We investigate different opinion formation models on adaptive network topologies. Depending on the dynamical process, rewiring can either (i) lead to the elimination of interactions between agents in different states, and accelerate the convergence to a consensus state or break the network in non-interacting groups or (ii) counter-intuitively, favor the existence of diverse interacting groups for exponentially long times. The mean-field analysis allows to elucidate the mechanisms at play. Strikingly, allowing the interacting agents to bear more than one opinion at the same time drastically changes the model's behavior and leads to fast consensus. In the recent years, agent based models have been more and more used in the area of social sciences. Through a rather simple modeling approach for the individual processes of social influence, these models focus on the emergence of social behavior at the global population level. Statistical physics models and tools provide therefore a natural framework for such studies, and have been widely applied, leading to the appearance of the field called sociophysics (see [1] for a recent review on the application of statistical physics models to social dynamics). The growing field of complex networks [2, 3, 4] has moreover allowed to obtain a better knowledge of social networks [5, 6] , and in particular to show that the typical topology of the networks on which social agents interact is not regular. Many studies have therefore considered the evolution of models of interacting agents when agents are embedded on more realistic networks, and studied the influence of various complex topologies on the corresponding dynamical behavior [7] . An additional feature of networks, that may have a strong impact on the model's behavior, lies in their dynamical nature. They may indeed evolve on various timescales, and the evolution of the topology and the dynamical processes can drive each other with complex feedback effects. Studies of this coevolution are more recent and still limited [8, 9, 10, 11, 12, 13, 14, 15, 16, 17] , with many open issues. In this Letter, we provide new insights into such feedback effects by an investigation of Voter-like models (VM), in which agents update their opinions by imitating their neighbors, and can also break and establish connections with other agents. More precisely, we show how slight modifications in the evolution rule, which have minor consequences if the topology of interactions is kept fixed, can change drastically the model's behavior as soon as the topology can evolve on the same timescale as the agents' opinions. On the other hand, the simple fact of allowing agents to have several opinions at the same time, in the spirit of the Naming Game [18] or of the AB model [19] , leads to more robust behavior. The Voter model [20] considers a population of size N in which each individual i has an opinion characterized by a binary variable s i = ±1: only two opposite opinions are here allowed (for example a political choice between two parties) [25] . Starting from a random configuration of opinions, the dynamical evolution of the direct VM (d-VM) is the following: at each elementary step, an agent (i) is randomly selected, chooses one of its neighbors (j) at random and adopts its opinion, i.e. s i is set equal to s j (one timestep consists of N such updates). In the reverse case (r-VM), the first agent i instead convinces its neighbor j (s j is set equal to s i ). The distinction between d-and r-VM is necessary since the two interacting nodes do not play the same role. Moreover, the degrees of the first and the second chosen nodes have different distributions, and the second is a large-degree node with larger probability [2] . The asymmetry in the opinion update between the two interacting nodes can then couple to the asymmetry between a randomly-chosen node and its randomly-chosen neighbor, leading to different dynamical properties. No important difference is expected on homogeneous networks but, on heterogeneous networks, the probability for a hub to update its state will vary strongly from one rule to the other. The basic imitation process of the VM mimics the homogenization of opinions but, since interactions are binary and random, do not guarantee the convergence to a uniform state. Since a consensus in which all individuals share the same opinion is an absorbing state of the dynamics, any finite population reaches a consensus, but the time needed t c (N ) depends on its size N and on the topology of interactions, and diverges as N → ∞. On static networks, t c (N ) grows as a power-law of N , with an exponent depending on the degree distribution, and on the updating rule [21, 22, 23] . On homogeneous networks in particular, t c (N ) ∝ N for both d-and r-VM. In this Letter, we consider the scenario in which agents can rewire their "unsatisfied" connections. More precisely, the initial configuration is given by a random homogeneous network of interacting agents, with average number of neighbors k and random opinions. At each timestep, an agent i and one of its neighbors j are cho-
al. REF studied the dynamics of NG on adaptive networks where the connections can be rewired with the evolution of the game.
Who's talking first? Consensus or lack thereof in coevolving opinion formation models
Physical Review Letters 100, 15 (2008) 158701
Physics Medicine