Incomplete ordering of the voter model on small-world networks

Abstract

Dynamics of social systems. PACS. 89.75.-k -Complex systems. PACS. 05.70.Ln -Nonequilibrium and irreversible thermodynamics.Abstract. -We investigate how the topology of small-world networks affects the dynamics of the voter model for opinion formation. We show that, contrary to what occurs on regular topologies with local interactions, the voter model on small-world networks does not display the emergence of complete order in the thermodynamic limit. The system settles in a stationary state with coexisting opinions whose lifetime diverges with the system size. Hence the nontrivial connectivity pattern leads to the counterintuitive conclusion that long-range connections inhibit the ordering process. However, for networks of finite size, for which full uniformity is reached, the ordering process takes a time shorter than on a regular lattice of the same size.

Solution of voter model dynamics on annealed small-world networks

“…By comparing the results reported in Ref. [9] and the theoretical approach presented here, we see that the annealed approximation correctly reproduces many of the important features of the original system, i. e. the existence of a regime with a disordered state before full order sets in and the linear dependence on L of the temporal scale τ between them. Concerning the shape of the final correlation function in the case of infinite quenched networks, Fig.…”

“…In Ref. [9] we have already performed a numerical investigation of the voter dynamics on the quenched WS topology. By comparing the results reported in Ref.…”

“…In a recent paper [9], we have studied numerically the dynamics of the voter model on a small-world network. This structure, more precisely the Watts-Strogatz (WS) network [10], is one of the simplest examples of complex topology.…”

Unanimity rule on networks

“…After mapping the above catalytic reactions onto a network structure, where nodes represent items and directed links show which items are necessary for the production of others, it is tempting to introduce a unanimity rule (UR): a node on the network is activated only if all the nodes arriving to it through a link are activated. Surprisingly, the dynamics of such an unanimity rule, that is a straightforward generalization of the usual majority rules of opinion dynamics [5][6][7][8][9], is poorly c EDP Sciences Fig. 1 -First two steps of UR starting from an initial network of 7 nodes, 2 of them being activated.…”

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

“…As a result, the two-opinion voter model exhibits only consensus absorbing configurations in regular lattices of arbitrary dimension [5] (see, however, [8] for a report of incomplete ordering on small-world networks). The same conclusion holds for the constrained voter model in an infinite onedimensional lattice (see, e.g., [9][10][11][12]) and since increasing the range of the agents' interactions favors the consensus regime [13,14], we expect this conclusion to hold for regular lattices of higher dimension as well.…”