The Axelrod model for cultural dissemination: role of noise, peer pressure and homophily

Claudio CastellanoCNR-INFM Statistical Mechanics and Complexity

andUniversita’ 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

• Do regularities exist? Universal laws? • What is the effect of the structure of social networks and

of mass-media?

Interactions between individuals tend to

favor homogeneization

Other mechanisms favor fragmentation

• How do opinions spread? • What affects the way consensus is reached? • What mechanisms are at work?

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, i,f , assuming q possible traits

1 <=i,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 i,f’ <> j,f’ and set

i,f’ = j,f’

• Iterate

Fragmentation-consensus transition

Low initial variability

High initial variability

The evolution depends on the number q of traits in the initial state

Phenomenology of Axelrod model: statics

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

Consensus-fragmentation transition

depending on the initial condition

Low q: consensus

High q: fragmentation (polarization)

C. Castellano et al., Phys. Rev. Lett, 85, 3536 (2000)

Phenomenology of Axelrod model: dynamicsObservable:

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

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

C. Castellano et al., Phys. Rev. Lett, 85, 3536 (2000)

F. Vazquez and S. Redner, EPL, 78, 18002 (2007)

On complex topologies

Small world (WS) networks: interpolations between regular

lattices and random networks.

• Randomness in topology increases order• Fragmented phase survives for any p

K. Klemm et al., Phys. Rev. E, 67, 026120 (2003)

On complex topologiesScale-free networks: P(k)~k--

• Scale-free topology increases order• Fragmented phase disappears for infinite systems

The effect of mass media

M = (1,……, F) is a

fixed external field

Mass media tend to reduce consensus

Y. Shibanai et al., J. Conflict Resol., 45, 80 (2001)

J. C. Gonzalez-Avella et al, Phys. Rev. E, 72, 065102 (2005)

Also valid for

global or local coupling

The role of noise

Cultural drift:

Each feature of each individual can spontaneously change at rate r

K. Klemm et al., Phys. Rev. E, 67, 045101 (2003)

The role of noise

What happens

as N changes?

Noise destroys the q-dependent transition

For large N the system is always disordered, for any q and r

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

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

Flache and Macy (preprint, 2007):

a threshold on prob. of interaction

Are there simple modifications

of Axelrod dynamics that preserve

under noise the existence of

a transition depending on q?

Fluctuations simply accumulate until the threshold is overcome.

Agent becomes equal tothe majority of neighbors

Agent becomes equal toa randomly chosen neighbor

Old relatives of Axelrod model

Voter model

Glauber-Ising dynamics

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, i,f’ , is shared by the majority of

neighbors then accept. Otherwise reject it.

For r=0 striped configurations

are reached.

A well known problem for

Glauber-Ising dynamics

[Spirin et al.,

Phys. Rev. E, 63, 036118 (2001)]

This introduces peer pressure (surface tension)

For r>0 there are long lived metastable striped configurations

that make the analysis difficult

Axelrod’s model with peer pressureWe 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

Consensus-fragmentation transition for qc(r).

Limit r0 does not coincide with case r=0.

Phase-diagram

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.