Diffusion processes on complex systems

Janusz Szwabiński

Lecture #10: Modelling social influence

Overview

● Historical background

● Social influence

● Opinion dynamics

● Models’ overview

● Our model(s) – assumptions and results

Note on social physics● Adolphe Quetelet, “On Man and the

Development of his Faculties, or Essays on Social Physics”, 1835

● goal - to understand the statistical laws underlying such phenomena as crime rates, marriage rates or suicide rates

● concept of the average man - characterized by the mean values of measured variables that follow a normal distribution

From social physics to sociology● Auguste Comte (1842) defined social

physics as the study of the laws of society:

– social structure

– social dynamics

● he discovered that Quetelet coined the term ‘social physics’ prior to him

● he invented the term sociology

From micro to macro

Wilhelm Lenz(1888-1957)

Ernst Ising(1900-1998)

Thomas Schelling(1921-2016)

Ising model● invented by Lenz (1920), solved by Ising (1925)● model of ferromagnetism

Agents

Interaction topology(e.g. square lattice in 2D)

Local interaction

External field

Dynamic rule spin fip, accepted →with probability:

Ising model● no magnetization in 1D● non-vanishing magnetization below a critical temperature

in 2D → phase transition

Schelling model of residential segregation

● Dynamic Models of Segregation, J. Math Sociology 1 (1971) 143)

Stay if a fractionof neighbors

is „kin”

Move to other location

otherwise

Thomas Schelling(1921-2016)

Schelling model of residential segregation

Hapiness threshold 30% Hapiness threshold 80%

Schelling model of residential segregation● one on the first agent-based models in social sciences

● mild preferences total segregation→

● a milestone in the study of emergent global phenomena based on local social interactions

● „(...) long time ago I discovered, somebody told me that, there were some physical models, I think something in crystal formation. Somebody was referring to ISING model, which was a well-known model of, I think, crystal formation”

Social psychology vs sociology

Social Psychology

Fundamental unit: a person

(micro scale)

Sociology

Fundamental unit:a social group (macro scale)

Social infuence

● Types of social infuence (Deutsch & Gerard 1955)

– normative – to conform with the positive expectations of others (need to be liked)

– informational – to accept information obtained from others as evidence of reality (need to be right)

Efforts by one or more individualsto change attitudes, beliefs, perceptions or behaviors of oneor more others

Diamond model● Nail, MacDonald & Levy, 2000

● possible responses to social infuence

Conformity

● the main mechanism of collective actions

● informational – when in doubt, imitate

● normative – when in Rome, do as the Romans do (aversion to standing out )

Matching attitudes, beliefsand behaviors to groupnorms

Congruence

● type of conformity

● occurs if the target of infuence is already aligned with the group

● requires no actions from the target

Tendency to stay aligned with the group of infuence

Independence

● motives:

– desire to be correct (Asch 1956)

– desire to accomplish group goals (Turner 1991)

– avoiding groupthink (Janis 1982)

– maintaining one’s self-concept or social identity

Unwillingness to yield to group pressure

Anticonformity

● motives include target’s desire:

– to provoke group confict (Hollander 1975)

– to distance the self from others (Hogg & Turner 1987)

– to project behavioral autonomy (Brehm & Brehm 1981)

– to project one’s uniqueness (Snyder & Fromkin 1980)

– to avoid groupthink (Janis 1982)

Conscious and deliberate challenging the position or actions of the group

Anticonformity vs conformity

– anticonformity is opposed to conformity...

– … but an anticonformist is still under the influence of a group → he is not independent

– conformity is relative, i.e. conformity to one group may be anticonformity to other (e.g. peer group vs parents)

– specific form of conformity

The power of social validation

321 X

Which line on the left matches the length of the line on the right?

The power of social validation

321 X

What would you answer in a group of 6 people with all others choosing the line 3?

The power of social validation

321 X

● majority of people (95%) gives right answer if asked individually

● 35% gives the wrong answer to go with the group

The power of social validation

The size of the group matters

Unanimity is the key!

● Participants were far more independent when they were opposed by a seven person majority and had a partner than when they were opposed by a three-person majority and did not have a partner

Conformity and independence

Opinion

● verbalized attitude (Trommsdorff, 1998)

● cannot be proven true or false

● measured in surveys

● discrete opinions:

– Ising spins

– Potts variables

● continuous opinions:

– not in surveys

– how to measure them?

Opinion dynamics

● opinions are the drivers of human behavior, and play a crucial role in many global challenges that societies are facing (e.g. pandemics, financial crises, migration patterns, ecological behavior)

● opinion formation is a process of collective intelligence evolving from the integrative tendencies of social infuence with the disintegrative effects of individualization

● often studied with computer simulations (agent-based models)

● field of complex system science, receiving a lot of attention from social scientists, physicists, mathematicians and computer scientists

Related phenomena

● diffusion of innovation - explaining how, why, and at what rate new ideas and technologies spread (Rogers, 1962)

● epidemic spreading – predicting various characteristics of a pathogen spreading in a community

● computer virus spreading – understanding virus propagation properties

Components of opinion spreading models (agent-based ones)

● agents:

– basic ontological unit of a model

– different „grain” sizes (individuals, social groupings, institutions)

– characterized by attributes describing their state (both internal and external)

● social network:

– interaction patterns between agents

● update rules:

– behavior of agents that may lead to a change in their state

– schedule of their actions

Models’ overview – Deffuant model

● N agents in the system

● each agent i is initially given an opinion xi , randomly chosen in the interval [0, 1]

● the dynamics is based on random binary encounters, i.e., at each time step, a randomly selected agent discusses with one of its neighbors, also chosen at random

● Deffuant dynamics is summarized as follows:

– if the difference of the opinions xi(t) and xj (t) exceeds the threshold ε, nothing happen

– if instead |xi(t) and xj (t)| < ε, then:

Models’ overview – Deffuant model

● a compromise strategy: after a constructive debate, the positions of agents get closer to each other

● for any value of ε and μ the average opinion of the agents’ pair is the same before and after the interaction, so the global average opinion (1/2) of the population is an invariant of Deffuant dynamics

● on complete graphs, regular lattices, random graphs and scale-free networks, for ε > εc = 1/2, all agents share the same opinion ½ (complete consensus)

● if ε is small, clusters of opinions emerge

Models’ overview – Hegselmann-Krause model● similar to Deffuant model● opinions take real values in an interval, say [0, 1]● an agent i with opinion xi interacts with neighboring

agents whose opinions lie in the range [xi − ε, xi + ε], where ε is the uncertainty

● the difference is given by the update rule:– agent i does not interact with one of its compatible neighbors,

like in Deffuant, but with all its compatible neighbors at once.

Models’ overview – Hegselmann-Krause model

● Deffuant’s prescription is suitable to describe the opinion dynamics of large populations, where people meet in small groups, like pairs

● HK rule is intended to describe formal meetings, where there is an effective interaction involving many people at the same time

● similar patterns in the time evolution

Models’ overview – Hegselmann-Krause model

Models’ overview – Sznajd model

● basic principle - convincing somebody is easier for two or more people than for a single individual

● in its original version agents occupy the sites of a linear chain and have binary opinions, denoted by Ising spin variables

● a pair of neighboring agents i and i+1 determines the opinions of their two nearest neighbors i−1 and i+2 according to these rules:

Models’ overview – Sznajd model

● in the most common version:– a pair of neighboring agents with the same

opinion convince all their neighbors, while they have no infuence if they disagree

Models’ overview – Sznajd model

● applications:– in politics, it has been used to describe voting behavior in

elections

– it was applied to study the interaction of economic and personal attitudes of individuals, which evolve according to different rules, but in a coupled manner

– competition of different products in an open market

– effects of aging, diffusion and a multi-layered society

– spread of opinions among a group of traders

Models’ overview – Voter model

● named in this way for the very natural interpretation of its rules in terms of opinion dynamics

● thoroughly investigated also in fields quite far from social dynamics, like probability theory and population genetics

● Voter dynamics was first considered in by Clifford and Sudbury (1973) as a model for the competition of species

● very popular because, despite being a rather crude description of any real process, it is one of the very few non-equilibrium stochastic processes that can be solved exactly in any dimension

Models’ overview – Voter model

● definition is extremely simple: – each agent is endowed with a binary variable s = ±1– at each time step, an agent i is selected along with one of its neighbors j and si =

sj , i.e., the agent takes the opinion of the neighbor

● update rule implies that agents imitate their neighbors● they feel the pressure of the majority of their peers only in an

average sense● bulk noise is absent in the model, so the states with all sites

equal (consensus) are absorbing● starting from a disordered initial condition, voter dynamics

tends to increase the order of the system, as in usual coarsening processes

● the question is whether full consensus is reached in a system of infinite size

Models’ overview – Voter model

Models’ overview – Galam majority rule model● N agents endowed with binary opinions● a fraction p+ of agents has opinion +1, while a fraction p− = 1 − p+ opinion −1● suppose that all agents can communicate with each other (complete graph)● at each iteration, a group of r agents is selected at random (discussion

group): – as a consequence of the interaction, all agents take the majority opinion inside the group

● group size r is not fixed, but is selected at each step from a given distribution

● if r is odd, there is always a majority in favor of either opinion● if r is even, there is the possibility of a tie, i.e., that either opinion is

supported by exactly r/2 agents– one introduces a bias in favor of one of the opinions, say +1

● inspired by the principle of social inertia, for which people are reluctant to accept a reform if there is no clear majority in its favor

Models’ overview – Galam majority rule model● if the initial fraction of agents with the opinion +1,

p0+, is greater than a critical value, all agents will

have opinion +1 in the long run (-1 otherwise)● the time to reach consensus (in number of

updates per spin) scales like logN● if the group sizes are odd, pc(r) = 1/2, due to the

symmetry of the two opinions● if there are groups with r even, pc < 1/2, i.e., the

favored opinion will eventually be the dominant one, even if it is initially shared by a minority of agents

Models’ overview – social impact model● psychological theory of social impact

(Latane) describes how individuals feel the presence of their peers and how they in turn influence other individuals

● the impact of a social group on a subject depends on:– the number of the individuals in the group– their convincing power– the distance from the subject, where the distance may

refer both to spatial proximity or to the closeness in an abstract space of personal relationships

Models’ overview – social impact model● population of N individuals● each individual i is characterized by:

– an opinion σi = ±1

– two real-valued parameters (random numbers), that estimate the strength of its action on the others: persuasiveness pi and supportiveness si

● the distance of a pair of agents i and j is dij

● the total impact Ii that an individual i experiences from his/her social environment is

Models’ overview – social impact model

● opinion dynamics expressed by the rule:

● hi - a random field representing all sources other than social impact that may affect the opinion (e.g. mass media)

● according to the update rule, a spin fips if the pressure in favor of the opinion change overcomes the pressure to keep the current opinion (Ii > 0 for vanishing hi)

● for a system of fully connected agents the model presents infinitely many stationary states

● the order parameter of the dynamics is a complex function of one variable, like in spin glasses

● in general, the dynamics leads to the dominance of one opinion on the other, but not to complete consensus

● if the initial magnetization is about zero, the system converges to configurations characterized by a large majority of spins in the same opinion state, and by stable domains of spins in the minority opinion state

Models’ overview - Q-voter model

● Castellano, Muñoz and Pastor-Satorras 2009● generalization of the voter model (Clifford and Sudbury 1973)● model of binary opinions● conformity as the only response to social infuence● q neighbors are consulted for a voter (i.e. target of infuence) to

change opinion– if the q neighbors agree, the voter takes their opinion– if they do not have an unanimous opinion, a voter can still fip

its state with probability ε

Our model(s) – basic assumptions

● very often opinions may be interpreted as simple "yes"/"no", "in favour of"/"against" or "adopted"/"not adopted" answers (Watts & Dodds 2007) → model of binary opinions

● conformity is the main driving force → q-voter model

● social interaction networks with different topologies

● additional responses to social infuence: independence and/or anticonformity

What helps an innovation to diffuse?● q-voter model with independence and an external field

(advertisement, mass media)

K. Sznajd-Weron, J. Szwabiński, R. Weron and T. Weron, Rewiring the network. What helps an innovation to diffuse?, J. Stat. Mech. 2014 (3) 3007

What helps an innovation to diffuse?

K. Sznajd-Weron, J. Szwabiński, R. Weron and T. Weron, Rewiring the network. What helps an innovation to diffuse?, J. Stat. Mech. 2014 (3) 3007

Contribution to person-situation debate

K. Sznajd-Weron, J. Szwabiński, R. Weron (2014), Is the Person-Situation Debate Important for Agent-Based Modeling and Vice-Versa? PLoS ONE 9(11): e112203

Do the details of the model matter?

A. Jędrzejewski, K. Sznajd-Weron, J. Szwabiński, Mapping the q-voter model: From a single chain to complex networks, Physica A 446 (2016) 110-119

Do the details of the model matter?

Conclusions

● q-voter model constitutes a nice framework for extensions

● diffusion of innovations rate increases with decreasing average path length of the social network

● the shorter the path length the smaller the impact of the details of the model on the opinion dynamics

● phase transitions complex systems→

● agent-based models may help to understand social systems