Effect of long-range interactions on the phase transition of Axelrod's model

Abstract

Axelrod's model with F = 2 cultural features, where each feature can assume k states drawn from a Poisson distribution of parameter q, exhibits a continuous nonequilibrium phase transition in the square lattice. Here we use extensive Monte Carlo simulations and finite size scaling to study the critical behavior of the order parameter ρ, which is the fraction of sites that belong to the largest domain of an absorbing configuration averaged over many runs. We find that it vanishes as ρ ∼ q 0 c − q β with β ≈ 0.25 at the critical point q 0 c ≈ 3.10 and that the exponent that measures the width of the critical region is ν 0 ≈ 2.1. In addition, we find that introduction of long-range links by rewiring the nearest-neighbors links of the square lattice with probability p turns the transition discontinuous, with the critical point q p c increasing from 3.1 to 27.17, approximately, as p increases from 0 to 1. The sharpness of the threshold, as measured by the exponent ν p ≈ 1 for p > 0, increases with the square root of the number of nodes of the resulting small-world network.

Comfort-driven mobility produces spatial fragmentation in Axelrod’s model

“…Once the dynamics reaches an absorbing configuration, we count the number of cultural domains (N d ) and record the size of the largest one (S d ), as usual [5][6][7][8][9]. In time, a cultural domain is defined as a connected subgraph where the agents have the same culture (i.e., they share all cultural features).…”

“…Somewhat surprisingly, in spite of this homogenizing mechanism, Axelrod's model exhibits global polarization (i.e., a stable multicultural regime) for large q in the case the agents are fixed to the sites of a square lattice and interact with their nearest neighbors only [4]. Variations of the original model have revealed that the relaxation of the homophilic interaction rules and the expansion of the interaction neighborhoods favor cultural homogenization (i.e., a stable monocultural regime) [5][6][7][8][9].…”

Topological transition in a coupled dynamic in random networks

“…This concept has its motivations from studies of cultural dissemination, using for example the Axelrod's model [27,28]. The cultural domains have been used as the order parameter in the study of cultural dissemination [8,[29][30][31].…”

Erdós-Rényi phase transition in the Axelrod model on complete graphs

“…The transition was reported to be continuous for one-dimensional networks and discontinuous for two dimensions (2D) when F > 2 [4], although a continuous transition is recovered when the topology becomes small world [3]. On the other hand, for F = 2, the type of the transition is the opposite: continuous for 2D, and discontinuous for small-world networks [5]. The case of F = 2 is important due to the possibility of taking an analytical approach to study the model [6].…”

“…* spinto@df.uba.ar Several scaling relationships have been found in the Axelrod model. For instance, in [7] and [5] a finite-size scaling analysis is performed for F = 2 in square lattices and smallworld networks. A scaling relation between Q and size N can be found in scale-free networks for F = 10 [3], a scaling relationship between the density of active bonds and time in a one-dimensional network in [8], and the finding of an effective noise rate is explored in [9].…”