The Axelrod model for cultural dissemination: role of noise, peer pressure and homophily Claudio Castellano CNR-INFM Statistical Mechanics and Complexity and Universita’ di Roma “La Sapienza” Outline: Phenomenology of the Axelrod model Variations of the original model The relevance of noise In search of robustness

Opinion dynamics How do opinions spread? What affects the way consensus is reached? What mechanisms are at work? Interactions between individuals tend to favor homogeneization Other mechanisms favor fragmentation Do regularities exist? Universal laws? What is the effect of the structure of social networks and of mass-media?

A model for cultural dissemination “If people tend to become more alike in their beliefs, attitudes and behaviors when they interact, why do not all differences disappear?” [R. Axelrod, J. of Conflict Resolution, 41, 203 (1997)] Culture is more complicated than opinions: several coupled features Two basic ingredients Social influence: Interactions make individuals more similar. Homophily: Likeliness of interaction grows with similarity.

Definition of Axelrod model Each individual is characterized by F integer variables, si,f , assuming q possible traits 1 <= si,f <= q Dynamics: Pick at random a site (i), a neighbor (j) Compute overlap = # of common features/F With prob. proportional to overlap: pick f’, such that si,f’ <> sj,f’ and set si,f’ = sj,f’ Iterate

Fragmentation-consensus transition The evolution depends on the number q of traits in the initial state Low initial variability High initial variability

Phenomenology of Axelrod model: statics C. Castellano et al., Phys. Rev. Lett, 85, 3536 (2000) Control parameter: q = number of possible traits for each feature Order parameter: smax/N fraction of system occupied by the largest domain smax/N @ N -1  disorder smax/N = O(1)  order Low q: consensus High q: fragmentation (polarization) Consensus-fragmentation transition depending on the initial condition

Phenomenology of Axelrod model: dynamics Observable: density of active links n(t) Active links: pairs of neighbors that are neither completely different (overlap=0) nor equal (overlap=1) Close to the transition (q<q_c) the density of active links almost goes to zero and then grows up to very large values before going to zero. Nontrivial interplay of different temporal scales

Mean-field approach Dynamics is mapped into a dynamics for links C. Castellano et al., Phys. Rev. Lett, 85, 3536 (2000) F. Vazquez and S. Redner, EPL, 78, 18002 (2007) Dynamics is mapped into a dynamics for links A transition between a state with active links and a state without them is found. The divergence of the temporal scales is computed: ~ |q-qc|-1/2 both above and below qc

On complex topologies K. Klemm et al., Phys. Rev. E, 67, 026120 (2003) Small world (WS) networks: interpolations between regular lattices and random networks. Randomness in topology increases order Fragmented phase survives for any p

On complex topologies Scale-free networks: P(k)~k-g Scale-free topology increases order Fragmented phase disappears for infinite systems

The effect of mass media Y. Shibanai et al., J. Conflict Resol., 45, 80 (2001) J. C. Gonzalez-Avella et al, Phys. Rev. E, 72, 065102 (2005) M = (m1,……, mF) is a fixed external field Also valid for global or local coupling Mass media tend to reduce consensus

The role of noise Cultural drift: K. Klemm et al., Phys. Rev. E, 67, 045101 (2003) Cultural drift: Each feature of each individual can spontaneously change at rate r

The role of noise What happens as N changes? Competition between temporal scale for noise (1/r) and for the relaxation of perturbation T(N). T(N) << 1/r consensus T(N) >> 1/r fragmentation Noise destroys the q-dependent transition For large N the system is always disordered, for any q and r

Another (dis)order parameter Ng = number of domains g = <Ng>/N r=0: Consensus g ~ N-1 Fragmentation g ~ const r>0: Consensus g ~ r Fragmentation g >> r

In search of robustness Are there simple modifications of Axelrod dynamics that preserve under noise the existence of a transition depending on q? Flache and Macy (preprint, 2007): a threshold on prob. of interaction Fluctuations simply accumulate until the threshold is overcome.

Axelrod model with (homophily)-1 Usual dynamics but when there is only one feature in common, make it different For small overlap, agents become even less similar when they interact Pattern qualitatively similar to normal Axelrod model

Old relatives of Axelrod model Voter model Agent becomes equal to a randomly chosen neighbor Glauber-Ising dynamics Agent becomes equal to the majority of neighbors Voter gives disorder for any noise rate. Glauber-Ising dynamics gives order for small noise.

Axelrod model with peer pressure Introduced by Kuperman (Phys. Rev. E, 73, 046139, 2006). Usual prescription for the interaction of agents + additional step: If the trait to be adopted, si,f’ , is shared by the majority of neighbors then accept. Otherwise reject it. This introduces peer pressure (surface tension) For r=0 striped configurations are reached. A well known problem for Glauber-Ising dynamics [Spirin et al., Phys. Rev. E, 63, 036118 (2001)] For r>0 there are long lived metastable striped configurations that make the analysis difficult

Axelrod’s model with peer pressure We can start from fully ordered configurations For any q, discontinuity in the asymptotic value of g: transition between consensus and fragmentation

Axelrod’s model with peer pressure The order parameter smax/N Phase-diagram Consensus-fragmentation transition for qc(r). Limit r0 does not coincide with case r=0.

Summary and outlook Axelrod model has rich and nontrivial phenomenology with a transition between consensus and fragmentation. Noise strongly perturbs the model behavior . If peer pressure is included the original Axelrod phenomenology is rather robust with respect to noise. Coevolution Theoretical understanding Empirical validation Thanks to: Matteo Marsili, Alessandro Vespignani, Daniele Vilone.