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Transcript of Complex Cooperative Networks from Evolutionary Preferential Attachment Complex Cooperative Networks...
Complex Cooperative Complex Cooperative Networks from Networks from
Evolutionary Preferential Evolutionary Preferential AttachmentAttachment
Jesús Gómez GardeñesJesús Gómez GardeñesUniversitat Rovira i VirgiliUniversitat Rovira i Virgili & & Scuola Superiore di Scuola Superiore di
CataniaCatania & & BIFIBIFI
Net-Works 08, Pamplona, June 9-11 2008 Net-Works 08, Pamplona, June 9-11 2008
Collaborators:- Luis Mario Floría (UZ, BIFI)- Luis Mario Floría (UZ, BIFI)- Yamir Moreno (BIFI)- Yamir Moreno (BIFI)- Anxo S- Anxo Sánchez (UC3M, BIFI)ánchez (UC3M, BIFI)- Julia Poncela (BIFI)Julia Poncela (BIFI)- Manuel CampilloManuel Campillo
10 years of net-working10 years of net-working1998 Structural studies of real complex systems1998 Structural studies of real complex systems
R. Albert, A.L. Barabàsi, Rev. Mod. Phys. (2002)
Statistical Characterization: Small-World, Scale-free, universality??
Models: clustering coefficient, degree-degree correlations,…
Community detection algorithms - Networks coarse graining
Function? Dynamics on networksFunction? Dynamics on networks
S. Boccaletti et al. Phys.Rep. (2006)
Diffusion (Technological/Social): Random Walks, Routing of data, Epidemic Spreading,…
Dynamical Systems (Biological): Synchronization transition & Linear Stability (MSF), Multistability (Dyn. reliability),..
Evolutionary Dynamics (Biology/Social): Survival of cooperation
10 years of net-working10 years of net-working
DYNAMICSDYNAMICSInfected, congested, synchronized,
evolutionary fitness
STRUCTURESTRUCTURESF, ER, clustering, correlations,
communities
OutlineOutline
FEEDBACK STRUCTURE- DYNAMICS:
Complex Networks from Evolutionary Preferential AttachmentComplex Networks from Evolutionary Preferential Attachment- Cooperative behavior
- Topological properties
STRUCTURE AFFECTS DYNAMICS:
Cooperative behavior in Complex NetworksCooperative behavior in Complex Networks- Regular versus homogeneous networks - Microscopic organization of Cooperation
Evolutionary Dynamics on graphsEvolutionary Dynamics on graphs
2-Strategies game:2-Strategies game: Fernado and Lewis have to chose one strategy: A or BFernado and Lewis have to chose one strategy: A or B They chose simultaneously and obtain a payoff given They chose simultaneously and obtain a payoff given
by the matrix:by the matrix:
⎟⎟⎠
⎞⎜⎜⎝
⎛db
ca
B
A
BA Fernando’s strategy
Lew
is’
stra
tegy
Lewis’payoff
Social DilemmasSocial Dilemmas
Population of N agents playing the game all-2-Population of N agents playing the game all-2-all:all:
Fraction Fraction x with strategy A ( with strategy A ( (1-x) with B) with B) Payoffs:Payoffs:
PA =ax+ c(1−x) PB =bx+ d(1−x)Natural selection:Natural selection: Evolution of strategies:Evolution of strategies:x
•=x PA − xPA + (1−x)PB( )⎡⎣ ⎤⎦
Social DilemmasSocial Dilemmas
Social Dilemmas:Social Dilemmas: A means A means cooperationcooperation
B means B means defectiondefection
⎟⎟⎠
⎞⎜⎜⎝
⎛
>>>
010
18
D
C
DC
dcab
Players prefer unilateral
defection to mutual cooperation
Prisoner’s DilemmaPrisoner’s DilemmaPrisoner’s DilemmaPrisoner’s DilemmaHawk-DoveHawk-DoveSnowdriftSnowdrift
Hawk-DoveHawk-DoveSnowdriftSnowdriftStag HuntStag HuntStag HuntStag Hunt
⎟⎟⎠
⎞⎜⎜⎝
⎛
>>>
18
010
D
C
DC
cdba
Players prefer mutual defection to unilateral
cooperation
⎟⎟⎠
⎞⎜⎜⎝
⎛
>>>
110
08
D
C
DC
cdab
Both tensions are incorporated
Social Dilemmas appear as a collection of Social Dilemmas appear as a collection of paradigmatic models accounting for diverse paradigmatic models accounting for diverse situations:situations:
Companies competing for a market Companies competing for a market (Economy)(Economy) Individuals cooperating for a common goal Individuals cooperating for a common goal
(Sociology)(Sociology) Animals hunting preys Animals hunting preys (Biology)(Biology)
Social DilemmasSocial Dilemmas
- In the well-mixed population hypothesis cooperation does not survive when prisoners dilemma is considered.
- The structure of interaction between players is set by a Graph.
- This is a realistic assumption, e.g. Social networks.
How does the structure of the graph affect the survival of cooperation???How does the structure of the graph affect the survival of cooperation???
Social DilemmasSocial Dilemmas
Numerical recipe for Replicator Dynamics on graphs:Numerical recipe for Replicator Dynamics on graphs:
At each time step (generation) each agent, i, plays once with all the agents in its neighborhood, .
The agents accumulate their obtained payoffs, Pi .
Each agent, i, compares its payoff with a single agent, i, picked up at random from its neighborhood.
Strategy update rule: - If Pi> Pj , i keeps its strategy.
- If Pi< Pj , i takes the strategy of j with probability
ij Γ∈
Social Dilemmas on Complex Social Dilemmas on Complex NetworksNetworks
bkk
PPPP
ji
ijijji },max{)(
−=−=Π → β
Homogeneous networks Heterogeneous networks
Social Dilemmas on Complex Social Dilemmas on Complex NetworksNetworks
… we let the system evolve and compute the average fraction of cooperators:
b b b b
c
⎟⎟⎠
⎞⎜⎜⎝
⎛
=>=
0
01
0;1
bD
C
DC
dba
⎟⎟⎠
⎞⎜⎜⎝
⎛
=>=
0
1
0;1
b
c
D
C
DC
dba
F.C. Santos et al., PNAS 103, 3490 (2006).F.C. Santos and J. Pacheco PRL 95, 098104 (2005).
Social Dilemmas on Complex NetworksSocial Dilemmas on Complex Networks
Again, we find surprinsing results when analyzing the impact of SF topology on wide variety of dynamics:
- Epidemic Spreading
- Synchronization
→ Absence of epidemic threshold
→ Enhancement of Synchronizability
Why???
Social dilemmas in synthetic networks show an extremely high promotion of cooperative behavior on Scale-Free networks
compared to that found in homogeneous (all-2-all or ER) graphs
Dynamical States
Star-like graph:
Linear Chain:
2N I.C.1
2N-1
c c c c c c
D D D DD D
2N I.C.
Central node+
P peripheral nodes
P ≥ Int[b]+1
Otherwise →
Prisoner’s Dilemma on CNPrisoner’s Dilemma on CN
⎟⎟⎠
⎞⎜⎜⎝
⎛
=>=
0
01
0;1
bD
C
DC
dba
F C DCs
- Nodes 1 & 2 are linked to ALL elements in F- Node 2 is also connected to ALL the nodes in B- The elements of F are arbitrarily linked between them
- The maximal degree of a node of F with other elements of F is kF
- The size of B is at least Int[kF(b-1)+b+1]
nF2 I.C.
Dynamical States
Prisoner’s Dilemma on CNPrisoner’s Dilemma on CN
⎟⎟⎠
⎞⎜⎜⎝
⎛
=>=
0
01
0;1
bD
C
DC
dba
Pure Cooperators Fluctuating Pure Defectors
Three asymptotic states:
No trajectory inside F evolves to an equilibrium configuration
b
Scale-Free Erdös-Rènyi
b
Fraction of Pure Cooperators
Fraction of Fluctuating <c>Fraction of Pure Defectors
Look at the contribution of each dynamical class to the asymptotic state of the population
Cooperation Evolution
Prisoner’s Dilemma on CNPrisoner’s Dilemma on CN
⎟⎟⎠
⎞⎜⎜⎝
⎛
=>=
0
01
0;1
bD
C
DC
dba
Emergent Clusters
- Clusters are defined by the nodes that share a common strategy (PC, F, PD) and the links among them.
Prisoner’s Dilemma on CNPrisoner’s Dilemma on CN
⎟⎟⎠
⎞⎜⎜⎝
⎛
=>=
0
01
0;1
bD
C
DC
dba
- There may coexist several disjoint clusters simultaneously
Number of simultaneous clusters of pure players?
Number of Pure Cooperators
Clusters
SF networks show a unique PC Cluster
Number of Pure Defectors
Clusters
PD Clusters collapse into a single
one in ER graphs at ρd<1
J. Gomez-Gardeñes et al., Phys. Rev. Lett. 98,108103 (2007).
Different internal organization of C and D cores
Two different paths from cooperation to defection
Emergent Clusters
Prisoner’s Dilemma on CNPrisoner’s Dilemma on CN
⎟⎟⎠
⎞⎜⎜⎝
⎛
=>=
0
01
0;1
bD
C
DC
dba
How are PC cluster exposed to How are PC cluster exposed to Fluctuating players?Fluctuating players?
Clusters’ Topology
Prisoner’s Dilemma on CNPrisoner’s Dilemma on CN
⎟⎟⎠
⎞⎜⎜⎝
⎛
=>=
0
01
0;1
bD
C
DC
dba
J. Gomez-Gardeñes et al., Phys. Rev. Lett. 98,108103 (2007).
Measure of the effective “surface” of PC clusters
Pure Cooperators are more frequently connected (exposed) to Fluctuating nodes
in ER graphs
log(
k)
b
FluctuatingsPure Cooperators
b
PCP / F
J. Gomez-Gardeñes et al., J. Theor. Biol. (2008).
All the nodes of an ER graph are topologically equivalent but…
where are PC, F, PD located in a SF heterogeneous network?
Highly connected nodes always plays as PC&
Fluctuating strategies spreads from low connectivity nodes
Clusters’ Topology
Prisoner’s Dilemma on CNPrisoner’s Dilemma on CN
⎟⎟⎠
⎞⎜⎜⎝
⎛
=>=
0
01
0;1
bD
C
DC
dba
Dynamical characterization of FluctuationsC
Dtime
Distribution of cooperationintervals’ times,
Distribution of total cooperationtimes of a fluctuating node, TC
cτ
J. Gomez-Gardeñes et al., J. Theor. Biol. (2008).
Prisoner’s Dilemma on CNPrisoner’s Dilemma on CN
SF (b=3.0):SF (b=2.1):
Clusters’ Topology
Prisoner’s Dilemma on CNPrisoner’s Dilemma on CN
PC PDPC
s F
ConclusionsConclusionsKnown Facts: SF topology promotes cooperation
Quantitative differences are observed <c>(b)
Three classes of agents:
Pure Cooperators Pure Defectors Fluctuating playersFluctuating players
They act as borders
between PC & PD They may occupy a macroscopic
part of the network
Once defined and identified, one can unveil the internal organization of the three classes of agents
Qualitative differences are observed PC(b), F(b), PD(b)
- NCC
Indicators: & - NDC
- Hubs roleConfirmed by: &
- Surface of PC clusters
• SF enhance the survival of cooperation
• The differences with homogeneous structure rely on the structural organization of strategies.
However:However: If SF networks are best suited to support cooperation, where did they come from?
What are the mechanisms that shape the system structure?
What have we learned?What have we learned?
One cannot think of an optimized design…
Social networks are the result of a collection of many local decisions based on local
interactions.
Function affects structure and the other way around!!Function affects structure and the other way around!!
Evolutionary Preferential AttachmentEvolutionary Preferential Attachment
• The network grows by adding new nodes every τT time steps:
Two channels of evolution: Network Growth and Evolutionary Dynamics
We explore two cases: τT =τD and τD =10τT
The new node is added as C or D with equal probability.
• A Prisoners Dilemma round robin is played within the nodes of the every τD time steps
Evolutionary Preferential AttachmentEvolutionary Preferential Attachment
• Coupling Dynamics And Growth:
The new node attaches to nodes following a preferential attachment to nodes with high evolutionary fitness, fi(t):
Two channels of evolution: Network Growth and Evolutionary Dynamics
Πi (t) =1 − ε + ε fi (t)
1 − ε + ε f j (t)j =1
N (t )
∑0: Weak selection limit.1: Strong selection limit.
Parameters: b, , τT/ τD Outputs: <c> + Topology
Πi (t) =1 − ε + ε fi (t)
1 − ε + ε fi (t)j =1
N (t )
∑
Degree DistributionDegree Distribution
b =1.5τD = 10τ T
⎟⎟⎠
⎞⎜⎜⎝
⎛
=>=
0
01
0;1
bD
C
DC
dba
τD = τ T τD = 10τ T
⎟⎟⎠
⎞⎜⎜⎝
⎛
=>=
0
01
0;1
bD
C
DC
dba
Degree DistributionDegree Distribution
→ 1τD = τ T τD = 10τ T
⎟⎟⎠
⎞⎜⎜⎝
⎛
=>=
0
01
0;1
bD
C
DC
dba
J. Poncela et al., PLoS ONE., (2008).
→ 1 τD = τ T τD = 10τ T⎟⎟⎠
⎞⎜⎜⎝
⎛
=>=
0
01
0;1
bD
C
DC
dba
• For a fixed b, strong selection (1) yields both the highest level of cooperation and scale-free behavior, all in one!
• Correlations are present, both in degree-degree and clustering-degree. A typical fingerprint in real networks.
Summary:
We have not imposed any maximization of cooperation level. The network is shaped by its dynamics.
Moreover, strong selection would seem to favor defective systems within the context of the PD game. It is just the opposite!!!
What about the organization of cooperation??
• Real hubs are defectors • Middle class are cooperators
J. Poncela et al., PLoS ONE., (2008).
Probability that a node of degree k plays as cooperator
This picture is radically different than that found for static SF networks
What would be the level of cooperation if the system stop growing?
J. Poncela et al., PLoS ONE., (2008).
Growing to Static Network
Cooperation is actually Cooperation is actually enhanced!!!enhanced!!!
Summary
• Structure is shaped by DynamicsStructure is shaped by Dynamics: It is possible to build up complex networks using a dynamical feedback mechanism that shapes the system’s structure.
The model provides an evolutionary explanation of the features of real networks: Scale-free and clustering
• Dynamics is affectedDynamics is affected: The model points out the many differences in the microscopic organization of strategist compared to the case in which the game evolves on static networks. Dynamics in a Growing is qualitatively different!
http://neptuno.unizar.es/jgg/
Related Publications:
• J. Gómez-Gardeñes et al., Phys. Rev. Lett. 98, 108103 (2007)• J. Poncela et al., New J. Phys. 9, 184 (2007).• J. Gómez-Gardeñes et al., J. Theor. Biol., in press (2008).• J. Poncela et al., PLoS ONE, in press (2008).
G. Szabó and G. Fáth, Physics Reports 446, 97 (2007).