Polarization inhibits the phase transition of Axelrod's model

Carlos Gracia-Lázaro, Edgardo Brigatti, Alexis R. Hernández, and Yamir Moreno

Phys. Rev. E 103, 062306 – Published 14 June 2021

Abstract

We study the effect of polarization in Axelrod's model of cultural dissemination. This is done through the introduction of a cultural feature that takes only two values, while the other features can present a larger number of possible traits. Our numerical results and mean-field approximations show that polarization reduces the characteristic phase transition of the original model to a finite-size effect, since at the thermodynamic limit only the ordered phase is present. Furthermore, for finite system sizes, the stationary state depends on the percolation threshold of the network where the model is implemented: a polarized phase is obtained for percolation thresholds below 1/2, and a fragmented multicultural one otherwise.

Received 13 February 2021
Accepted 26 May 2021

DOI:https://doi.org/10.1103/PhysRevE.103.062306

Physics Subject Headings (PhySH)

Complex systems Phase transitions Social systems

Statistical Physics & ThermodynamicsInterdisciplinary PhysicsNetworks

Authors & Affiliations

Carlos Gracia-Lázaro ¹, Edgardo Brigatti ^2,*, Alexis R. Hernández ², and Yamir Moreno ^1,3,4

¹Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), Universidad de Zaragoza, 50018 Zaragoza, Spain
²Instituto de Física, Universidade Federal do Rio de Janeiro, 22452-970 Rio de Janeiro, Brazil
³Departamento de Física Teórica. Universidad de Zaragoza, Zaragoza E-50009, Spain
⁴ISI Foundation, via Chisola 5, 10126 Torino, Italy

^*Corresponding author: edgardo@if.ufrj.br

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Vol. 103, Iss. 6 — June 2021

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Images

Figure 1
(a) Mean of the normalized largest cultural cluster size $〈 S_{max} / N 〉$ as a function of the initial cultural diversity $q$ for different system sizes $N$ . (b) The same plot after rescaling the $x$ axis for $q / N$ . (c) Variance $χ$ of the order parameter $〈 S_{max} / N 〉$ versus $q$ for different system sizes. Inset: Value of $q_{c}$ from the maximum variance $χ$ (blue squares) and from the jump location in the normalized size of the largest cluster (red $\times$ 's) as a function of the system size $N$ , showing the tendency of the finite-size transition points $q_{c} (N)$ in the thermodynamic limit: $q_{c} = {lim}_{N \to \infty} q_{c} (N) = \infty$ . (d) $χ$ versus $q / N$ for different system sizes $N$ . It is shown that the maximum of the fluctuations in the order parameter takes place for the same value of $q / N$ regardless of the system size. All these results correspond to a lattice ( $k = 4$ ) and $F = 10$ . Each point is averaged over 1000 simulations. Note that as $q_{c}$ grows linearly with $L$ , for characterizing the theoretical behavior of the phase transition we must consider a very large number of cultural traits, even if these specific high values cannot be considered realistic, taking into account the typical situation of a social community.
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Figure 2
Evolution of the density $ρ$ of active links in a regular lattice ( $N = 1600, k = 4$ ) for $q = 100$ (a), $q = 5 \times 10^{5}$ (b). Each color corresponds to a realization among ten randomly selected.
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Figure 3
Evolution of the mean overlap for the binary feature versus the overlap for the nonbinary features. Solid lines correspond to the numerical results for a regular lattice ( $k = 4, N = 1600$ ) and $q = 10^{2}$ (a), $10^{4}$ (b), and $10^{12}$ (c). Different colors correspond to ten different characteristic realizations. Dashed blue lines show the results corresponding to the mean-field approximation given in Eq. (2).
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Figure 4
Schematic description of the different processes that contribute to Eq. (2). (a)–(c) correspond to direct processes while (d)–(i) correspond to indirect ones. To illustrate the meaning of these diagrams let us consider two of them. The subdiagram (b) corresponds to the direct process where a $(m - 1, s)$ link evolves to a $(m, s)$ link. The probability for this process to happen is given, in a mean-field approximation, by the probability of a $(m - 1, s)$ link to be sorted ( $P_{m - 1, s}$ ), times, the probability of the imitation process to happen ( $m - 1 + s / F$ ), times, the probability for the imitation to occur on a nonbinary feature $[(F - m + s) / (F - m + 1)]$ , which produces the second term of Eq. (2). The subdiagram (e) corresponds to the indirect process where the state change of a link, from $(n, s^{'})$ to $(n + 1, s^{'})$ , also modifies one of the ( $k - 1$ ) neighbor links from $(m, s)$ to $(m - 1, s)$ . The probability for this process to occur is approximated by the probability of a $n (n, s^{'})$ link to be sorted ( $P_{n, s^{'}}$ ), times, the probability to have a $(m, s)$ as a neighbor link $[(k - 1) \cdot P_{m, s}]$ , times, the probability for the imitation to happen $[(n + s^{'}) / F]$ , times, the probability to imitate a nonbinary feature $[(F - 1 - n) / (F - n - s^{'})]$ , times, the probability that the chosen feature was one of the $m$ shared features $[m / (F - 1)]$ , which produce the fifth term of Eq. (2).
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Figure 5
Solid lines show the evolution of the overlap for the binary feature versus the overlap for the nonbinary features according to simulations run on a RRN ( $N = 1600$ ) with $k = 4$ (a), 6 (b), and 8 (c); different colors correspond to ten different characteristic runs. Dashed blue lines show the results corresponding to the mean-field approximation. $q = 10^{3}$ .
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Figure 6
Normalized size of the communities corresponding to the two largest cultures as a function of $q$ for the model implemented on a regular square lattice [panel (a)] and on a RRN (b), both with connectivity $k = 4$ . Averages are taken over 100 independent simulations and $N = 1600$ .
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