Analytical and numerical study of the non-linear noisy voter model on complex networks

Journal article

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.

Imitation models where individuals copy the actions or opinions of others are the basis to understand the transition to consensus and organized behavior in societies. The voter model incorporates the simplest mechanism of blind imitation and has become one of the accepted paradigms in this field. To make the model closer to reality, one needs to take into account the addition of spontaneous changes of state, the characteristics of the network of interactions, and the details of the imitation mechanism. All three ingredients have been considered in the present paper. The resulting non-equilibrium model turns out to offer a rich phenomenology, and its phase diagram includes tricritical points, catastrophes, and a non-trivial scaling behavior that can be analyzed using some tools of equilibrium statistical mechanics.

Authors: Peralta, A; Carro, A; San Miguel, M; Toral, R

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: American Institute of Physics
• Journal: Chaos: An Interdisciplinary Journal of Nonlinear Science
• Volume: 28
• Issue: 7
• Article number: 075516
• Publication date: 2018-07-24
• Acceptance date: 2018-06-22
• DOI: 10.1063/1.5030112
• EISSN: 1089-7682
• ISSN: 1054-1500
• Keywords: physics.soc-ph; cond-mat.stat-mech