We study a coevolution voter model on a complex network. A mean-field approximation reveals an absorbing transition from an active to a frozen phase at a critical value p c ÿ2 ÿ1 that only depends on the average degree of the network. In finite-size systems, the active and frozen phases correspond to a connected and a fragmented network, respectively. The transition can be seen as the sudden change in the trajectory of an equivalent random walk at the critical point, resulting in an approach to the final frozen state whose time scale diverges as jp c ÿ pj ÿ1 near p c . DOI: 10.1103/PhysRevLett.100.108702 PACS numbers: 89.75.Fb, 05.40.ÿa, 05.65.+b, 89.75.Hc The dynamics of collective phenomena in a system of interacting units depends on both the topology of the network of interactions and the interaction rule among connected units. The effects of these two ingredients on the emergent phenomena in a fixed network have been extensively studied. However, in many instances, both the structure of the network and the dynamical processes on it evolve in a coupled manner [1,2]. In particular, in the dynamics of social systems (Refs. [1,[3][4][5] and references therein), the network of interactions is not an exogenous structure, but it evolves and adapts driven by the changes in the state of the nodes that form the network. In recent models implementing this type of coevolution dynamics [2,4 -12] a transition is often observed from a phase where all nodes are in the same state forming a single connected network to a phase where the network is fragmented into disconnected components, each composed by nodes in the same state [13].In this Letter we address the question of how generic this type of transition is and the mechanism behind it. For this purpose, we introduce a minimal model of interacting binary state nodes that incorporates two basic features shared by many models displaying a fragmentation transition: (i) two or more absorbing states in a fixed connected network, and (ii) a rewiring rule that does not increase the number of links between nodes in the opposite state. The state dynamics consists of nodes copying the state of a random neighbor, while the network dynamics results from nodes dropping their links with opposite-state neighbors and creating new links with randomly selected same-state nodes. This model can be thought of as a coevolution version of the voter model [14] in which agents may select their interacting partners according to their states. It has the advantage of being analytically tractable and allows a fundamental understanding of the network fragmentation, explaining the transition numerically observed in related models [5,[8][9][10][11][12]. The mechanism responsible for the transition is the competition between two internal time scales, happening at a critical value that controls the relative ratio of these scales.We consider a network with N nodes. Initially, each node is endowed with a state 1 or ÿ1 with the same probability 1=2, and it is randomly connected to exactly neighbors, formi...

We present a mathematical description of the voter model dynamics on heterogeneous networks. When the average degree of the graph is µ ≤ 2 the system reaches complete order exponentially fast. For µ > 2, a finite system falls, before it fully orders, in a quasistationary state in which the average density of active links (links between opposite-state nodes) in surviving runs is constant and equal to (µ−2) 3(µ−1) , while an infinite large system stays ad infinitum in a partially ordered stationary active state. The mean life time of the quasistationary state is proportional to the mean time to reach the fully ordered state T , which scales as T ∼ (µ−1)µ 2 N (µ−2) µ2 , where N is the number of nodes of the network, and µ 2 is the second moment of the degree distribution. We find good agreement between these analytical results and numerical simulations on random networks with various degree distributions.PACS numbers: ‡ http://ifisc.uib.es

We study opinion formation in a population of leftists, centrists, and rightist. In an interaction between neighboring agents, a centrist and a leftist can become both centrists or leftists (and similarly for a centrist and a rightist), while leftists and rightists do not affect each other. For any spatial dimension the final state is either consensus (of one of three possible opinions), or a frozen population of leftists and rightists. In one dimension, the opinion evolution is mapped onto a constrained spin-1 Ising model with zero-temperature Glauber kinetics. The approach to the final state is governed by a t −ψ long-time tail, with ψ a non-universal exponent that depends on the initial densities. In the frozen state, the length distribution of single-opinion domains has an algebraic small-size tail x −2(1−ψ) with average domain length L 2ψ , where L is the length of the system.PACS numbers: 64.60. My, 05.50.+q, 75.40.Gb One of the basic issues in opinion dynamics is to understand the conditions under which consensus or diversity is reached from an initial population of individuals (agents) with different opinions. Models for such evolution are typically based on each agent freely adopting a new state in response to opinions in a local neighborhood [1]. The attribute of incompatibility -in which agents with sufficiently disparate opinions do not interact -has recently been found to prevent ultimate consensus from being reached [2,3]. Related phenomenology arises in the Axelrod model [4,5], a simple model for the formation and evolution of cultural domains. The goal of the present paper is to investigate the role of incompatibility within a minimal model for opinion dynamics. This constraint has a profound effect on the nature of the final state. Moreover, there is anomalously slow relaxation to the final state. While we primarily frame our discussion in terms of opinion dynamics, our results apply equally well to the coarsening of spin systems. In the latter context, we obtain a new non-universal kinetic exponent in one dimension that originates from topological constraints on the arrangement of spins.We consider a ternary system in which each agent can adopt the opinions of leftist, centrist, and rightist. The agents populate a lattice and in a single microscopic event an agent adopts the opinion of a randomly-chosen neighbor, but with the crucial proviso that that leftists and rightists are considered to be so incompatible that they do not interact. While a leftist cannot directly become a rightist (and vice versa) in a single step, the indirect evolution leftist ⇒ centrist ⇒ rightist is possible. Our model is similar to the classical voter model [6] and also turns out to be isomorphic to the 2-trait 2-state Axelrod model [4,5]. Due to the incompatibility constraint in our * Electronic address: fvazquez@buphy.bu.edu † Electronic address: paulk@bu.edu ‡ Electronic address: redner@bu.edu model, the final opinion outcome can be either consensus or a frozen mixture of extremists with no centrists. Figure 1 sh...

Moment-closure approximations are an important tool in the analysis of the dynamics on both static and adaptive networks. Here, we provide a broad survey over different approximation schemes by applying each of them to the adaptive voter model. While already the simplest schemes provide reasonable qualitative results, even very complex and sophisticated approximations fail to capture the dynamics quantitatively. We then perform a detailed analysis that identifies the emergence of specific correlations as the reason for the failure of established approaches, before presenting a simple approximation scheme that works best in the parameter range where all other approaches fail. By combining a focused review of published results with new analysis and illustrations, we seek to build up an intuition regarding the situations when existing approaches work, when they fail, and how new approaches can be tailored to specific problems.

We investigate the dynamics of two agent based models of language competition. In the first model, each individual can be in one of two possible states, either using language X or language Y , while the second model incorporates a third state XY , representing individuals that use both languages (bilinguals). We analyze the models on complex networks and two-dimensional square lattices by analytical and numerical methods, and show that they exhibit a transition from one-language dominance to language coexistence. We find that the coexistence of languages is more difficult to maintain in the Bilinguals model, where the presence of bilinguals in use facilitates the ultimate dominance of one of the two languages. A stability analysis reveals that the coexistence is more unlikely to happen in poorly-connected than in fully connected networks, and that the dominance of only one language is enhanced as the connectivity decreases. This dominance effect is even stronger in a two-dimensional space, where domain coarsening tends to drive the system towards language consensus.