Nonequilibrium Opinion Spreading on 2D Small-World Networks

Four decades ago, seminal observations by Milgram [21] showed that the mean social distance between a pair of randomly selected individuals was astonishingly short, typically of a few degrees. Since then, the phenomenon of average shortest path-lengths that scale logarithmically with the system size, known as small-world effect, has ubiquitously been found in a huge diversity of different real networks, such as the Internet and the World Wide Web, ecological and food webs, power grids and electronic circuits, genome and metabolic reactions, collaboration among scientists and among Hollywood actors, and many others. Furthermore, empirical observations also showed that local neighborhoods are generally highly interconnected [22,23,24,25,26].

The well-studied classical random graphs, which are networks built by linking nodes at random, display the small-world effect but have much lower connectivities than usually observed in real networks. In this context, the small-world networks were proposed few years ago [22,27] as a realization of complex networks having short mean path-lengths (and hence showing the small-world effect) as well as large connectivities. Starting from a regular lattice, a small-world network is built by randomly adding or rewiring a fraction p of the initial number of links. Even a small fraction of added or rewired links provides the shortcuts needed to produce the small-world effect, thus displaying a global behavior close to that of a random graph, while preserving locally the ordered, highly connected structure of a regular lattice. It has been shown that this small-world regime is reached for any given disorder probability p > 0, provided only that the system size N is large enough (i.e. N > N c , where the critical system size is N c ∝ 1/p) [28,29].

Small-world networks, showing the appropriate topological features observed in real social networks, can thus be meaningfully used as substrates in order to investigate different social phenomena. Indeed, this motivated the study of spin models defined on small-world networks, in which spin states denote different opinions or preferences [12,17,29,30,31,32,33]. Under this interpretation, the coupling constant describes the convincing power between interacting individuals, which is in competition with the “free will” given by the thermal noise [12], while a magnetic field could be used to add a bias that can be interpreted as “prejudice” or “stubbornness” [31].

Very recently, irreversible opinion spreading phenomena have been studied in 1D small-world networks by means of the so-called magnetic Eden model (MEM) [17], a nonequilibrium kinetic growth model in which the deposited particles have an intrinsic spin and grow in contact with a thermal bath [34,35]. According to the growth rules of the MEM, which are given in the next Section, the opinion or decision of an individual would be affected by those of their acquaintances, but opinion changes (analogous to spin flips in the Ising model) would not occur. The MEM defined on a 1D small-world network presents a second-order phase transition taking place at a finite critical temperature for any value of the rewiring probability p > 0, a phenomenon analogous to observations reported previously in the investigation of equilibrium spin systems [17].

Within the context of these recent developments, the aim of this work is to study the behavior of the MEM growing on small-world networks generated from 2D regular lattices. Particular emphasis is put on aspects inherent to the underlying 2D nature of the substrate, such as domain growth and cluster size distributions. Moreover, the critical behavior associated to the observed thermally-driven orderdisorder phase transitions is also studied and discussed in detail. Notice that, although this work is mainly motivated by social phenomena, a magnetic language is adopted throughout. As commented above, physical concepts such as temperature and magnetization, spin growth and clustering, ferromagnetic-paramagnetic phase transitions, etc, can be meaningfully re-interpreted in sociological/sociophysical contexts.

This paper is organized as follows: in Section 2, details on the model definition and the simulation method are given; Section 3 is devoted to the presentation and discussion of the results, and Section 4 contains the conclusions.

In this work, we consider the 2D, nearest-neighbor, adding-type small-world network model [36,37]. Starting with a 2D lattice of N sites and 2N bonds, a network realizatio

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