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 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 connectivity, decreasing with both; however it seems not to depend on network size and degree heterogeneity.

The voter model has been studied extensively as a paradigmatic opinion dynamics' model. However, its ability for modeling real opinion dynamics has not been addressed. We introduce a noisy voter model (accounting for social influence) with agents' recurrent mobility (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 the US 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. Opinion dynamics focuses on the way different options compete in a population, giving raise to either consensus (every individual holding the same opinion or option) or coexistence of several opinions. Many theoretical efforts have been devoted to clarify the implications on the macroscopic outcome, among other aspects, of different interaction mechanisms, different topologies of the interaction networks, the inclusion of opinion leaders or of zealots, external fields [1,2]. To advance our understanding on social phenomena these theoretical efforts need to be complemented with empirical [3-5] and experimental results [6][7][8][9]. In this context elections offer an opportunity for contrasting opinion dynamics' models with empirical results [10]. On one hand, the data are publicly available in many countries, with a good level of spatial resolution and several temporal observations. On the other hand, there is evidence that voting behavior is strongly influenced by the social context of the individuals [6,7,9,[11][12][13][14][15][16][17][18][19][20]. Thus it is natural to model electoral processes as systems of interacting agents with the aim of explaining the statistical regularities [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38], as for example, the universal scaling of the distribution of votes in proportional elections [21,22] or signatures of irregularities in the democratic process [23,26].In this work we asses the capacity of the voter model to capture real voter choices and propose a microscopic foundation for modeling voting behavior in elections. The model is based on social influence and recurrent mobility (SIRM): social influence will be modeled as a noisy voter model, while recurrent mobility serves as a proxy of the social context. In the voter model each agent updates its state by randomly copying the opinion of one of its neighbors [39][40][41]. We will consider that agents interact a...

We consider the voter model dynamics in random networks with an arbitrary distribution of the degree of the nodes. We find that for the usual node-update dynamics the average magnetization is not conserved, while an average magnetization weighted by the degree of the node is conserved. However, for a link-update dynamics the average magnetization is still conserved. For the particular case of a Barabasi-Albert scale-free network the voter model dynamics leads to a partially ordered metastable state with a finite size survival time. This characteristic time scales linearly with system size only when the updating rule respects the conservation law of the average magnetization. This scaling identifies a universal or generic property of the voter model dynamics associated with the conservation law of the magnetization.

The horizontal axis of Fig. 2(a) of the Letter was wrongly scaled. The correct Fig. 2(a) is presented below. This affects the assessment of the critical value D Ã but not any other statement or conclusion drawn in the Letter.Summary of corrections: Figure 2(a) of the original manuscript should look as Fig. 2 of this communication. Page 3, left column should say: "[…]Best agreement is obtained for D ¼ 0.02[…] For D ¼ 0.02 the standard deviation of the vote-share distribution[…]" Page 3, right column should say: "[…]Sets of electoral results are produced with the model, with D ¼ 0.02 and with a fixed number of Monte Carlo steps between elections.[…]" Caption of Fig. 3 should start: "Parameters of the simulation are α ¼ 1=2, D ¼ 0.02[…]" FIG. 2 (color online). (a) Vote-share standard deviation versus noise intensity D. The dashed black line marks the dispersion of the empirical data (σ e ¼ 0.11).Boxes surrounding the main plot (not shown here, they stay the same as in the Letter) display results obtained with the level of noise marked as squares and include the distribution of vote-shares shifted to have zero mean, and their spatial correlations. Black curves are initial conditions. In the red box, the red curve is for 10 MC steps, and the orange for 20 MC steps; in the green box, the times are 100 MC steps (red) and 200 MC steps (orange); in the purple box, 40 MC steps (red) and 140 MC steps (orange).

Universal scaling of distances between vertices of Erdos-Rényi random graphs, scale-free Barabási-Albert models, science collaboration networks, biological networks, Internet Autonomous Systems and public transport networks are observed. A mean distance between two nodes of degrees k(i) and k(j) equals to (l(ij)) = A - B log(k(i)k(j)). The scaling is valid over several decades. A simple theory for the appearance of this scaling is presented. Parameters A and B depend on the mean value of a node degree (k)nn calculated for the nearest neighbors and on network clustering coefficients.