1. 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

2. 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?

3. 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.

4. 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

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

6. 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
N -1  disorder
smax/N = O(1)  order
Low q: consensus
High q: fragmentation (polarization)
Consensus-fragmentation transition
depending on the initial condition

7. 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

8. 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, (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

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

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

11. 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, (2005)
M = (m1,……, mF) is a
fixed external field
Also valid for
global or local coupling
Mass media tend to reduce consensus

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

13. 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

14. 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

15. 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.

16. 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

17. 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.

18. Axelrod model with peer pressure
Introduced by Kuperman (Phys. Rev. E, 73, , 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, (2001)]
For r>0 there are long lived metastable striped configurations
that make the analysis difficult

19. 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

20. 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.

21. 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.