Solution of voter model dynamics on annealed small-world networks

Abstract

An analytical study of the behavior of the voter model on the small-world topology is performed. In order to solve the equations for the dynamics, we consider an annealed version of the WattsStrogatz (WS) network, where long-range connections are randomly chosen at each time step. The resulting dynamics is as rich as on the original WS network. A temporal scale τ separates a quasistationary disordered state with coexisting domains from a fully ordered frozen configuration. τ is proportional to the number of nodes in the network, so that the system remains asymptotically disordered in the thermodynamic limit.

Dynamics of opinion formation in a small-world network

“…The additional of links in a lattice of fixed spatial dimension is found to have similar effects as increasing the spatial dimensionality of a regular lattice and to bring the system closer to the mean field results previously obtained in the literature [9,10]. It should be noted that the model studied here is based on a local majority rule, which is different from another popular model of opinion formation called the voter model [11,13]. The latter has the advantage of being analytically manageable, while analytic treatments of the local majority model on complex networks remains a challenging task.…”

“…Comparing with results in the voter model [11], there are similarities as well as qualitative differences. For both models, the addition of shortcuts leads to a shorter consensus time when compared with a regular lattice (q = 0).…”

“…This model of small-world network is analogous to that of Watts and Strogatz [20], and this underlying network has been used to investigate voting processes [11,12] and the Ising model [21]. We found that the time to reach a consensus drops sensitively with the additional of a small fraction of links to an otherwise regular lattice.…”

“…(see Fig.1(a)). Interestingly, recent studies on the dynamics of voter models [11,12] in 1D small-world networks revealed that ordering processes become harder to achieve with the addition of shortcuts. The ordering processes in Ising model was also found to be slower [21] in small-world networks.…”

“…Taking the nodes as individuals, such dynamical processes may be epidemics in a population, cultural assimilation, opinion formation, voting or election, decision making on competing for limited resources [7,8,9,10,11,12,13,14,15,16]. An up-to-date review on dynamical processes in complex networks can be found in the recent article by Bocaletti et al [17].…”