744ndash750 2010
[10] M G Kitzbichler M L Smith S R Christensen and EBullmore ldquoBroadband criticality of human brain networksynchronizationrdquo PLoS Computational Biology vol 5 no 3Article ID e1000314 2009
[11] P Bak andM Paczuski ldquoComplexity contingency and critical-ityrdquo Proceedings of the National Academy of Sciences vol 92 pp6689ndash6696 1995
[12] W Truccolo L R Hochberg and J P Donoghue ldquoCollectivedynamics in human and monkey sensorimotor cortex Predict-ing single neuron spikesrdquoNature Neuroscience vol 13 no 1 pp105ndash111 2010
[13] E Schneidman M J Berry II R Segev and W Bialek ldquoWeakpairwise correlations imply strongly correlated network statesin a neural populationrdquoNature vol 440 no 7087 pp 1007ndash10122006
[14] B J West M Turalska and P Grigolini Networks of EchoesImitation Innovation and Invisible Leaders Springer 2014
[15] M Turalska B J West and P Grigolini ldquoTemporal complexityof the order parameter at the phase transitionrdquo Physical ReviewE vol 91 Article ID 012907 2015
[16] P Paradisi P Allegrini A Gemignani M Laurino DMenicucci and A Piarulli ldquoScaling and intermittency of brainevents as a manifestation of consciousnessrdquo AIP ConferenceProceedings vol 1510 pp 151ndash161 2013
[17] P Paradisi and P Allegrini ldquoScaling law of diffusivity generatedby a noisy telegraph signal with fractal intermittencyrdquo ChaosSolitons amp Fractals vol 81 part B pp 451ndash462 2015
[18] E Tagliazucchi D R ChialvoM Siniatchkin et al ldquoLarge-scalesignatures of unconsciousness are consistent with a departurefrom critical dynamicsrdquo Journal of the Royal Society Interfacevol 13 no 114 Article ID 20151027 2016
[19] L Cocchi L L Gollo A Zalesky and M Breakspear ldquoCrit-icality in the brain A synthesis of neurobiology models andcognitionrdquo Progress in Neurobiology 2017
[20] P Allegrini P Paradisi D Menicucci and A GemignanildquoFractal complexity in spontaneous EEGmetastable-state tran-sitions new vistas on integrated neural dynamicsrdquo Frontiers inPhysiology vol 1 2010
[21] P Allegrini P Paradisi D Menicucci and A GemignanildquoRenewal Processes in the Critical Brainrdquo Studies of NonlinearPhenomena in Life Science vol 15 pp 161ndash179 2010
[22] M Lukovic F Vanni A Svenkeson and P Grigolini ldquoTrans-mission of information at criticalityrdquo Physica A StatisticalMechanics and its Applications vol 416 pp 430ndash438 2014
[23] M T Beig A Svenkeson M Bologna B J West and PGrigolini ldquoCritical slowing down in networks generating tem-poral complexityrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 91 no 1 Article ID 012907 2015
[24] J U Mafahim D Lambert M Zare and P Grigolini ldquoCom-plexity matching in neural networksrdquo New Journal of Physics vol 17 Article ID 015003 2015
[25] M Turalska B J West and P Grigolini ldquoRole of committedminorities in times of crisisrdquo Scientific Reports vol 3 article no1371 2013
[26] B Gert Morality a New Justification of Moral Rules OxfordUniversity Press New York NY USA 1988
10 Complexity
[27] H A Simon Models of Bounded Rationality vol 1 and 2 MITPress 1982
[28] J GMarch ldquoRationality foolishness and adaptive intelligencerdquoStrategic Management Journal vol 27 no 3 pp 201ndash214 2006
[29] D KahnemanThinking Fast and Slow Macmillan 2012[30] D G Rand and M A Nowak ldquoHuman cooperationrdquo Trends in
Cognitive Sciences vol 17 no 8 pp 413ndash425 2013[31] D G Rand ldquoCooperation Fast and Slow Meta-Analytic Evi-
dence for a Theory of Social Heuristics and Self-InterestedDeliberationrdquo Psychological Science vol 27 no 9 pp 1192ndash12062016
[32] G Gigerenzer Risk Savvy how to make good decisions PenguinBooks New York NY USA 2014
[33] G Gigerenzer Gut Feelings The Intelligence of the UnconsciousPenguin Books New York NY USA 2007
[34] P Grigolini N Piccinini A Svenkeson P Pramukkul D Lam-bert and B J West ldquoFrom Neural and Social Cooperation totheGlobal Emergence of Cognitionrdquo Frontiers in Bioengineeringand Biotechnology vol 3 article 78 2015
[35] HGintisThebounds of reasonmdashgame theory and the unificationof the behavioral sciences PrincetonUniversity Press PrincetonNJ USA 2014
[36] A Pease K Mahmoodi and B J West ldquoComplexity measuresof musicrdquo Chaos Solitons amp Fractals vol 108 pp 82ndash86 2018
[37] F VanniM Lukovic and PGrigolini ldquoCriticality and transmis-sion of information in a swarm of cooperative unitsrdquo PhysicalReview Letters vol 107 no 7 Article ID 078103 2011
[38] M Turalska E Geneston B J West P Allegrini and PGrigolini ldquoCooperation-induced topological complexity Apromising road to fault tolerance and Hebbian learningrdquoFrontiers in Physiology vol 3 article 52 2012
[39] KMahmoodi P Grigolini and B JWest ldquoOn Social Sensitivityto either Zealot or IndependentMinoritiesrdquo submitted to ChaosSolitons and Fractals
[40] P Allegrini M Bologna P Grigolini and B J WestldquoFluctuation-dissipation theorem for event-dominated pro-cessesrdquo Physical Review Letters vol 99 no 1 Article ID 0106032007
[41] N Khalil M San Miguel and R Toral ldquoZealots in themean-field noisy voter modelrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 97 no 1 Article ID012310 2018
[42] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[43] M A Nowak and R M May ldquoEvolutionary games and spatialchaosrdquo Nature vol 359 no 6398 pp 826ndash829 1992
[44] K Mahmoodi and P Grigolini ldquoEvolutionary game theory andcriticalityrdquo Journal of Physics A Mathematical and Theoreticalvol 50 no 1 Article ID 015101 2017
[45] C Gracia-Lazaro J A Cuesta A Sanchez and Y MorenoldquoHuman behavior in Prisonerrsquos Dilemma experiments sup-presses network reciprocityrdquo Scientific Reports vol 2 p 3252012
[46] W Bialek ldquoPerspectives on theory at the interface of physics andbiologyrdquo Reports on Progress in Physics vol 81 no 1 Article ID012601 2018
[47] M Munoz ldquoColloquium Criticality and dynamical scaling inliving systemsrdquo httpsarxivorgabs171204499
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Complexity 5
cong 13
()
1E minus 6
1E minus 5
1E minus 4
0001
001
01
10 100 10001
Figure 2 For bottom-up SOTC the waiting-time PDF of thetime interval between two consecutive crossings of the averagevalue of the mean value of 119870(119905) which is asymp15 is graphed 120595(120591) isexponentially truncated and has an intermediate asymptotic regimewith an index of 120583 asymp 13 This figure is similar to Figure 4 of [3]
This is evidence that the spontaneous transition to criticalityalso generates the crucial events responsible for informationtransport Earlier work [3] confirms that the crucial eventsfacilitate the transport of information from one complexsocial system to another if the two systems are at criticalityas a result of the spontaneous process of self-organization
In spite of the exponential truncation that may lead to themisleading conclusion that the Poisson-like character of thelong-time regime quenches the manifestations of complexitythe transport of information is determined by the interme-diate asymptotics In the earlier work of [3] we studied theefficiency of transport of information from one bottom-upSOTC to another identical SOTC and we found that themaximal efficiency of the information transport correspondsto about 119872 = 100 which is the condition studied inthis paper The explanation of these interesting effects is thefollowing The intensity of fluctuations generating crucialevents decreases with the size increase according to thefollowing formula (see Eq (14) of [3])
Δ120577 prop 1119872] (10)
with ] = 05 Note that Δ120577 denotes the intensity of thefluctuations of the variables 119870Π and 119909(119905) around theirmean values Therefore the systems with 119872 gt 100 havecrucial fluctuations of smaller intensity thereby explainingthe reduction of the process of information transport Forvalues 119872 lt 100 the role of the exponential truncationbecomes more important and the time extension of thecomplex intermediate asymptotics is reduced and eventuallythe intermediate asymptotics regime vanishes turning thesystem into a Poisson system with no complexity This hasthe effect of significantly reducing the efficiency of the processof information transport As far as the resilience of thebottom-up SOTC is concerned the theory of this paper rests
on the connection between resilience and the efficiency ofinformation transport As a consequence the results on theresilience of the bottom-up SOTC for119872 = 100 automaticallycorrespond on the condition of maximal resilience [39]
3 Top-Down Approach to Self-OrganizedTemporal Criticality
The top-down approach to self-organization is done usingagain (4) and (5) The adoption of the top-down perspectiveis realized by replacing (6) with
119870 (119905) = 119870 (119905 minus Δ119905) + 120594Π (119905 minus Δ119905) minus Π (119905 minus 2Δ119905)Π (119905 minus Δ119905) + Π (119905 minus 2Δ119905) (11)
The top-down origin of this process is made evident by thefact that all individuals in the network are forced to adoptthe same time-dependent imitation strength Furthermorerational choice is made on the basis of the collective payoffΠ(119905) using PDGThe conceptual difference with the bottom-up approach of Section 2 is impressive In fact with (11) all theindividuals of this society must change their social sensitivityat the same time and the information about the increase ordecrease of the global payoff implies that all the individualsare given this information from a central source such as thegovernment suggesting that a form of organization alreadyexists and is not created by the interaction between theindividuals In [3] the assumptionwasmade that a benevolentdictator exists and leads such a process Using Paretorsquos socialtheory we make the assumption that this process implies theleadership of an elite [2]
The second term on the right-hand side of (11) is theratio between two quantities that similarly to the bottom-upmodel for special cases vanishes In these cases as done inSection 2 we set the condition
119870119903 (119905) = 119877119870119903 (119905 minus Δ119905) (12)
with 119877 lt 1 We selected 119877 = 0 When 119870119903(119905) goes to negativevalues we set it equal to zero
For the top-down case discussed in this section thecalculation is done with the parameters119872 = 100 1198920 = 001119879 = 19 and 120594 = 4 with the social benefit imitation strengthand mean field starting from zero 120594 is chosen to be largerthan in the bottom-up case because of the fact that transitionto criticality in the top-down case is much slower (see Figures1 and 3)
Under the leadership of an elite see Figure 3 the controlparameter 119870(119905) shows a behavior totally different from thatof Figure 1 In this section we focus on the behavior of 119870(119905)in the absence of perturbation and discuss the effects ofperturbation in Section 4 With no perturbation there is atransient from 119905 = 0 to 119905 asymp 40000 after which time a sequenceof rises and falls occurs The value of 119870 adjusts according to(11) from small values around 02 to a maximal value of 18which is known to correspond to a supercritical conditionin the case of the conventional DMM When using the finetuning control parameter approach we set119870 = 18 the socialsystem is far from the intelligence condition that accordingto a widely accepted opinion [9 16ndash19] requires criticality
6 Complexity
Time100000800006000040000200000
minus02
00
02
04
06
08
10
12
14
16
18
20
K(t)
Figure 3 Black line the time evolution 119870(119905) of (11) for the sameregular two-dimensional network of Figure 1 red line the timeevolution of119870(119905) of the self-organized system described by the blackunder the influence of the weak noisy perturbation described inSection 4
The mean field 119909(119905) has very fast fluctuations around a meanvalue close to 1 but these fluctuations are Poisson and theconventional DMM system loses its complexity [22]
The falls to the small values of119870 are interpreted as falls ofelites The subcritical condition as well as the supercriticalis characterized by a lack of intelligence We have to remarkalso that values of 119870 significantly smaller than 119870 asymp 15indicate that there are many units with119870119903 = 0 like the singleunit of Figure 1 at a time close to 119905 asymp 2000 In conclusionboth small and maximal values of 119870 are affected by a lack ofconsciousness and the transitions through 119870 asymp 15 are toofast for the social system to benefit from the intelligence of thecritical condition This lack of intelligence is responsible forthe lack of resilience The sojourn times in the supercriticalstate correspond to the time durations of elites We do nothave to confuse the fluctuations of 119870(119905) with those of themean field 119909(119905) that are not shown here The fluctuations of119909(119905) are always Poisson around mean values close to 1 when119870(119905) is close to 18 and around the vanishing mean valuewhen 119870(119905) drops
It is interesting to notice that also the time intervalbetween consecutive falls of elite is a complex dynamicalprocess characterized by an IPL with 120583 asymp 2 in this case asshown by Figure 4 However the system is not resilient Thesojourn in a supercritical state with a 119870 significantly largerthan119870 asymp 15 is characterized by fast Poisson fluctuations andan external perturbation can easily affect the time durationof this regime [40] In fact the big difference betweenPoisson events and crucial events is that the former eventsobey conventional linear response theory and any formsof perturbation can deeply affect their dynamics therebyundermining the social resilience as we show in the nextsection
1 10 100 1000
1
10
100
()
1000
10000
Figure 4 For bottom-up SOTC the waiting-time PDF of the timeintervals between two consecutive crossings of the horizontal linewith 119870 = 07 (black and red line) and 119870 = 17 (blue line)for the top-down SOTC The blue and the black lines describethe unperturbed case and the red line describes the perturbationdescribed in Section 4
4 Perturbing the Self-Organized Society
To substantiate the arguments of the earlier section with theresults of a numerical simulation we devote this section toillustrating some numerical experiments on the effects of aperturbation on the process of societal self-organization
First of all let us define two different sources of pertur-bation the independent and the committed minorities Weassume that a minority of independent individual exists Anindependent individual is a unit that is characterized by119870119903 =0 As a consequence this unit does not adopt (6) and iscompletely insensitive to the connection between individualand societal benefit that yields the emergence of cooperation[3] The perturbing nature of this independent individualis realized by the fact that while the independent keeping119870119903 = 0 is completely independent of the choices made bythe other units her nearest neighbors are influenced by thechoices of the independent through the DMM and throughthe evaluation of the payoff 119875119903(119905) of (6)
In the top-down SOTCmodel the independent influencesthe process through his vanishing contribution to 119870(119905) andthrough his contribution to the global payoff of (11) Theperturbation of independents ismademore devastatingwhenthe independents are allowed to move randomly through thesocial network
The other kind of perturbation produced by committedminorities has already been studied elsewhere [25] Theseare minorities that keep selecting the state119863 The committedminorities are also called zealots and have been the subject ofmany publications see [41] for a wide set of referencesThesepublications emphasize the dramatic consequences that thezealots have on their societies thereby implying that theirmodels of organization are not resilient The experiment onthe perturbation of zealots done herein shows that the top-down SOTC model shares the lack of resilience observed in
Complexity 7
time10000080000600004000020000
00
02
04
06
08
10
12
14
16
18
K(t)
Figure 5119870(119905) as a function of time for the bottom-up SOTC in theno perturbation (black line) and perturbed case (red line)
these earlier studies on the social influences of zealots Thebottom-up SOTC model seems to be the only fully resilientmodel
Let us discuss first the strongest source of perturbationthe randomly moving independents At any time step oneof the 119872 = 100 is randomly selected to play the role ofindependent namely we force her to adopt the value 120585 = 1or the value 120585 = minus1 with equal probability In the case of thebottom-up SOTC this perturbation does not have significanteffect on the time evolution of119870(119905) as shown by Figure 5
In the case of the top-down SOTC the effects of thisperturbation are impressive The black line of Figure 3 showsthe rise and the fall of an elite The weak noisy perturbationmakes 119870(119905) evolve as illustrated by the red line of the samefigure which shows that the sojourn times of unperturbedelites are filled with many falls that are a clear manifestationof the lack of societal resilience
Important information on the lack of resilience of thetop-down SOTC is afforded by Figure 4 showing 120595(119905) fordifferent values of the threshold used to find the statistics ofcrucial events When the threshold is 17 close to the topsupercritical region reached by the system the intermediateasymptotics has a power index 120583 asymp 145 larger than that ofthe bottom-up SOTCThe adoption of the threshold119870 = 07as an effect of the collapse of elites cancels the intermediateasymptotic temporal complexity and favors the birth of aPoisson shoulder The noisy perturbation of independentsmakes this behavior even more pronounced This strongexponential shoulder is a signature of the death of dynamicalcomplexity and of the transition fromnon-Poisson to Poissonbehavior [3]
The perturbing action of independents is weaker if theindependent individuals do not move This response to thisform of perturbation is illustrated in Figure 6 showing thateven in this case the bottom-up SOTC model is more robustthan the top-down
We establish the perturbation of independent individ-uals in a different way We assume that all the units are
02 0800 100604
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
Figure 6 Dependence of mean field (at time 106) on the ratio 120588 ofindependent units (which are fixed on the lattice) for the bottom-up SOTCmodel (black line) top-down SOTCmodel (red line) andordinary DMMwith tuned control parameter119870119888 = 15 (green line)Ensemble average is done over 10 experiments
independent for a fraction 120578 of their time The results aredepicted in Figure 7 It is clear from the figure that thebottom-up SOTCmodel ismore resilient than the two-downeven to this most violent form of disruption
Finally in Figure 8 we show the action of committedminorities We see that only the bottom-up SOTC model isresilient The top-down SOTC model shares the same lack ofresilience shown by ordinary DMM at criticality
5 Conclusions
It is remarkable that according to SOTC the crucial eventsmay be harmful as well as beneficial If the global parameter119870(119905) fluctuating around the long-range correlation generat-ingmean value returns to the vanishing value the temporarycollapse is turned into a societal disaster The collapse into119870 = 0 would correspond to a new initial condition andas shown by Figure 1 119870(119905) would start increasing againgenerating a new organization led by a new elite [2] Howeverthe genuinely bottom-up process leading the time evolutionillustrated by Figure 1 is expected to keep forever the socialsystem in the condition of weak fluctuations around119870 asymp 15In other words there is an impressive difference between thecrucial events hosted by the weak fluctuation around119870 asymp 15and the regressions of 119870(119905) to values of 119870 ≪ 15 generatedby the adoption of a top-down process led by an elite
The main conclusion of this paper concerning resilienceis that criticality is necessary for resilience but it is notsufficient The top-down SOTC model generates criticalitybut it is not resilient Therefore information transport fromone top-down SOTC model system to another top-downSOTC model system is expected to occur by means ofcomplexity matching in spite of the fact that the two systemsare not resilient and the information transport may be easilyquenched by stray perturbing noises
8 Complexity
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10times(106)
0005 0010 0015 00200000
(a)
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
0005 0010 0015 00200000
(b)
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
0005 0010 0015 00200000
(c)
Figure 7 Dependence of mean field (at time 106) on 120578 (the ratio of time in which units behave individually) for bottom-up SOTCmodel (a)top-down SOTC model (b) and ordinary DMMwith tuned parameter119870119888 = 15 (c) Each set of colored dots corresponds to one experimentand the black solid lines are the average of them
An attractive interpretation of the resilient nature of thebottom-up SOTC model is that the ideal condition of fulldemocracy is the most robust form of social organization
In this paper the social payoff is evaluated using thePDG [35] The prisonerrsquos dilemma game is frequently usedin the field of EGT [42 43] EGTs aim at solving thealtruism paradox using the concept of network reciprocity[43] A game is played many times on a network where eachindividual is surrounded by a set of nearest neighbors andadopts the strategy of the most successful nearest neighborSince the clusters of cooperators are richer than the clustersof defectors it is plausible that the most successful nearestneighbor is a cooperator However this attempt at mimickingthe action of a collective intelligence failed because the socialactivity of the units being subcritical disrupts the beneficialeffects of network reciprocity [44 45] We note that SOTCmodeling represents an attempt to amend the field of EGT
by the limitations preventing for instance the concept ofnetwork reciprocity from yielding a satisfactory resolution ofthe altruism paradox
The human inclination to cooperate is the result of bio-logical evolution and of the spontaneous evolution towardscriticality The time appears ripe to unify the models ofbiology and physics made necessary to reach the ambitiousgoal of achieving a rigorous scientific foundation of thisimportant human characteristic [46 47] The spontaneoustransition to criticality of SOTC contributes to bypassing thecurrent limitations of the field of EGT SOTC models asshown in this paper can be adapted to take into account thetop-downprocesses connectedwith the nonresilient action ofelites It is possible to supplement the nonrational decision-making process based on (4) and (5) with self-righteousbiases [1] taking into account the influence of religion or otherpolarizing influences We expect that such generalizations
Complexity 9
minus12
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
06 0804 1000 02
Figure 8 Dependence of mean field (at time 106) on the ratio 120588of fanatics (having fixed position on the lattice) for the bottom-upSOTC model (red line) top-down SOTC model (black line) andordinary DMMwith tuned control parameter119870119888 = 15 (green line)Ensemble average is done over 10 experiments
of SOTC theory will lead to a lack of societal resilienceHowever this is left as a subject for future research
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
Korosh Mahmoodi and Paolo Grigolini warmly thank AROandWelch for support throughGrants noW911NF-15-1-0245and no B-1577 respectively
References
[1] J HaidtThe Righteous Mind Why Good People Are Divided byPolitics and Religion Pantheon Books 2012
[2] V ParetoTheRise and fall of Elites Transaction Publishers NewBrunswick Canada 1991
[3] K Mahmoodi B J West and P Grigolini ldquoSelf-organizingcomplex networks Individual versus global rulesrdquo Frontiers inPhysiology vol 8 article no 478 2017
[4] H E Stanley Introduction to Phase Transition and CriticalPhenomena OxfordUniversity Press NewYork NY USA 1987
[5] E Ising ldquoBeitrag zur theorie des ferromagnetismusrdquo Zeitschriftfur Physik vol 31 no 1 pp 253ndash258 1925
[6] L Onsager ldquoCrystal statistics I A two-dimensional modelwith an order-disorder transitionrdquo Physical Review A AtomicMolecular and Optical Physics vol 65 no 3-4 pp 117ndash149 1944
[7] T Mora and W Bialek ldquoAre biological systems poised atcriticalityrdquo Journal of Statistical Physics vol 144 no 2 pp 268ndash302 2011
[8] D Fraiman P Balenzuela J Foss and D R Chialvo ldquoIsing-like dynamics in large-scale functional brain networksrdquoPhysical
Review E Statistical Nonlinear and Soft Matter Physics vol 79no 6 Article ID 061922 2009
[9] D R Chialvo ldquoEmergent complex neural dynamicsrdquo NaturePhysics vol 6 no 10 pp 744ndash750 2010
[10] M G Kitzbichler M L Smith S R Christensen and EBullmore ldquoBroadband criticality of human brain networksynchronizationrdquo PLoS Computational Biology vol 5 no 3Article ID e1000314 2009
[11] P Bak andM Paczuski ldquoComplexity contingency and critical-ityrdquo Proceedings of the National Academy of Sciences vol 92 pp6689ndash6696 1995
[12] W Truccolo L R Hochberg and J P Donoghue ldquoCollectivedynamics in human and monkey sensorimotor cortex Predict-ing single neuron spikesrdquoNature Neuroscience vol 13 no 1 pp105ndash111 2010
[13] E Schneidman M J Berry II R Segev and W Bialek ldquoWeakpairwise correlations imply strongly correlated network statesin a neural populationrdquoNature vol 440 no 7087 pp 1007ndash10122006
[14] B J West M Turalska and P Grigolini Networks of EchoesImitation Innovation and Invisible Leaders Springer 2014
[15] M Turalska B J West and P Grigolini ldquoTemporal complexityof the order parameter at the phase transitionrdquo Physical ReviewE vol 91 Article ID 012907 2015
[16] P Paradisi P Allegrini A Gemignani M Laurino DMenicucci and A Piarulli ldquoScaling and intermittency of brainevents as a manifestation of consciousnessrdquo AIP ConferenceProceedings vol 1510 pp 151ndash161 2013
[17] P Paradisi and P Allegrini ldquoScaling law of diffusivity generatedby a noisy telegraph signal with fractal intermittencyrdquo ChaosSolitons amp Fractals vol 81 part B pp 451ndash462 2015
[18] E Tagliazucchi D R ChialvoM Siniatchkin et al ldquoLarge-scalesignatures of unconsciousness are consistent with a departurefrom critical dynamicsrdquo Journal of the Royal Society Interfacevol 13 no 114 Article ID 20151027 2016
[19] L Cocchi L L Gollo A Zalesky and M Breakspear ldquoCrit-icality in the brain A synthesis of neurobiology models andcognitionrdquo Progress in Neurobiology 2017
[20] P Allegrini P Paradisi D Menicucci and A GemignanildquoFractal complexity in spontaneous EEGmetastable-state tran-sitions new vistas on integrated neural dynamicsrdquo Frontiers inPhysiology vol 1 2010
[21] P Allegrini P Paradisi D Menicucci and A GemignanildquoRenewal Processes in the Critical Brainrdquo Studies of NonlinearPhenomena in Life Science vol 15 pp 161ndash179 2010
[22] M Lukovic F Vanni A Svenkeson and P Grigolini ldquoTrans-mission of information at criticalityrdquo Physica A StatisticalMechanics and its Applications vol 416 pp 430ndash438 2014
[23] M T Beig A Svenkeson M Bologna B J West and PGrigolini ldquoCritical slowing down in networks generating tem-poral complexityrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 91 no 1 Article ID 012907 2015
[24] J U Mafahim D Lambert M Zare and P Grigolini ldquoCom-plexity matching in neural networksrdquo New Journal of Physics vol 17 Article ID 015003 2015
[25] M Turalska B J West and P Grigolini ldquoRole of committedminorities in times of crisisrdquo Scientific Reports vol 3 article no1371 2013
[26] B Gert Morality a New Justification of Moral Rules OxfordUniversity Press New York NY USA 1988
10 Complexity
[27] H A Simon Models of Bounded Rationality vol 1 and 2 MITPress 1982
[28] J GMarch ldquoRationality foolishness and adaptive intelligencerdquoStrategic Management Journal vol 27 no 3 pp 201ndash214 2006
[29] D KahnemanThinking Fast and Slow Macmillan 2012[30] D G Rand and M A Nowak ldquoHuman cooperationrdquo Trends in
Cognitive Sciences vol 17 no 8 pp 413ndash425 2013[31] D G Rand ldquoCooperation Fast and Slow Meta-Analytic Evi-
dence for a Theory of Social Heuristics and Self-InterestedDeliberationrdquo Psychological Science vol 27 no 9 pp 1192ndash12062016
[32] G Gigerenzer Risk Savvy how to make good decisions PenguinBooks New York NY USA 2014
[33] G Gigerenzer Gut Feelings The Intelligence of the UnconsciousPenguin Books New York NY USA 2007
[34] P Grigolini N Piccinini A Svenkeson P Pramukkul D Lam-bert and B J West ldquoFrom Neural and Social Cooperation totheGlobal Emergence of Cognitionrdquo Frontiers in Bioengineeringand Biotechnology vol 3 article 78 2015
[35] HGintisThebounds of reasonmdashgame theory and the unificationof the behavioral sciences PrincetonUniversity Press PrincetonNJ USA 2014
[36] A Pease K Mahmoodi and B J West ldquoComplexity measuresof musicrdquo Chaos Solitons amp Fractals vol 108 pp 82ndash86 2018
[37] F VanniM Lukovic and PGrigolini ldquoCriticality and transmis-sion of information in a swarm of cooperative unitsrdquo PhysicalReview Letters vol 107 no 7 Article ID 078103 2011
[38] M Turalska E Geneston B J West P Allegrini and PGrigolini ldquoCooperation-induced topological complexity Apromising road to fault tolerance and Hebbian learningrdquoFrontiers in Physiology vol 3 article 52 2012
[39] KMahmoodi P Grigolini and B JWest ldquoOn Social Sensitivityto either Zealot or IndependentMinoritiesrdquo submitted to ChaosSolitons and Fractals
[40] P Allegrini M Bologna P Grigolini and B J WestldquoFluctuation-dissipation theorem for event-dominated pro-cessesrdquo Physical Review Letters vol 99 no 1 Article ID 0106032007
[41] N Khalil M San Miguel and R Toral ldquoZealots in themean-field noisy voter modelrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 97 no 1 Article ID012310 2018
[42] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[43] M A Nowak and R M May ldquoEvolutionary games and spatialchaosrdquo Nature vol 359 no 6398 pp 826ndash829 1992
[44] K Mahmoodi and P Grigolini ldquoEvolutionary game theory andcriticalityrdquo Journal of Physics A Mathematical and Theoreticalvol 50 no 1 Article ID 015101 2017
[45] C Gracia-Lazaro J A Cuesta A Sanchez and Y MorenoldquoHuman behavior in Prisonerrsquos Dilemma experiments sup-presses network reciprocityrdquo Scientific Reports vol 2 p 3252012
[46] W Bialek ldquoPerspectives on theory at the interface of physics andbiologyrdquo Reports on Progress in Physics vol 81 no 1 Article ID012601 2018
[47] M Munoz ldquoColloquium Criticality and dynamical scaling inliving systemsrdquo httpsarxivorgabs171204499
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
6 Complexity
Time100000800006000040000200000
minus02
00
02
04
06
08
10
12
14
16
18
20
K(t)
Figure 3 Black line the time evolution 119870(119905) of (11) for the sameregular two-dimensional network of Figure 1 red line the timeevolution of119870(119905) of the self-organized system described by the blackunder the influence of the weak noisy perturbation described inSection 4
The mean field 119909(119905) has very fast fluctuations around a meanvalue close to 1 but these fluctuations are Poisson and theconventional DMM system loses its complexity [22]
The falls to the small values of119870 are interpreted as falls ofelites The subcritical condition as well as the supercriticalis characterized by a lack of intelligence We have to remarkalso that values of 119870 significantly smaller than 119870 asymp 15indicate that there are many units with119870119903 = 0 like the singleunit of Figure 1 at a time close to 119905 asymp 2000 In conclusionboth small and maximal values of 119870 are affected by a lack ofconsciousness and the transitions through 119870 asymp 15 are toofast for the social system to benefit from the intelligence of thecritical condition This lack of intelligence is responsible forthe lack of resilience The sojourn times in the supercriticalstate correspond to the time durations of elites We do nothave to confuse the fluctuations of 119870(119905) with those of themean field 119909(119905) that are not shown here The fluctuations of119909(119905) are always Poisson around mean values close to 1 when119870(119905) is close to 18 and around the vanishing mean valuewhen 119870(119905) drops
It is interesting to notice that also the time intervalbetween consecutive falls of elite is a complex dynamicalprocess characterized by an IPL with 120583 asymp 2 in this case asshown by Figure 4 However the system is not resilient Thesojourn in a supercritical state with a 119870 significantly largerthan119870 asymp 15 is characterized by fast Poisson fluctuations andan external perturbation can easily affect the time durationof this regime [40] In fact the big difference betweenPoisson events and crucial events is that the former eventsobey conventional linear response theory and any formsof perturbation can deeply affect their dynamics therebyundermining the social resilience as we show in the nextsection
1 10 100 1000
1
10
100
()
1000
10000
Figure 4 For bottom-up SOTC the waiting-time PDF of the timeintervals between two consecutive crossings of the horizontal linewith 119870 = 07 (black and red line) and 119870 = 17 (blue line)for the top-down SOTC The blue and the black lines describethe unperturbed case and the red line describes the perturbationdescribed in Section 4
4 Perturbing the Self-Organized Society
To substantiate the arguments of the earlier section with theresults of a numerical simulation we devote this section toillustrating some numerical experiments on the effects of aperturbation on the process of societal self-organization
First of all let us define two different sources of pertur-bation the independent and the committed minorities Weassume that a minority of independent individual exists Anindependent individual is a unit that is characterized by119870119903 =0 As a consequence this unit does not adopt (6) and iscompletely insensitive to the connection between individualand societal benefit that yields the emergence of cooperation[3] The perturbing nature of this independent individualis realized by the fact that while the independent keeping119870119903 = 0 is completely independent of the choices made bythe other units her nearest neighbors are influenced by thechoices of the independent through the DMM and throughthe evaluation of the payoff 119875119903(119905) of (6)
In the top-down SOTCmodel the independent influencesthe process through his vanishing contribution to 119870(119905) andthrough his contribution to the global payoff of (11) Theperturbation of independents ismademore devastatingwhenthe independents are allowed to move randomly through thesocial network
The other kind of perturbation produced by committedminorities has already been studied elsewhere [25] Theseare minorities that keep selecting the state119863 The committedminorities are also called zealots and have been the subject ofmany publications see [41] for a wide set of referencesThesepublications emphasize the dramatic consequences that thezealots have on their societies thereby implying that theirmodels of organization are not resilient The experiment onthe perturbation of zealots done herein shows that the top-down SOTC model shares the lack of resilience observed in
Complexity 7
time10000080000600004000020000
00
02
04
06
08
10
12
14
16
18
K(t)
Figure 5119870(119905) as a function of time for the bottom-up SOTC in theno perturbation (black line) and perturbed case (red line)
these earlier studies on the social influences of zealots Thebottom-up SOTC model seems to be the only fully resilientmodel
Let us discuss first the strongest source of perturbationthe randomly moving independents At any time step oneof the 119872 = 100 is randomly selected to play the role ofindependent namely we force her to adopt the value 120585 = 1or the value 120585 = minus1 with equal probability In the case of thebottom-up SOTC this perturbation does not have significanteffect on the time evolution of119870(119905) as shown by Figure 5
In the case of the top-down SOTC the effects of thisperturbation are impressive The black line of Figure 3 showsthe rise and the fall of an elite The weak noisy perturbationmakes 119870(119905) evolve as illustrated by the red line of the samefigure which shows that the sojourn times of unperturbedelites are filled with many falls that are a clear manifestationof the lack of societal resilience
Important information on the lack of resilience of thetop-down SOTC is afforded by Figure 4 showing 120595(119905) fordifferent values of the threshold used to find the statistics ofcrucial events When the threshold is 17 close to the topsupercritical region reached by the system the intermediateasymptotics has a power index 120583 asymp 145 larger than that ofthe bottom-up SOTCThe adoption of the threshold119870 = 07as an effect of the collapse of elites cancels the intermediateasymptotic temporal complexity and favors the birth of aPoisson shoulder The noisy perturbation of independentsmakes this behavior even more pronounced This strongexponential shoulder is a signature of the death of dynamicalcomplexity and of the transition fromnon-Poisson to Poissonbehavior [3]
The perturbing action of independents is weaker if theindependent individuals do not move This response to thisform of perturbation is illustrated in Figure 6 showing thateven in this case the bottom-up SOTC model is more robustthan the top-down
We establish the perturbation of independent individ-uals in a different way We assume that all the units are
02 0800 100604
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
Figure 6 Dependence of mean field (at time 106) on the ratio 120588 ofindependent units (which are fixed on the lattice) for the bottom-up SOTCmodel (black line) top-down SOTCmodel (red line) andordinary DMMwith tuned control parameter119870119888 = 15 (green line)Ensemble average is done over 10 experiments
independent for a fraction 120578 of their time The results aredepicted in Figure 7 It is clear from the figure that thebottom-up SOTCmodel ismore resilient than the two-downeven to this most violent form of disruption
Finally in Figure 8 we show the action of committedminorities We see that only the bottom-up SOTC model isresilient The top-down SOTC model shares the same lack ofresilience shown by ordinary DMM at criticality
5 Conclusions
It is remarkable that according to SOTC the crucial eventsmay be harmful as well as beneficial If the global parameter119870(119905) fluctuating around the long-range correlation generat-ingmean value returns to the vanishing value the temporarycollapse is turned into a societal disaster The collapse into119870 = 0 would correspond to a new initial condition andas shown by Figure 1 119870(119905) would start increasing againgenerating a new organization led by a new elite [2] Howeverthe genuinely bottom-up process leading the time evolutionillustrated by Figure 1 is expected to keep forever the socialsystem in the condition of weak fluctuations around119870 asymp 15In other words there is an impressive difference between thecrucial events hosted by the weak fluctuation around119870 asymp 15and the regressions of 119870(119905) to values of 119870 ≪ 15 generatedby the adoption of a top-down process led by an elite
The main conclusion of this paper concerning resilienceis that criticality is necessary for resilience but it is notsufficient The top-down SOTC model generates criticalitybut it is not resilient Therefore information transport fromone top-down SOTC model system to another top-downSOTC model system is expected to occur by means ofcomplexity matching in spite of the fact that the two systemsare not resilient and the information transport may be easilyquenched by stray perturbing noises
8 Complexity
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10times(106)
0005 0010 0015 00200000
(a)
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
0005 0010 0015 00200000
(b)
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
0005 0010 0015 00200000
(c)
Figure 7 Dependence of mean field (at time 106) on 120578 (the ratio of time in which units behave individually) for bottom-up SOTCmodel (a)top-down SOTC model (b) and ordinary DMMwith tuned parameter119870119888 = 15 (c) Each set of colored dots corresponds to one experimentand the black solid lines are the average of them
An attractive interpretation of the resilient nature of thebottom-up SOTC model is that the ideal condition of fulldemocracy is the most robust form of social organization
In this paper the social payoff is evaluated using thePDG [35] The prisonerrsquos dilemma game is frequently usedin the field of EGT [42 43] EGTs aim at solving thealtruism paradox using the concept of network reciprocity[43] A game is played many times on a network where eachindividual is surrounded by a set of nearest neighbors andadopts the strategy of the most successful nearest neighborSince the clusters of cooperators are richer than the clustersof defectors it is plausible that the most successful nearestneighbor is a cooperator However this attempt at mimickingthe action of a collective intelligence failed because the socialactivity of the units being subcritical disrupts the beneficialeffects of network reciprocity [44 45] We note that SOTCmodeling represents an attempt to amend the field of EGT
by the limitations preventing for instance the concept ofnetwork reciprocity from yielding a satisfactory resolution ofthe altruism paradox
The human inclination to cooperate is the result of bio-logical evolution and of the spontaneous evolution towardscriticality The time appears ripe to unify the models ofbiology and physics made necessary to reach the ambitiousgoal of achieving a rigorous scientific foundation of thisimportant human characteristic [46 47] The spontaneoustransition to criticality of SOTC contributes to bypassing thecurrent limitations of the field of EGT SOTC models asshown in this paper can be adapted to take into account thetop-downprocesses connectedwith the nonresilient action ofelites It is possible to supplement the nonrational decision-making process based on (4) and (5) with self-righteousbiases [1] taking into account the influence of religion or otherpolarizing influences We expect that such generalizations
Complexity 9
minus12
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
06 0804 1000 02
Figure 8 Dependence of mean field (at time 106) on the ratio 120588of fanatics (having fixed position on the lattice) for the bottom-upSOTC model (red line) top-down SOTC model (black line) andordinary DMMwith tuned control parameter119870119888 = 15 (green line)Ensemble average is done over 10 experiments
of SOTC theory will lead to a lack of societal resilienceHowever this is left as a subject for future research
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
Korosh Mahmoodi and Paolo Grigolini warmly thank AROandWelch for support throughGrants noW911NF-15-1-0245and no B-1577 respectively
References
[1] J HaidtThe Righteous Mind Why Good People Are Divided byPolitics and Religion Pantheon Books 2012
[2] V ParetoTheRise and fall of Elites Transaction Publishers NewBrunswick Canada 1991
[3] K Mahmoodi B J West and P Grigolini ldquoSelf-organizingcomplex networks Individual versus global rulesrdquo Frontiers inPhysiology vol 8 article no 478 2017
[4] H E Stanley Introduction to Phase Transition and CriticalPhenomena OxfordUniversity Press NewYork NY USA 1987
[5] E Ising ldquoBeitrag zur theorie des ferromagnetismusrdquo Zeitschriftfur Physik vol 31 no 1 pp 253ndash258 1925
[6] L Onsager ldquoCrystal statistics I A two-dimensional modelwith an order-disorder transitionrdquo Physical Review A AtomicMolecular and Optical Physics vol 65 no 3-4 pp 117ndash149 1944
[7] T Mora and W Bialek ldquoAre biological systems poised atcriticalityrdquo Journal of Statistical Physics vol 144 no 2 pp 268ndash302 2011
[8] D Fraiman P Balenzuela J Foss and D R Chialvo ldquoIsing-like dynamics in large-scale functional brain networksrdquoPhysical
Review E Statistical Nonlinear and Soft Matter Physics vol 79no 6 Article ID 061922 2009
[9] D R Chialvo ldquoEmergent complex neural dynamicsrdquo NaturePhysics vol 6 no 10 pp 744ndash750 2010
[10] M G Kitzbichler M L Smith S R Christensen and EBullmore ldquoBroadband criticality of human brain networksynchronizationrdquo PLoS Computational Biology vol 5 no 3Article ID e1000314 2009
[11] P Bak andM Paczuski ldquoComplexity contingency and critical-ityrdquo Proceedings of the National Academy of Sciences vol 92 pp6689ndash6696 1995
[12] W Truccolo L R Hochberg and J P Donoghue ldquoCollectivedynamics in human and monkey sensorimotor cortex Predict-ing single neuron spikesrdquoNature Neuroscience vol 13 no 1 pp105ndash111 2010
[13] E Schneidman M J Berry II R Segev and W Bialek ldquoWeakpairwise correlations imply strongly correlated network statesin a neural populationrdquoNature vol 440 no 7087 pp 1007ndash10122006
[14] B J West M Turalska and P Grigolini Networks of EchoesImitation Innovation and Invisible Leaders Springer 2014
[15] M Turalska B J West and P Grigolini ldquoTemporal complexityof the order parameter at the phase transitionrdquo Physical ReviewE vol 91 Article ID 012907 2015
[16] P Paradisi P Allegrini A Gemignani M Laurino DMenicucci and A Piarulli ldquoScaling and intermittency of brainevents as a manifestation of consciousnessrdquo AIP ConferenceProceedings vol 1510 pp 151ndash161 2013
[17] P Paradisi and P Allegrini ldquoScaling law of diffusivity generatedby a noisy telegraph signal with fractal intermittencyrdquo ChaosSolitons amp Fractals vol 81 part B pp 451ndash462 2015
[18] E Tagliazucchi D R ChialvoM Siniatchkin et al ldquoLarge-scalesignatures of unconsciousness are consistent with a departurefrom critical dynamicsrdquo Journal of the Royal Society Interfacevol 13 no 114 Article ID 20151027 2016
[19] L Cocchi L L Gollo A Zalesky and M Breakspear ldquoCrit-icality in the brain A synthesis of neurobiology models andcognitionrdquo Progress in Neurobiology 2017
[20] P Allegrini P Paradisi D Menicucci and A GemignanildquoFractal complexity in spontaneous EEGmetastable-state tran-sitions new vistas on integrated neural dynamicsrdquo Frontiers inPhysiology vol 1 2010
[21] P Allegrini P Paradisi D Menicucci and A GemignanildquoRenewal Processes in the Critical Brainrdquo Studies of NonlinearPhenomena in Life Science vol 15 pp 161ndash179 2010
[22] M Lukovic F Vanni A Svenkeson and P Grigolini ldquoTrans-mission of information at criticalityrdquo Physica A StatisticalMechanics and its Applications vol 416 pp 430ndash438 2014
[23] M T Beig A Svenkeson M Bologna B J West and PGrigolini ldquoCritical slowing down in networks generating tem-poral complexityrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 91 no 1 Article ID 012907 2015
[24] J U Mafahim D Lambert M Zare and P Grigolini ldquoCom-plexity matching in neural networksrdquo New Journal of Physics vol 17 Article ID 015003 2015
[25] M Turalska B J West and P Grigolini ldquoRole of committedminorities in times of crisisrdquo Scientific Reports vol 3 article no1371 2013
[26] B Gert Morality a New Justification of Moral Rules OxfordUniversity Press New York NY USA 1988
10 Complexity
[27] H A Simon Models of Bounded Rationality vol 1 and 2 MITPress 1982
[28] J GMarch ldquoRationality foolishness and adaptive intelligencerdquoStrategic Management Journal vol 27 no 3 pp 201ndash214 2006
[29] D KahnemanThinking Fast and Slow Macmillan 2012[30] D G Rand and M A Nowak ldquoHuman cooperationrdquo Trends in
Cognitive Sciences vol 17 no 8 pp 413ndash425 2013[31] D G Rand ldquoCooperation Fast and Slow Meta-Analytic Evi-
dence for a Theory of Social Heuristics and Self-InterestedDeliberationrdquo Psychological Science vol 27 no 9 pp 1192ndash12062016
[32] G Gigerenzer Risk Savvy how to make good decisions PenguinBooks New York NY USA 2014
[33] G Gigerenzer Gut Feelings The Intelligence of the UnconsciousPenguin Books New York NY USA 2007
[34] P Grigolini N Piccinini A Svenkeson P Pramukkul D Lam-bert and B J West ldquoFrom Neural and Social Cooperation totheGlobal Emergence of Cognitionrdquo Frontiers in Bioengineeringand Biotechnology vol 3 article 78 2015
[35] HGintisThebounds of reasonmdashgame theory and the unificationof the behavioral sciences PrincetonUniversity Press PrincetonNJ USA 2014
[36] A Pease K Mahmoodi and B J West ldquoComplexity measuresof musicrdquo Chaos Solitons amp Fractals vol 108 pp 82ndash86 2018
[37] F VanniM Lukovic and PGrigolini ldquoCriticality and transmis-sion of information in a swarm of cooperative unitsrdquo PhysicalReview Letters vol 107 no 7 Article ID 078103 2011
[38] M Turalska E Geneston B J West P Allegrini and PGrigolini ldquoCooperation-induced topological complexity Apromising road to fault tolerance and Hebbian learningrdquoFrontiers in Physiology vol 3 article 52 2012
[39] KMahmoodi P Grigolini and B JWest ldquoOn Social Sensitivityto either Zealot or IndependentMinoritiesrdquo submitted to ChaosSolitons and Fractals
[40] P Allegrini M Bologna P Grigolini and B J WestldquoFluctuation-dissipation theorem for event-dominated pro-cessesrdquo Physical Review Letters vol 99 no 1 Article ID 0106032007
[41] N Khalil M San Miguel and R Toral ldquoZealots in themean-field noisy voter modelrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 97 no 1 Article ID012310 2018
[42] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[43] M A Nowak and R M May ldquoEvolutionary games and spatialchaosrdquo Nature vol 359 no 6398 pp 826ndash829 1992
[44] K Mahmoodi and P Grigolini ldquoEvolutionary game theory andcriticalityrdquo Journal of Physics A Mathematical and Theoreticalvol 50 no 1 Article ID 015101 2017
[45] C Gracia-Lazaro J A Cuesta A Sanchez and Y MorenoldquoHuman behavior in Prisonerrsquos Dilemma experiments sup-presses network reciprocityrdquo Scientific Reports vol 2 p 3252012
[46] W Bialek ldquoPerspectives on theory at the interface of physics andbiologyrdquo Reports on Progress in Physics vol 81 no 1 Article ID012601 2018
[47] M Munoz ldquoColloquium Criticality and dynamical scaling inliving systemsrdquo httpsarxivorgabs171204499
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Submit your manuscripts atwwwhindawicom
Complexity 7
time10000080000600004000020000
00
02
04
06
08
10
12
14
16
18
K(t)
Figure 5119870(119905) as a function of time for the bottom-up SOTC in theno perturbation (black line) and perturbed case (red line)
these earlier studies on the social influences of zealots Thebottom-up SOTC model seems to be the only fully resilientmodel
Let us discuss first the strongest source of perturbationthe randomly moving independents At any time step oneof the 119872 = 100 is randomly selected to play the role ofindependent namely we force her to adopt the value 120585 = 1or the value 120585 = minus1 with equal probability In the case of thebottom-up SOTC this perturbation does not have significanteffect on the time evolution of119870(119905) as shown by Figure 5
In the case of the top-down SOTC the effects of thisperturbation are impressive The black line of Figure 3 showsthe rise and the fall of an elite The weak noisy perturbationmakes 119870(119905) evolve as illustrated by the red line of the samefigure which shows that the sojourn times of unperturbedelites are filled with many falls that are a clear manifestationof the lack of societal resilience
Important information on the lack of resilience of thetop-down SOTC is afforded by Figure 4 showing 120595(119905) fordifferent values of the threshold used to find the statistics ofcrucial events When the threshold is 17 close to the topsupercritical region reached by the system the intermediateasymptotics has a power index 120583 asymp 145 larger than that ofthe bottom-up SOTCThe adoption of the threshold119870 = 07as an effect of the collapse of elites cancels the intermediateasymptotic temporal complexity and favors the birth of aPoisson shoulder The noisy perturbation of independentsmakes this behavior even more pronounced This strongexponential shoulder is a signature of the death of dynamicalcomplexity and of the transition fromnon-Poisson to Poissonbehavior [3]
The perturbing action of independents is weaker if theindependent individuals do not move This response to thisform of perturbation is illustrated in Figure 6 showing thateven in this case the bottom-up SOTC model is more robustthan the top-down
We establish the perturbation of independent individ-uals in a different way We assume that all the units are
02 0800 100604
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
Figure 6 Dependence of mean field (at time 106) on the ratio 120588 ofindependent units (which are fixed on the lattice) for the bottom-up SOTCmodel (black line) top-down SOTCmodel (red line) andordinary DMMwith tuned control parameter119870119888 = 15 (green line)Ensemble average is done over 10 experiments
independent for a fraction 120578 of their time The results aredepicted in Figure 7 It is clear from the figure that thebottom-up SOTCmodel ismore resilient than the two-downeven to this most violent form of disruption
Finally in Figure 8 we show the action of committedminorities We see that only the bottom-up SOTC model isresilient The top-down SOTC model shares the same lack ofresilience shown by ordinary DMM at criticality
5 Conclusions
It is remarkable that according to SOTC the crucial eventsmay be harmful as well as beneficial If the global parameter119870(119905) fluctuating around the long-range correlation generat-ingmean value returns to the vanishing value the temporarycollapse is turned into a societal disaster The collapse into119870 = 0 would correspond to a new initial condition andas shown by Figure 1 119870(119905) would start increasing againgenerating a new organization led by a new elite [2] Howeverthe genuinely bottom-up process leading the time evolutionillustrated by Figure 1 is expected to keep forever the socialsystem in the condition of weak fluctuations around119870 asymp 15In other words there is an impressive difference between thecrucial events hosted by the weak fluctuation around119870 asymp 15and the regressions of 119870(119905) to values of 119870 ≪ 15 generatedby the adoption of a top-down process led by an elite
The main conclusion of this paper concerning resilienceis that criticality is necessary for resilience but it is notsufficient The top-down SOTC model generates criticalitybut it is not resilient Therefore information transport fromone top-down SOTC model system to another top-downSOTC model system is expected to occur by means ofcomplexity matching in spite of the fact that the two systemsare not resilient and the information transport may be easilyquenched by stray perturbing noises
8 Complexity
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10times(106)
0005 0010 0015 00200000
(a)
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
0005 0010 0015 00200000
(b)
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
0005 0010 0015 00200000
(c)
Figure 7 Dependence of mean field (at time 106) on 120578 (the ratio of time in which units behave individually) for bottom-up SOTCmodel (a)top-down SOTC model (b) and ordinary DMMwith tuned parameter119870119888 = 15 (c) Each set of colored dots corresponds to one experimentand the black solid lines are the average of them
An attractive interpretation of the resilient nature of thebottom-up SOTC model is that the ideal condition of fulldemocracy is the most robust form of social organization
In this paper the social payoff is evaluated using thePDG [35] The prisonerrsquos dilemma game is frequently usedin the field of EGT [42 43] EGTs aim at solving thealtruism paradox using the concept of network reciprocity[43] A game is played many times on a network where eachindividual is surrounded by a set of nearest neighbors andadopts the strategy of the most successful nearest neighborSince the clusters of cooperators are richer than the clustersof defectors it is plausible that the most successful nearestneighbor is a cooperator However this attempt at mimickingthe action of a collective intelligence failed because the socialactivity of the units being subcritical disrupts the beneficialeffects of network reciprocity [44 45] We note that SOTCmodeling represents an attempt to amend the field of EGT
by the limitations preventing for instance the concept ofnetwork reciprocity from yielding a satisfactory resolution ofthe altruism paradox
The human inclination to cooperate is the result of bio-logical evolution and of the spontaneous evolution towardscriticality The time appears ripe to unify the models ofbiology and physics made necessary to reach the ambitiousgoal of achieving a rigorous scientific foundation of thisimportant human characteristic [46 47] The spontaneoustransition to criticality of SOTC contributes to bypassing thecurrent limitations of the field of EGT SOTC models asshown in this paper can be adapted to take into account thetop-downprocesses connectedwith the nonresilient action ofelites It is possible to supplement the nonrational decision-making process based on (4) and (5) with self-righteousbiases [1] taking into account the influence of religion or otherpolarizing influences We expect that such generalizations
Complexity 9
minus12
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
06 0804 1000 02
Figure 8 Dependence of mean field (at time 106) on the ratio 120588of fanatics (having fixed position on the lattice) for the bottom-upSOTC model (red line) top-down SOTC model (black line) andordinary DMMwith tuned control parameter119870119888 = 15 (green line)Ensemble average is done over 10 experiments
of SOTC theory will lead to a lack of societal resilienceHowever this is left as a subject for future research
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
Korosh Mahmoodi and Paolo Grigolini warmly thank AROandWelch for support throughGrants noW911NF-15-1-0245and no B-1577 respectively
References
[1] J HaidtThe Righteous Mind Why Good People Are Divided byPolitics and Religion Pantheon Books 2012
[2] V ParetoTheRise and fall of Elites Transaction Publishers NewBrunswick Canada 1991
[3] K Mahmoodi B J West and P Grigolini ldquoSelf-organizingcomplex networks Individual versus global rulesrdquo Frontiers inPhysiology vol 8 article no 478 2017
[4] H E Stanley Introduction to Phase Transition and CriticalPhenomena OxfordUniversity Press NewYork NY USA 1987
[5] E Ising ldquoBeitrag zur theorie des ferromagnetismusrdquo Zeitschriftfur Physik vol 31 no 1 pp 253ndash258 1925
[6] L Onsager ldquoCrystal statistics I A two-dimensional modelwith an order-disorder transitionrdquo Physical Review A AtomicMolecular and Optical Physics vol 65 no 3-4 pp 117ndash149 1944
[7] T Mora and W Bialek ldquoAre biological systems poised atcriticalityrdquo Journal of Statistical Physics vol 144 no 2 pp 268ndash302 2011
[8] D Fraiman P Balenzuela J Foss and D R Chialvo ldquoIsing-like dynamics in large-scale functional brain networksrdquoPhysical
Review E Statistical Nonlinear and Soft Matter Physics vol 79no 6 Article ID 061922 2009
[9] D R Chialvo ldquoEmergent complex neural dynamicsrdquo NaturePhysics vol 6 no 10 pp 744ndash750 2010
[10] M G Kitzbichler M L Smith S R Christensen and EBullmore ldquoBroadband criticality of human brain networksynchronizationrdquo PLoS Computational Biology vol 5 no 3Article ID e1000314 2009
[11] P Bak andM Paczuski ldquoComplexity contingency and critical-ityrdquo Proceedings of the National Academy of Sciences vol 92 pp6689ndash6696 1995
[12] W Truccolo L R Hochberg and J P Donoghue ldquoCollectivedynamics in human and monkey sensorimotor cortex Predict-ing single neuron spikesrdquoNature Neuroscience vol 13 no 1 pp105ndash111 2010
[13] E Schneidman M J Berry II R Segev and W Bialek ldquoWeakpairwise correlations imply strongly correlated network statesin a neural populationrdquoNature vol 440 no 7087 pp 1007ndash10122006
[14] B J West M Turalska and P Grigolini Networks of EchoesImitation Innovation and Invisible Leaders Springer 2014
[15] M Turalska B J West and P Grigolini ldquoTemporal complexityof the order parameter at the phase transitionrdquo Physical ReviewE vol 91 Article ID 012907 2015
[16] P Paradisi P Allegrini A Gemignani M Laurino DMenicucci and A Piarulli ldquoScaling and intermittency of brainevents as a manifestation of consciousnessrdquo AIP ConferenceProceedings vol 1510 pp 151ndash161 2013
[17] P Paradisi and P Allegrini ldquoScaling law of diffusivity generatedby a noisy telegraph signal with fractal intermittencyrdquo ChaosSolitons amp Fractals vol 81 part B pp 451ndash462 2015
[18] E Tagliazucchi D R ChialvoM Siniatchkin et al ldquoLarge-scalesignatures of unconsciousness are consistent with a departurefrom critical dynamicsrdquo Journal of the Royal Society Interfacevol 13 no 114 Article ID 20151027 2016
[19] L Cocchi L L Gollo A Zalesky and M Breakspear ldquoCrit-icality in the brain A synthesis of neurobiology models andcognitionrdquo Progress in Neurobiology 2017
[20] P Allegrini P Paradisi D Menicucci and A GemignanildquoFractal complexity in spontaneous EEGmetastable-state tran-sitions new vistas on integrated neural dynamicsrdquo Frontiers inPhysiology vol 1 2010
[21] P Allegrini P Paradisi D Menicucci and A GemignanildquoRenewal Processes in the Critical Brainrdquo Studies of NonlinearPhenomena in Life Science vol 15 pp 161ndash179 2010
[22] M Lukovic F Vanni A Svenkeson and P Grigolini ldquoTrans-mission of information at criticalityrdquo Physica A StatisticalMechanics and its Applications vol 416 pp 430ndash438 2014
[23] M T Beig A Svenkeson M Bologna B J West and PGrigolini ldquoCritical slowing down in networks generating tem-poral complexityrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 91 no 1 Article ID 012907 2015
[24] J U Mafahim D Lambert M Zare and P Grigolini ldquoCom-plexity matching in neural networksrdquo New Journal of Physics vol 17 Article ID 015003 2015
[25] M Turalska B J West and P Grigolini ldquoRole of committedminorities in times of crisisrdquo Scientific Reports vol 3 article no1371 2013
[26] B Gert Morality a New Justification of Moral Rules OxfordUniversity Press New York NY USA 1988
10 Complexity
[27] H A Simon Models of Bounded Rationality vol 1 and 2 MITPress 1982
[28] J GMarch ldquoRationality foolishness and adaptive intelligencerdquoStrategic Management Journal vol 27 no 3 pp 201ndash214 2006
[29] D KahnemanThinking Fast and Slow Macmillan 2012[30] D G Rand and M A Nowak ldquoHuman cooperationrdquo Trends in
Cognitive Sciences vol 17 no 8 pp 413ndash425 2013[31] D G Rand ldquoCooperation Fast and Slow Meta-Analytic Evi-
dence for a Theory of Social Heuristics and Self-InterestedDeliberationrdquo Psychological Science vol 27 no 9 pp 1192ndash12062016
[32] G Gigerenzer Risk Savvy how to make good decisions PenguinBooks New York NY USA 2014
[33] G Gigerenzer Gut Feelings The Intelligence of the UnconsciousPenguin Books New York NY USA 2007
[34] P Grigolini N Piccinini A Svenkeson P Pramukkul D Lam-bert and B J West ldquoFrom Neural and Social Cooperation totheGlobal Emergence of Cognitionrdquo Frontiers in Bioengineeringand Biotechnology vol 3 article 78 2015
[35] HGintisThebounds of reasonmdashgame theory and the unificationof the behavioral sciences PrincetonUniversity Press PrincetonNJ USA 2014
[36] A Pease K Mahmoodi and B J West ldquoComplexity measuresof musicrdquo Chaos Solitons amp Fractals vol 108 pp 82ndash86 2018
[37] F VanniM Lukovic and PGrigolini ldquoCriticality and transmis-sion of information in a swarm of cooperative unitsrdquo PhysicalReview Letters vol 107 no 7 Article ID 078103 2011
[38] M Turalska E Geneston B J West P Allegrini and PGrigolini ldquoCooperation-induced topological complexity Apromising road to fault tolerance and Hebbian learningrdquoFrontiers in Physiology vol 3 article 52 2012
[39] KMahmoodi P Grigolini and B JWest ldquoOn Social Sensitivityto either Zealot or IndependentMinoritiesrdquo submitted to ChaosSolitons and Fractals
[40] P Allegrini M Bologna P Grigolini and B J WestldquoFluctuation-dissipation theorem for event-dominated pro-cessesrdquo Physical Review Letters vol 99 no 1 Article ID 0106032007
[41] N Khalil M San Miguel and R Toral ldquoZealots in themean-field noisy voter modelrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 97 no 1 Article ID012310 2018
[42] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[43] M A Nowak and R M May ldquoEvolutionary games and spatialchaosrdquo Nature vol 359 no 6398 pp 826ndash829 1992
[44] K Mahmoodi and P Grigolini ldquoEvolutionary game theory andcriticalityrdquo Journal of Physics A Mathematical and Theoreticalvol 50 no 1 Article ID 015101 2017
[45] C Gracia-Lazaro J A Cuesta A Sanchez and Y MorenoldquoHuman behavior in Prisonerrsquos Dilemma experiments sup-presses network reciprocityrdquo Scientific Reports vol 2 p 3252012
[46] W Bialek ldquoPerspectives on theory at the interface of physics andbiologyrdquo Reports on Progress in Physics vol 81 no 1 Article ID012601 2018
[47] M Munoz ldquoColloquium Criticality and dynamical scaling inliving systemsrdquo httpsarxivorgabs171204499
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
8 Complexity
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10times(106)
0005 0010 0015 00200000
(a)
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
0005 0010 0015 00200000
(b)
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
0005 0010 0015 00200000
(c)
Figure 7 Dependence of mean field (at time 106) on 120578 (the ratio of time in which units behave individually) for bottom-up SOTCmodel (a)top-down SOTC model (b) and ordinary DMMwith tuned parameter119870119888 = 15 (c) Each set of colored dots corresponds to one experimentand the black solid lines are the average of them
An attractive interpretation of the resilient nature of thebottom-up SOTC model is that the ideal condition of fulldemocracy is the most robust form of social organization
In this paper the social payoff is evaluated using thePDG [35] The prisonerrsquos dilemma game is frequently usedin the field of EGT [42 43] EGTs aim at solving thealtruism paradox using the concept of network reciprocity[43] A game is played many times on a network where eachindividual is surrounded by a set of nearest neighbors andadopts the strategy of the most successful nearest neighborSince the clusters of cooperators are richer than the clustersof defectors it is plausible that the most successful nearestneighbor is a cooperator However this attempt at mimickingthe action of a collective intelligence failed because the socialactivity of the units being subcritical disrupts the beneficialeffects of network reciprocity [44 45] We note that SOTCmodeling represents an attempt to amend the field of EGT
by the limitations preventing for instance the concept ofnetwork reciprocity from yielding a satisfactory resolution ofthe altruism paradox
The human inclination to cooperate is the result of bio-logical evolution and of the spontaneous evolution towardscriticality The time appears ripe to unify the models ofbiology and physics made necessary to reach the ambitiousgoal of achieving a rigorous scientific foundation of thisimportant human characteristic [46 47] The spontaneoustransition to criticality of SOTC contributes to bypassing thecurrent limitations of the field of EGT SOTC models asshown in this paper can be adapted to take into account thetop-downprocesses connectedwith the nonresilient action ofelites It is possible to supplement the nonrational decision-making process based on (4) and (5) with self-righteousbiases [1] taking into account the influence of religion or otherpolarizing influences We expect that such generalizations
Complexity 9
minus12
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
06 0804 1000 02
Figure 8 Dependence of mean field (at time 106) on the ratio 120588of fanatics (having fixed position on the lattice) for the bottom-upSOTC model (red line) top-down SOTC model (black line) andordinary DMMwith tuned control parameter119870119888 = 15 (green line)Ensemble average is done over 10 experiments
of SOTC theory will lead to a lack of societal resilienceHowever this is left as a subject for future research
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
Korosh Mahmoodi and Paolo Grigolini warmly thank AROandWelch for support throughGrants noW911NF-15-1-0245and no B-1577 respectively
References
[1] J HaidtThe Righteous Mind Why Good People Are Divided byPolitics and Religion Pantheon Books 2012
[2] V ParetoTheRise and fall of Elites Transaction Publishers NewBrunswick Canada 1991
[3] K Mahmoodi B J West and P Grigolini ldquoSelf-organizingcomplex networks Individual versus global rulesrdquo Frontiers inPhysiology vol 8 article no 478 2017
[4] H E Stanley Introduction to Phase Transition and CriticalPhenomena OxfordUniversity Press NewYork NY USA 1987
[5] E Ising ldquoBeitrag zur theorie des ferromagnetismusrdquo Zeitschriftfur Physik vol 31 no 1 pp 253ndash258 1925
[6] L Onsager ldquoCrystal statistics I A two-dimensional modelwith an order-disorder transitionrdquo Physical Review A AtomicMolecular and Optical Physics vol 65 no 3-4 pp 117ndash149 1944
[7] T Mora and W Bialek ldquoAre biological systems poised atcriticalityrdquo Journal of Statistical Physics vol 144 no 2 pp 268ndash302 2011
[8] D Fraiman P Balenzuela J Foss and D R Chialvo ldquoIsing-like dynamics in large-scale functional brain networksrdquoPhysical
Review E Statistical Nonlinear and Soft Matter Physics vol 79no 6 Article ID 061922 2009
[9] D R Chialvo ldquoEmergent complex neural dynamicsrdquo NaturePhysics vol 6 no 10 pp 744ndash750 2010
[10] M G Kitzbichler M L Smith S R Christensen and EBullmore ldquoBroadband criticality of human brain networksynchronizationrdquo PLoS Computational Biology vol 5 no 3Article ID e1000314 2009
[11] P Bak andM Paczuski ldquoComplexity contingency and critical-ityrdquo Proceedings of the National Academy of Sciences vol 92 pp6689ndash6696 1995
[12] W Truccolo L R Hochberg and J P Donoghue ldquoCollectivedynamics in human and monkey sensorimotor cortex Predict-ing single neuron spikesrdquoNature Neuroscience vol 13 no 1 pp105ndash111 2010
[13] E Schneidman M J Berry II R Segev and W Bialek ldquoWeakpairwise correlations imply strongly correlated network statesin a neural populationrdquoNature vol 440 no 7087 pp 1007ndash10122006
[14] B J West M Turalska and P Grigolini Networks of EchoesImitation Innovation and Invisible Leaders Springer 2014
[15] M Turalska B J West and P Grigolini ldquoTemporal complexityof the order parameter at the phase transitionrdquo Physical ReviewE vol 91 Article ID 012907 2015
[16] P Paradisi P Allegrini A Gemignani M Laurino DMenicucci and A Piarulli ldquoScaling and intermittency of brainevents as a manifestation of consciousnessrdquo AIP ConferenceProceedings vol 1510 pp 151ndash161 2013
[17] P Paradisi and P Allegrini ldquoScaling law of diffusivity generatedby a noisy telegraph signal with fractal intermittencyrdquo ChaosSolitons amp Fractals vol 81 part B pp 451ndash462 2015
[18] E Tagliazucchi D R ChialvoM Siniatchkin et al ldquoLarge-scalesignatures of unconsciousness are consistent with a departurefrom critical dynamicsrdquo Journal of the Royal Society Interfacevol 13 no 114 Article ID 20151027 2016
[19] L Cocchi L L Gollo A Zalesky and M Breakspear ldquoCrit-icality in the brain A synthesis of neurobiology models andcognitionrdquo Progress in Neurobiology 2017
[20] P Allegrini P Paradisi D Menicucci and A GemignanildquoFractal complexity in spontaneous EEGmetastable-state tran-sitions new vistas on integrated neural dynamicsrdquo Frontiers inPhysiology vol 1 2010
[21] P Allegrini P Paradisi D Menicucci and A GemignanildquoRenewal Processes in the Critical Brainrdquo Studies of NonlinearPhenomena in Life Science vol 15 pp 161ndash179 2010
[22] M Lukovic F Vanni A Svenkeson and P Grigolini ldquoTrans-mission of information at criticalityrdquo Physica A StatisticalMechanics and its Applications vol 416 pp 430ndash438 2014
[23] M T Beig A Svenkeson M Bologna B J West and PGrigolini ldquoCritical slowing down in networks generating tem-poral complexityrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 91 no 1 Article ID 012907 2015
[24] J U Mafahim D Lambert M Zare and P Grigolini ldquoCom-plexity matching in neural networksrdquo New Journal of Physics vol 17 Article ID 015003 2015
[25] M Turalska B J West and P Grigolini ldquoRole of committedminorities in times of crisisrdquo Scientific Reports vol 3 article no1371 2013
[26] B Gert Morality a New Justification of Moral Rules OxfordUniversity Press New York NY USA 1988
10 Complexity
[27] H A Simon Models of Bounded Rationality vol 1 and 2 MITPress 1982
[28] J GMarch ldquoRationality foolishness and adaptive intelligencerdquoStrategic Management Journal vol 27 no 3 pp 201ndash214 2006
[29] D KahnemanThinking Fast and Slow Macmillan 2012[30] D G Rand and M A Nowak ldquoHuman cooperationrdquo Trends in
Cognitive Sciences vol 17 no 8 pp 413ndash425 2013[31] D G Rand ldquoCooperation Fast and Slow Meta-Analytic Evi-
dence for a Theory of Social Heuristics and Self-InterestedDeliberationrdquo Psychological Science vol 27 no 9 pp 1192ndash12062016
[32] G Gigerenzer Risk Savvy how to make good decisions PenguinBooks New York NY USA 2014
[33] G Gigerenzer Gut Feelings The Intelligence of the UnconsciousPenguin Books New York NY USA 2007
[34] P Grigolini N Piccinini A Svenkeson P Pramukkul D Lam-bert and B J West ldquoFrom Neural and Social Cooperation totheGlobal Emergence of Cognitionrdquo Frontiers in Bioengineeringand Biotechnology vol 3 article 78 2015
[35] HGintisThebounds of reasonmdashgame theory and the unificationof the behavioral sciences PrincetonUniversity Press PrincetonNJ USA 2014
[36] A Pease K Mahmoodi and B J West ldquoComplexity measuresof musicrdquo Chaos Solitons amp Fractals vol 108 pp 82ndash86 2018
[37] F VanniM Lukovic and PGrigolini ldquoCriticality and transmis-sion of information in a swarm of cooperative unitsrdquo PhysicalReview Letters vol 107 no 7 Article ID 078103 2011
[38] M Turalska E Geneston B J West P Allegrini and PGrigolini ldquoCooperation-induced topological complexity Apromising road to fault tolerance and Hebbian learningrdquoFrontiers in Physiology vol 3 article 52 2012
[39] KMahmoodi P Grigolini and B JWest ldquoOn Social Sensitivityto either Zealot or IndependentMinoritiesrdquo submitted to ChaosSolitons and Fractals
[40] P Allegrini M Bologna P Grigolini and B J WestldquoFluctuation-dissipation theorem for event-dominated pro-cessesrdquo Physical Review Letters vol 99 no 1 Article ID 0106032007
[41] N Khalil M San Miguel and R Toral ldquoZealots in themean-field noisy voter modelrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 97 no 1 Article ID012310 2018
[42] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[43] M A Nowak and R M May ldquoEvolutionary games and spatialchaosrdquo Nature vol 359 no 6398 pp 826ndash829 1992
[44] K Mahmoodi and P Grigolini ldquoEvolutionary game theory andcriticalityrdquo Journal of Physics A Mathematical and Theoreticalvol 50 no 1 Article ID 015101 2017
[45] C Gracia-Lazaro J A Cuesta A Sanchez and Y MorenoldquoHuman behavior in Prisonerrsquos Dilemma experiments sup-presses network reciprocityrdquo Scientific Reports vol 2 p 3252012
[46] W Bialek ldquoPerspectives on theory at the interface of physics andbiologyrdquo Reports on Progress in Physics vol 81 no 1 Article ID012601 2018
[47] M Munoz ldquoColloquium Criticality and dynamical scaling inliving systemsrdquo httpsarxivorgabs171204499
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Complexity 9
minus12
minus10
minus08
minus06
minus04
minus02
00
02
04
06
08
10
times(106)
06 0804 1000 02
Figure 8 Dependence of mean field (at time 106) on the ratio 120588of fanatics (having fixed position on the lattice) for the bottom-upSOTC model (red line) top-down SOTC model (black line) andordinary DMMwith tuned control parameter119870119888 = 15 (green line)Ensemble average is done over 10 experiments
of SOTC theory will lead to a lack of societal resilienceHowever this is left as a subject for future research
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
Korosh Mahmoodi and Paolo Grigolini warmly thank AROandWelch for support throughGrants noW911NF-15-1-0245and no B-1577 respectively
References
[1] J HaidtThe Righteous Mind Why Good People Are Divided byPolitics and Religion Pantheon Books 2012
[2] V ParetoTheRise and fall of Elites Transaction Publishers NewBrunswick Canada 1991
[3] K Mahmoodi B J West and P Grigolini ldquoSelf-organizingcomplex networks Individual versus global rulesrdquo Frontiers inPhysiology vol 8 article no 478 2017
[4] H E Stanley Introduction to Phase Transition and CriticalPhenomena OxfordUniversity Press NewYork NY USA 1987
[5] E Ising ldquoBeitrag zur theorie des ferromagnetismusrdquo Zeitschriftfur Physik vol 31 no 1 pp 253ndash258 1925
[6] L Onsager ldquoCrystal statistics I A two-dimensional modelwith an order-disorder transitionrdquo Physical Review A AtomicMolecular and Optical Physics vol 65 no 3-4 pp 117ndash149 1944
[7] T Mora and W Bialek ldquoAre biological systems poised atcriticalityrdquo Journal of Statistical Physics vol 144 no 2 pp 268ndash302 2011
[8] D Fraiman P Balenzuela J Foss and D R Chialvo ldquoIsing-like dynamics in large-scale functional brain networksrdquoPhysical
Review E Statistical Nonlinear and Soft Matter Physics vol 79no 6 Article ID 061922 2009
[9] D R Chialvo ldquoEmergent complex neural dynamicsrdquo NaturePhysics vol 6 no 10 pp 744ndash750 2010
[10] M G Kitzbichler M L Smith S R Christensen and EBullmore ldquoBroadband criticality of human brain networksynchronizationrdquo PLoS Computational Biology vol 5 no 3Article ID e1000314 2009
[11] P Bak andM Paczuski ldquoComplexity contingency and critical-ityrdquo Proceedings of the National Academy of Sciences vol 92 pp6689ndash6696 1995
[12] W Truccolo L R Hochberg and J P Donoghue ldquoCollectivedynamics in human and monkey sensorimotor cortex Predict-ing single neuron spikesrdquoNature Neuroscience vol 13 no 1 pp105ndash111 2010
[13] E Schneidman M J Berry II R Segev and W Bialek ldquoWeakpairwise correlations imply strongly correlated network statesin a neural populationrdquoNature vol 440 no 7087 pp 1007ndash10122006
[14] B J West M Turalska and P Grigolini Networks of EchoesImitation Innovation and Invisible Leaders Springer 2014
[15] M Turalska B J West and P Grigolini ldquoTemporal complexityof the order parameter at the phase transitionrdquo Physical ReviewE vol 91 Article ID 012907 2015
[16] P Paradisi P Allegrini A Gemignani M Laurino DMenicucci and A Piarulli ldquoScaling and intermittency of brainevents as a manifestation of consciousnessrdquo AIP ConferenceProceedings vol 1510 pp 151ndash161 2013
[17] P Paradisi and P Allegrini ldquoScaling law of diffusivity generatedby a noisy telegraph signal with fractal intermittencyrdquo ChaosSolitons amp Fractals vol 81 part B pp 451ndash462 2015
[18] E Tagliazucchi D R ChialvoM Siniatchkin et al ldquoLarge-scalesignatures of unconsciousness are consistent with a departurefrom critical dynamicsrdquo Journal of the Royal Society Interfacevol 13 no 114 Article ID 20151027 2016
[19] L Cocchi L L Gollo A Zalesky and M Breakspear ldquoCrit-icality in the brain A synthesis of neurobiology models andcognitionrdquo Progress in Neurobiology 2017
[20] P Allegrini P Paradisi D Menicucci and A GemignanildquoFractal complexity in spontaneous EEGmetastable-state tran-sitions new vistas on integrated neural dynamicsrdquo Frontiers inPhysiology vol 1 2010
[21] P Allegrini P Paradisi D Menicucci and A GemignanildquoRenewal Processes in the Critical Brainrdquo Studies of NonlinearPhenomena in Life Science vol 15 pp 161ndash179 2010
[22] M Lukovic F Vanni A Svenkeson and P Grigolini ldquoTrans-mission of information at criticalityrdquo Physica A StatisticalMechanics and its Applications vol 416 pp 430ndash438 2014
[23] M T Beig A Svenkeson M Bologna B J West and PGrigolini ldquoCritical slowing down in networks generating tem-poral complexityrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 91 no 1 Article ID 012907 2015
[24] J U Mafahim D Lambert M Zare and P Grigolini ldquoCom-plexity matching in neural networksrdquo New Journal of Physics vol 17 Article ID 015003 2015
[25] M Turalska B J West and P Grigolini ldquoRole of committedminorities in times of crisisrdquo Scientific Reports vol 3 article no1371 2013
[26] B Gert Morality a New Justification of Moral Rules OxfordUniversity Press New York NY USA 1988
10 Complexity
[27] H A Simon Models of Bounded Rationality vol 1 and 2 MITPress 1982
[28] J GMarch ldquoRationality foolishness and adaptive intelligencerdquoStrategic Management Journal vol 27 no 3 pp 201ndash214 2006
[29] D KahnemanThinking Fast and Slow Macmillan 2012[30] D G Rand and M A Nowak ldquoHuman cooperationrdquo Trends in
Cognitive Sciences vol 17 no 8 pp 413ndash425 2013[31] D G Rand ldquoCooperation Fast and Slow Meta-Analytic Evi-
dence for a Theory of Social Heuristics and Self-InterestedDeliberationrdquo Psychological Science vol 27 no 9 pp 1192ndash12062016
[32] G Gigerenzer Risk Savvy how to make good decisions PenguinBooks New York NY USA 2014
[33] G Gigerenzer Gut Feelings The Intelligence of the UnconsciousPenguin Books New York NY USA 2007
[34] P Grigolini N Piccinini A Svenkeson P Pramukkul D Lam-bert and B J West ldquoFrom Neural and Social Cooperation totheGlobal Emergence of Cognitionrdquo Frontiers in Bioengineeringand Biotechnology vol 3 article 78 2015
[35] HGintisThebounds of reasonmdashgame theory and the unificationof the behavioral sciences PrincetonUniversity Press PrincetonNJ USA 2014
[36] A Pease K Mahmoodi and B J West ldquoComplexity measuresof musicrdquo Chaos Solitons amp Fractals vol 108 pp 82ndash86 2018
[37] F VanniM Lukovic and PGrigolini ldquoCriticality and transmis-sion of information in a swarm of cooperative unitsrdquo PhysicalReview Letters vol 107 no 7 Article ID 078103 2011
[38] M Turalska E Geneston B J West P Allegrini and PGrigolini ldquoCooperation-induced topological complexity Apromising road to fault tolerance and Hebbian learningrdquoFrontiers in Physiology vol 3 article 52 2012
[39] KMahmoodi P Grigolini and B JWest ldquoOn Social Sensitivityto either Zealot or IndependentMinoritiesrdquo submitted to ChaosSolitons and Fractals
[40] P Allegrini M Bologna P Grigolini and B J WestldquoFluctuation-dissipation theorem for event-dominated pro-cessesrdquo Physical Review Letters vol 99 no 1 Article ID 0106032007
[41] N Khalil M San Miguel and R Toral ldquoZealots in themean-field noisy voter modelrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 97 no 1 Article ID012310 2018
[42] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[43] M A Nowak and R M May ldquoEvolutionary games and spatialchaosrdquo Nature vol 359 no 6398 pp 826ndash829 1992
[44] K Mahmoodi and P Grigolini ldquoEvolutionary game theory andcriticalityrdquo Journal of Physics A Mathematical and Theoreticalvol 50 no 1 Article ID 015101 2017
[45] C Gracia-Lazaro J A Cuesta A Sanchez and Y MorenoldquoHuman behavior in Prisonerrsquos Dilemma experiments sup-presses network reciprocityrdquo Scientific Reports vol 2 p 3252012
[46] W Bialek ldquoPerspectives on theory at the interface of physics andbiologyrdquo Reports on Progress in Physics vol 81 no 1 Article ID012601 2018
[47] M Munoz ldquoColloquium Criticality and dynamical scaling inliving systemsrdquo httpsarxivorgabs171204499
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
10 Complexity
[27] H A Simon Models of Bounded Rationality vol 1 and 2 MITPress 1982
[28] J GMarch ldquoRationality foolishness and adaptive intelligencerdquoStrategic Management Journal vol 27 no 3 pp 201ndash214 2006
[29] D KahnemanThinking Fast and Slow Macmillan 2012[30] D G Rand and M A Nowak ldquoHuman cooperationrdquo Trends in
Cognitive Sciences vol 17 no 8 pp 413ndash425 2013[31] D G Rand ldquoCooperation Fast and Slow Meta-Analytic Evi-
dence for a Theory of Social Heuristics and Self-InterestedDeliberationrdquo Psychological Science vol 27 no 9 pp 1192ndash12062016
[32] G Gigerenzer Risk Savvy how to make good decisions PenguinBooks New York NY USA 2014
[33] G Gigerenzer Gut Feelings The Intelligence of the UnconsciousPenguin Books New York NY USA 2007
[34] P Grigolini N Piccinini A Svenkeson P Pramukkul D Lam-bert and B J West ldquoFrom Neural and Social Cooperation totheGlobal Emergence of Cognitionrdquo Frontiers in Bioengineeringand Biotechnology vol 3 article 78 2015
[35] HGintisThebounds of reasonmdashgame theory and the unificationof the behavioral sciences PrincetonUniversity Press PrincetonNJ USA 2014
[36] A Pease K Mahmoodi and B J West ldquoComplexity measuresof musicrdquo Chaos Solitons amp Fractals vol 108 pp 82ndash86 2018
[37] F VanniM Lukovic and PGrigolini ldquoCriticality and transmis-sion of information in a swarm of cooperative unitsrdquo PhysicalReview Letters vol 107 no 7 Article ID 078103 2011
[38] M Turalska E Geneston B J West P Allegrini and PGrigolini ldquoCooperation-induced topological complexity Apromising road to fault tolerance and Hebbian learningrdquoFrontiers in Physiology vol 3 article 52 2012
[39] KMahmoodi P Grigolini and B JWest ldquoOn Social Sensitivityto either Zealot or IndependentMinoritiesrdquo submitted to ChaosSolitons and Fractals
[40] P Allegrini M Bologna P Grigolini and B J WestldquoFluctuation-dissipation theorem for event-dominated pro-cessesrdquo Physical Review Letters vol 99 no 1 Article ID 0106032007
[41] N Khalil M San Miguel and R Toral ldquoZealots in themean-field noisy voter modelrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 97 no 1 Article ID012310 2018
[42] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[43] M A Nowak and R M May ldquoEvolutionary games and spatialchaosrdquo Nature vol 359 no 6398 pp 826ndash829 1992
[44] K Mahmoodi and P Grigolini ldquoEvolutionary game theory andcriticalityrdquo Journal of Physics A Mathematical and Theoreticalvol 50 no 1 Article ID 015101 2017
[45] C Gracia-Lazaro J A Cuesta A Sanchez and Y MorenoldquoHuman behavior in Prisonerrsquos Dilemma experiments sup-presses network reciprocityrdquo Scientific Reports vol 2 p 3252012
[46] W Bialek ldquoPerspectives on theory at the interface of physics andbiologyrdquo Reports on Progress in Physics vol 81 no 1 Article ID012601 2018
[47] M Munoz ldquoColloquium Criticality and dynamical scaling inliving systemsrdquo httpsarxivorgabs171204499
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Dierential EquationsInternational Journal of
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Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
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