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Register a new userOrdering in voter models on networks: Exact reduction to a single-coordinate diffusion
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Authors
R. A. Blythe
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We study the voter model and related random-copying processes on arbitrarily complex network structures. Through a representation of the dynamics as a particle reaction process, we show that a quantity measuring the degree of order in a finite system is, under certain conditions, exactly governed by a universal diffusion equation. Whenever this reduction occurs, the details of the network structure and random-copying process affect only a single parameter in the diffusion equation. The validity of the reduction can be established with considerably less information than one might expect: it suffices to know just two characteristic timescales within the dynamics of a single pair of reacting particles. We develop methods to identify these timescales, and apply them to deterministic and random network structures. We focus in particular on how the ordering time is affected by degree correlations, since such effects are hard to access by existing theoretical approaches.
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We study a generalization of the voter model on complex networks, focusing on the scaling of mean exit time. Previous work has defined the voter model in terms of an initially chosen node and a randomly chosen neighbor, which makes it difficult to disentangle the effects of the stochastic process itself relative to the network structure. We introduce a process with two steps, one that selects a pair of interacting nodes and one that determines the direction of interaction as a function of the degrees of the two nodes and a parameter $alpha$ which sets the likelihood of the higher degree node giving its state. Traditional voter model behavior can be recovered within the model. We find that on a complete bipartite network, the traditional voter model is the fastest process. On a random network with power law degree distribution, we observe two regimes. For modest values of $alpha$, exit time is dominated by diffusive drift of the system state, but as the high nodes become more influential, the exit time becomes becomes dominated by frustration effects. For certain selection processes, a short intermediate regime occurs where exit occurs after exponential mixing.
We propose a modified voter model with locally conserved magnetization and investigate its phase ordering dynamics in two dimensions in numerical simulations. Imposing a local constraint on the dynamics has the surprising effect of speeding up the phase ordering process. The system is shown to exhibit a scaling regime characterized by algebraic domain growth, at odds with the logarithmic coarsening of the standard voter model. A phenomenological approach based on cluster diffusion and similar to Smoluchowski ripening correctly predicts the observed scaling regime. Our analysis exposes unexpected complexity in the phase ordering dynamics without thermodynamic potential.
We analyze a nonlinear $q$-voter model with stochastic noise, interpreted in the social context as independence, on a duplex network. The size of the lobby $q$ (i.e., the pressure group) is a crucial parameter that changes the behavior of the system. The $q$-voter model has been applied on multiplex networks in a previous work [Phys. Rev E. 92. 052812. (2015)], and it has been shown that the character of the phase transition depends on the number of levels in the multiplex network as well as the value of $q$. Here we study phase transition character in the case when on each level of the network the lobby size is different, resulting in two parameters $q_1$ and $q_2$. We find evidence of successive phase transitions when a continuous phase transition is followed by a discontinuous one or two consecutive discontinuous phases appear, depending on the parameter. When analyzing this system, we even encounter mixed-order (or hybrid) phase transition. We perform simulations and obtain supporting analytical solutions on a simple multiplex case - a duplex clique, which consists of two fully overlapped complete graphs (cliques).
By considering three different spin models belonging to the generalized voter class for ordering dynamics in two dimensions [I. Dornic, textit{et al.} Phys. Rev. Lett. textbf{87}, 045701 (2001)], we show that they behave differently from the linear voter model when the initial configuration is an unbalanced mixture up and down spins. In particular we show that for nonlinear voter models the exit probability (probability to end with all spins up when starting with an initial fraction $x$ of them) assumes a nontrivial shape. The change is traced back to the strong nonconservation of the average magnetization during the early stages of dynamics. Also the time needed to reach the final consensus state $T_N(x)$ has an anomalous nonuniversal dependence on $x$.
The exact formula for the average path length of Apollonian networks is found. With the help of recursion relations derived from the self-similar structure, we obtain the exact solution of average path length, $bar{d}_t$, for Apollonian networks. In contrast to the well-known numerical result $bar{d}_t propto (ln N_t)^{3/4}$ [Phys. Rev. Lett. textbf{94}, 018702 (2005)], our rigorous solution shows that the average path length grows logarithmically as $bar{d}_t propto ln N_t$ in the infinite limit of network size $N_t$. The extensive numerical calculations completely agree with our closed-form solution.
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