Voter models on heterogeneous networks.
We study simple interacting particle systems on heterogeneous networks, including the voter model and the invasion process. These are both two-state models in which in an update event an individual changes state to agree with a neighbor. For the voter model, an individual "imports" its state from a randomly chosen neighbor. Here the average time TN to reach consensus for a network of N nodes with an uncorrelated degree distribution scales as N mu1 2/mu2, where mu k is the kth moment of the degree distribution. Quick consensus thus arises on networks with broad degree distributions. We also identify the conservation law that characterizes the route by which consensus is reached. Parallel results are derived for the invasion process, in which the state of an agent is "exported" to a random neighbor. We further generalize to biased dynamics in which one state is favored. The probability for a single fitter mutant located at a node of degree k to overspread the population-the fixation probability--is proportional to k for the voter model and to 1k for the invasion process.
Voter model dynamics in complex networks: Role of dimensionality, disorder, and degree distribution.
We analyze the ordering dynamics of the voter model in different classes of complex networks. We observe that whether the voter dynamics orders the system depends on the effective dimensionality of the interaction networks. We also find that when there is no ordering in the system, the average survival time of metastable states in finite networks decreases with network disorder and degree heterogeneity. The existence of hubs, i.e., highly connected nodes, in the network modifies the linear system size scaling law of the survival time. The size of an ordered domain is sensitive to the network disorder and the average degree, decreasing with both; however, it seems not to depend on network size and on the heterogeneity of the degree distribution.
Incomplete ordering of the voter model on small-world networks
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.
Is the Voter Model a model for voters?
The voter model has been studied extensively as a paradigmatic opinion dynamics model. However, its ability to model real opinion dynamics has not been addressed. We introduce a noisy voter model (accounting for social influence) with recurrent mobility of agents (as a proxy for social context), where the spatial and population diversity are taken as inputs to the model. We show that the dynamics can be described as a noisy diffusive process that contains the proper anisotropic coupling topology given by population and mobility heterogeneity. The model captures statistical features of U.S. presidential elections as the stationary vote-share fluctuations across counties and the long-range spatial correlations that decay logarithmically with the distance. Furthermore, it recovers the behavior of these properties when the geographical space is coarse grained at different scales-from the county level through congressional districts, and up to states. Finally, we analyze the role of the mobility range and the randomness in decision making, which are consistent with the empirical observations.
The Median Voter Model
Most analytical work in public choice is based upon relatively simple models of majority decision making. These models are widely used even though the researchers know that real political settings are more complex than the models seem to imply. The use of such simple models can be defended for a variety of reasons: First, simple models allow knowledge to be transmitted more economically from one person to another than possible with more complex models. Second, simple models provide us with engines of analysis that allow a variety of hypotheses about more complex phenomena to be developed, many of which would be impossible (or uninteresting) without the frame of reference provided by models. Third, it is possible that simple models are all that is necessary to understand the main features of the world. The world may be less complex that it appears; in which case simple models that extract the essential from the observed will serve us well.
Ordering dynamics with two non-excluding options: bilingualism in language competition
We consider an extension of the voter model in which a set of interacting elements (agents) can be in either of two equivalent states (A or B) or in a third additional mixed (AB) state. The model is motivated by studies of language competition dynamics, where the AB state is associated with bilingualism. We study the ordering process and associated interface and coarsening dynamics in regular lattices and small world networks. Agents in the AB state define the interfaces, changing the interfacial noise driven coarsening of the voter model to curvature driven coarsening. This change in the coarsening mechanism is also shown to originate for a class of perturbations of the voter model dynamics. When interaction is through a small world network the AB agents restore coarsening, eliminating the metastable states of the voter model. The characteristic time to reach the absorbing state scales with system size as τ ∼ lnN to be compared with the result τ ∼ N for the voter model in a small world network.
Analytical solution of the voter model on uncorrelated networks
We present a mathematical description of the voter model dynamics on uncorrelated networks. When the average degree of the graph is µ 6 2 the system reaches complete order exponentially fast. For µ > 2, a finite system falls, before it fully orders, in a quasi-stationary 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 infinitely large system stays ad infinitum in a partially ordered stationary active state. The mean lifetime of the quasi- stationary 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 µ2 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.
Nonlinear q-voter model.
We introduce a nonlinear variant of the voter model, the q-voter model, in which q neighbors (with possible repetition) are consulted for a voter to change opinion. If the q neighbors agree, the voter takes their opinion; if they do not have a unanimous opinion, still a voter can flip its state with probability epsilon . We solve the model on a fully connected network (i.e., in mean field) and compute the exit probability as well as the average time to reach consensus by employing the backward Fokker-Planck formalism and scaling arguments. We analyze the results in the perspective of a recently proposed Langevin equation aimed at describing generic phase transitions in systems with two ( Z2-symmetric) absorbing states. In particular, by deriving explicitly the coefficients of such a Langevin equation as a function of the microscopic flipping probabilities, we find that in mean field the q-voter model exhibits a disordered phase for high epsilon and an ordered one for low epsilon with three possible ways to go from one to the other: (i) a unique (generalized-voter-like) transition, (ii) a series of two consecutive transitions, one (Ising-like) in which the Z2 symmetry is broken and a separate one (in the directed-percolation class) in which the system falls into an absorbing state, and (iii) a series of two transitions, including an intermediate regime in which the final state depends on initial conditions. This third (so far unexplored) scenario, in which a type of ordering dynamics emerges, is rationalized and found to be specific of mean field, i.e., fluctuations are explicitly shown to wash it out in spatially extended systems.
The noisy voter model
The noisy voter model is a spin system on a graph which may be obtained from the basic voter model by adding spontaneous flipping from 0 to 1 and from 1 to 0 at each site. Using duality, we obtain exact formulas for some important time-dependent and equilibrium functionals of this process. By letting the spontaneous flip rates tend to zero, we get the basic voter model, and we calculate the exact critical exponents associated with this "phase transition". Finally, we use the noisy voter model to present an alternate view of a result due to Cox and Griffeath on clustering in the two-dimensional basic voter model.
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