Diagramas de fase do modelo de Ising definido sobre dois grafos acoplados

Abstract

In this work we study a model of two coupled complex networks, with the purpose to study the phase diagrams and the metastability of the system. The networks have internal ferromagnetic interactions and are coupled to each other through antiferromagnetic interactions. The model has finite connectivity, i.e., each Ising spin interacts with a finite number of other spins. The number of connections per site in each network is a random variable that follows a Poisson distribution, which characterizes Erdös-Rényi random graphs. The main objective of this work is to obtain the phase diagrams and the curves that limit the region of metastability as a function of the model parameters, such as the average connectivity between the networks, the intensity of the antiferromagnetic interactions between the networks and the temperature. Using the replica method, we derive the self-consistent equations for the distributions of effective fields, from which we can calculate the magnetization of each network and the free energy of the system. The self-consistent equations have been solved numerically through the population dynamics algorithm. We calculate numerically the magnetization of each network and the free energy, from which we construct the phase diagrams. In the first part of the results, we consider a vanishing average connectivity between the networks and we recover some known results for the Ising model on an Erdös-Rényi random graph. For the case of two coupled networks, we construct the phase diagrams and we calculate the free energy. The model has a paramagnetic phase, where the magnetization of each network is zero, and an antiferromagnetic phase, where the graphs have magnetizations with opposite signs. Based on the calculation of the free-energy, we show that this model has a metastable solution, where the ferromagnetic state corresponds to a local minimum of the free energy. We study the stability limit of the ferromagnetic solution as a function of the parameters of the model. Besides that, we observe the presence of a paramagnetic phase at low temperatures that is related to the low connectivity between the two networks and inside them. The theoretical results for the model of coupled networks have been compared with Monte-Carlo simulations, showing a very good agreement.