Consensus in the two-state Axelrod model

Lanchier, Nicolas; Schweinsberg, Jason

doi:10.1016/j.spa.2012.06.010

Cited by 20 publications

(31 citation statements)

References 6 publications

(9 reference statements)

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“…As Axelrod's model can be seen as F coupled voter models [9], most of the information we have on the behavior of the model in regular lattices was obtained us-ing Monte Carlo simulations of lattices of finite linear size L and then properly extrapolating the results to the thermodynamic limit L → ∞ within a well-established framework in statistical physics (see, e.g., [10]). Hence our surprise with the recent claim by Lanchier [11] (see also [12]) that in the particular case F = q = 2 of the one-dimensional system, which is isomorphic to the constrained voter model [13,14], the Monte Carlo simulations [15] yielded predictions that seemed to disagree with his analytical results, leading to the assertions that 'spatial simulations are usually difficult to interpret' and that 'there is a need for rigorous analytical results' [11]. In particular, whereas the Monte Carlo results indicate the presence of multicultural absorbing configurations in the thermodynamic limit, Lanchier's analysis shows that only the consensus configurations exist in that limit.…”

Section: Introductionmentioning

confidence: 79%

The consensus in the two-feature two-state one-dimensional Axelrod model revisited

Biral¹,

Tilles²,

Fontanari³

2015

J. Stat. Mech.

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The Axelrod model for the dissemination of culture exhibits a rich spatial distribution of cultural domains, which depends on the values of the two model parameters: F , the number of cultural features and q, the common number of states each feature can assume. In the one-dimensional model with F = q = 2, which is closely related to the constrained voter model, Monte Carlo simulations indicate the existence of multicultural absorbing configurations in which at least one macroscopic domain coexist with a multitude of microscopic ones in the thermodynamic limit. However, rigorous analytical results for the infinite system starting from the configuration where all cultures are equally likely show convergence to only monocultural or consensus configurations. Here we show that this disagreement is due simply to the order that the time-asymptotic limit and the thermodynamic limit are taken in the simulations. In addition, we show how the consensus-only result can be derived using Monte Carlo simulations of finite chains.

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Section: Introductionmentioning

confidence: 79%

The consensus in the two-feature two-state one-dimensional Axelrod model revisited

Biral¹,

Tilles²,

Fontanari³

2015

J. Stat. Mech.

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“…The analysis of the infinite system is more challenging. The first key to all our proofs is to use the formal machinery introduced in [13,15,17] that consists in keeping track of the disagreements along the edges of the spatial structure. This technique has also been used in [14,16] to study related models.…”

Section: Introductionmentioning

confidence: 99%

Limiting behavior for a general class of voter models with confidence threshold

Lanchier

Scarlatos

2017

ALEA

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This article is concerned with a general class of stochastic spatial models for the dynamics of opinions. Like in the voter model, individuals are located on the vertex set of a connected graph and update their opinion at a constant rate based on the opinion of their neighbors. However, unlike in the voter model, the set of opinions is represented by the set of vertices of another connected graph that we call the opinion graph: when an individual interacts with a neighbor, she imitates this neighbor if and only if the distance between their opinions, defined as the graph distance induced by the opinion graph, does not exceed a certain confidence threshold. When the confidence threshold is at least equal to the radius of the opinion graph, we prove that the one-dimensional process fluctuates and clusters and give a universal lower bound for the probability of consensus of the process on finite connected graphs. We also establish a general sufficient condition for fixation of the infinite system based on the structure of the opinion graph, which we then significantly improve for opinion graphs which are distance-regular. Our general results are used to understand the dynamics of the system for various examples of opinion graphs: paths and stars, which are not distance-regular, and cycles, hypercubes and the five Platonic solids, which are distance-regular.

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“…In this more general setting, the particles that keep track of the disagreements between neighbors again evolve like annihilating-coalescing random walks where active particles jump at rates that depend on the number of disagreements. The techniques we develop to study the generalized model requires active particles to all jump at the same rate, which is the case only when there are two [7,9] whereas the white dots represent the set of parameters for which fixation has been proved in [8].…”

Section: Introductionmentioning

confidence: 99%

Fixation Results for the Two-Feature Axelrod Model with a Variable Number of Opinions

Lanchier

Moisson

2015

J Theor Probab

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The Axelrod model is a spatial stochastic model for the dynamics of cultures that includes two key social mechanisms: homophily and social influence, respectively defined as the tendency of individuals to interact more frequently with individuals who are more similar and the tendency of individuals to become more similar when they interact. The original model assumes that individuals are located on the vertex set of an interaction network and are characterized by their culture, a vector of opinions about F cultural features, each of which offering the same number q of alternatives. Pairs of neighbors interact at a rate proportional to the number of cultural features for which they agree, which results in one more agreement between the two neighbors. In this article, we study a more general and more realistic version of the standard Axelrod model that allows for a variable number of opinions across cultural features, say qi possible alternatives for the ith cultural feature. Our main result shows that the one-dimensional system with two cultural features fixates when q1 + q2 ≥ 6.

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Consensus in the two-state Axelrod model

Cited by 20 publications

References 6 publications

The consensus in the two-feature two-state one-dimensional Axelrod model revisited

The consensus in the two-feature two-state one-dimensional Axelrod model revisited

Limiting behavior for a general class of voter models with confidence threshold

Fixation Results for the Two-Feature Axelrod Model with a Variable Number of Opinions

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