The noisy voter model on complex networks

Abstract

We propose a new analytical method to study stochastic, binary-state models on complex networks. Moving beyond the usual mean-field theories, this alternative approach is based on the introduction of an annealed approximation for uncorrelated networks, allowing to deal with the network structure as parametric heterogeneity. As an illustration, we study the noisy voter model, a modification of the original voter model including random changes of state. The proposed method is able to unfold the dependence of the model not only on the mean degree (the mean-field prediction) but also on more complex averages over the degree distribution. In particular, we find that the degree heterogeneity--variance of the underlying degree distribution--has a strong influence on the location of the critical point of a noise-induced, finite-size transition occurring in the model, on the local ordering of the system, and on the functional form of its temporal correlations. Finally, we show how this latter point opens the possibility of inferring the degree heterogeneity of the underlying network by observing only the aggregate behavior of the system as a whole, an issue of interest for systems where only macroscopic, population level variables can be measured.

Figure 1. Fraction of nodes in state 1 on a Barabási-Albert scale-free network.

Single realizations. The interaction parameter is fixed as h = 1, the system size as N = 2500 and the mean degree as .

Figure 2. Steady state variance of n as a function of the noise parameter a, for three different types of networks: Erdös-Rényi random network, Barabási-Albert scale-free network and dichotomous network.

Symbols: Numerical results (averages over 20 networks, 10 realizations per network and 50000 time steps per realization). Solid lines: Analytical results [see equation 14]. Dash-dotted lines: Analytical results for the critical points [see equation 18]. Dashed line: Mean-field approximation (see42). The interaction parameter is fixed as h = 1, the system size as N = 2500 and the mean degree as .

Figure 3. Steady state variance of n as a function of the noise parameter a for a Barabási-Albert scale-free network.

Symbols: Numerical results (averages over 20 networks, 10 realizations per network and 50000 time steps per realization). Solid line: Analytical results [see equation (14)]. Dotted line: asymptotic approximation for small a [see equation (16)]. Dash-dotted line: asymptotic approximation for large a [see equation (17)]. Dashed line: Crossover point between both asymptotic approximations (a* = 0.014157). The interaction parameter is fixed as h = 1, the system size as N = 2500 and the mean degree as .

Figure 4. Steady state variance of n as a function of the variance of the degree distribution for two values of the noise parameter a.

In order to keep all parameters constant except the variance of the degree distribution, a different network type is used for each point (in order of increasing : Erdös-Rényi random network, Barabási-Albert scale-free network and dichotomous network). Circles with error bars: Numerical results (averages over 20 networks, 10 realizations per network and 50000 time steps per realization). Solid line and squares: Analytical results [see equation (14)]. Dotted line: asymptotic approximation for small a [see equation (16)]. Dash-dotted line: asymptotic approximation for large a [see equation (17)]. Dashed line: Mean-field approximation (see42). The interaction parameter is fixed as h = 1, the system size as N = 2500 and the mean degree as .

Figure 5. Critical value of the noise parameter a as a function of the variance of the degree distribution of the underlying network,.

In order to keep all parameters constant except the variance of the degree distribution, a different network type is used for each point (in order of increasing : Erdös-Rényi random network, Barabási-Albert scale-free network and dichotomous network). Symbols: Numerical results (averages over 20 networks, 10 realizations per network and 50000 time steps per realization). Solid line: Analytical results [see equation (18)]. Dashed line: Mean-field approximation (see4244). The interaction parameter is fixed as h = 1, the system size as N = 2500 and the mean degree as .

Figure 6. Interface density on a Barabási-Albert scale-free network.

Single realizations (the same realizations shown in Fig. 1). The interaction parameter is fixed as h = 1, the system size as N = 2500 and the mean degree as .

Figure 7. Steady state of the average interface density as a function of the noise parameter a in a linear-logarithmic scale and for three different types of networks: Erdös-Rényi random network, Barabási-Albert scale-free network and dichotomous network.

Symbols: Numerical results (averages over 20 networks, 10 realizations per network and 50000 time steps per realization). Solid lines: Analytical results [see equation (21)]. Dashed line: Mean-field pair-approximation (see44). The interaction parameter is fixed as h = 1, the system size as N = 2500 and the mean degree as .

Figure 8. Autocorrelation function of n in log-linear scale for a dichotomous network and a regular 2D lattice.

Symbols: Numerical results (averages over 10 networks, 2 realizations per network and 200000 time steps per realization). Solid lines: Analytical results [see equation (23)]. Parameter values are fixed as a = 0.01, h = 1, the system size as N = 2500 and the mean degree as .