We study the noisy voter model using a specific non-linear dependence of the rates that takes into account collective interaction between individuals. The resulting model is solved exactly under the all-to-all coupling configuration and approximately in some random network environments. In the all-to-all setup, we find that the non-linear interactions induce bona fide phase transitions that, contrary to the linear version of the model, survive in the thermodynamic limit. The main effect of the complex network is to shift the transition lines and modify the finite-size dependence, a modification that can be captured with the introduction of an effective system size that decreases with the degree heterogeneity of the network. While a non-trivial finite-size dependence of the moments of the probability distribution is derived from our treatment, mean-field exponents are nevertheless obtained in the thermodynamic limit. These theoretical predictions are well confirmed by numerical simulations of the stochastic process.

We present a full stochastic description of the pair approximation scheme to study binary-state dynamics on heterogeneous networks. Within this general approach, we obtain a set of equations for the dynamical correlations, fluctuations and finite-size effects, as well as for the temporal evolution of all relevant variables. We test this scheme for a prototypical model of opinion dynamics known as the noisy voter model that has a finite-size critical point. Using a closure approach based on a system size expansion around a stochastic dynamical attractor we obtain very accurate results, as compared with numerical simulations, for stationary and time-dependent quantities whether below, within or above the critical region. We also show that finite-size effects in complex networks cannot be captured, as often suggested, by merely replacing the actual system size N by an effective network dependent size N eff . IntroductionFrom classical problems in statistical physics [1, 2] to questions in biology and ecology [3][4][5], and over to the spreading of opinions and diseases in social systems [6-10], stochastic binary-state models have been widely used to study the emergence of collective phenomena in systems of stochastically interacting components. In general, these components are modeled as binary-state variables -spin up or down-sitting at the nodes of a network whose links represent the possible interactions among them. While initial research focused on the limiting cases of a well-mixed population, where each of the components is allowed to interact with any other, and regular lattice structures, later works turned to more complex and heterogeneous topologies [11][12][13][14]. A most important insight derived from these more recent works is that the macroscopic dynamics of the system can be greatly affected by the particular topology of the underlying network. In the case of systems with critical behavior, different network characteristics have been shown to have a significant impact on the critical values of the model parameters [15][16][17], such as the critical temperature of the Ising model [18][19][20] and the epidemic threshold in models of infectious disease transmission [21][22][23][24][25]. Thus, the identification of the particular network characteristics that have an impact on the dynamics of these models, as well as the quantification of their effect, are of paramount importance.The first theoretical treatments introduced for the study of stochastic, binary-state dynamics on networks relied on a global-state approach [2, 26], i.e. they focused on a single variable-for example, the number of nodes in one of the two possible states-assumed to represent the whole state of the system. In order to write a master equation for this global-state variable, some approximation is required to move from the individual particle transition rates defining the model to some effective transition rates depending only on the chosen global variable. Within this global-state approach, the effective-field approximation assumes...

Aging is considered as the property of the elements of a system to be less prone to change states as they get older. We incorporate aging into the noisy voter model, a stochastic model in which the agents modify their binary state by means of noise and pair-wise interactions. Interestingly, due to aging the system passes from a finite-size discontinuous transition between ordered (ferromagnetic) and disordered (paramagnetic) phases to a second order phase transition, well defined in the thermodynamic limit, belonging to the Ising universality class. We characterize it analytically by finding the stationary solution of an infinite set of mean field equations. The theoretical predictions are tested with extensive numerical simulations in low dimensional lattices and complex networks. We finally employ the aging properties to understand the symmetries broken in the phase transition.