## The Annals of Applied Probability

### Fixation in the one-dimensional Axelrod model

#### Abstract

The Axelrod model is a spatial stochastic model for the dynamics of cultures which includes two important social factors: social influence, the tendency of individuals to become more similar when they interact, and homophily, the tendency of individuals to interact more frequently with individuals who are more similar. Each vertex of the interaction network is characterized by its culture, a vector of $F$ cultural features that can each assumes $q$ different states. Pairs of neighbors interact at a rate proportional to the number of cultural features they have in common, which results in the interacting pair having one more cultural feature in common. In this article, we continue the analysis of the Axelrod model initiated by the first author by proving that the one-dimensional system fixates when $F\leq cq$ where the slope satisfies the equation $e^{-c}=c$. In addition, we show that the two-feature model with at least three states fixates. This last result is sharp since it is known from previous works that the one-dimensional two-feature two-state Axelrod model clusters.

#### Article information

Source
Ann. Appl. Probab., Volume 23, Number 6 (2013), 2538-2559.

Dates
First available in Project Euclid: 22 October 2013

https://projecteuclid.org/euclid.aoap/1382447697

Digital Object Identifier
doi:10.1214/12-AAP910

Mathematical Reviews number (MathSciNet)
MR3127944

Zentralblatt MATH identifier
1286.60096

#### Citation

Lanchier, Nicolas; Scarlatos, Stylianos. Fixation in the one-dimensional Axelrod model. Ann. Appl. Probab. 23 (2013), no. 6, 2538--2559. doi:10.1214/12-AAP910. https://projecteuclid.org/euclid.aoap/1382447697

#### References

• [1] Axelrod, R. (1997). The dissemination of culture: A model with local convergence and global polarization. J. Conflict. Resolut. 41 203–226.
• [2] Bramson, M. and Griffeath, D. (1980). On the Williams–Bjerknes tumour growth model. II. Math. Proc. Cambridge Philos. Soc. 88 339–357.
• [3] Bramson, M. and Griffeath, D. (1981). On the Williams–Bjerknes tumour growth model. I. Ann. Probab. 9 173–185.
• [4] Bramson, M. and Griffeath, D. (1989). Flux and fixation in cyclic particle systems. Ann. Probab. 17 26–45.
• [5] Clifford, P. and Sudbury, A. (1973). A model for spatial conflict. Biometrika 60 581–588.
• [6] Harris, T. E. (1972). Nearest-neighbor Markov interaction processes on multidimensional lattices. Adv. Math. 9 66–89.
• [7] Holley, R. A. and Liggett, T. M. (1975). Ergodic theorems for weakly interacting infinite systems and the voter model. Ann. Probab. 3 643–663.
• [8] Lanchier, N. (2012). The Axelrod model for the dissemination of culture revisited. Ann. Appl. Probab. 22 860–880.
• [9] Lanchier, N. and Schweinsberg, J. (2012). Consensus in the two-state Axelrod model. Stochastic Process. Appl. 122 3701–3717.
• [10] Scarlatos, S. (2012). Behavior of social dynamical models I: Fixation in the symmetric cyclic system (with paradoxical effect in the six-color automaton). In ACRI 2012. LNCS, (G. C. Sirakoulis and S. Bandini, eds.) 7495 141–150. Springer, Heidelberg.
• [11] Vilone, D., Vespignani, A. and Castellano, C. (2002). Ordering phase transition in the one-dimensional Axelrod model. Eur. Phys. J. B 30 399–406.