Exponential random graphs are important to model the structure of real-world complex networks. Here we solve the two-star model with degree-degree correlations in the sparse regime. The model constraints the average correlation between the degrees of adjacent nodes (nearest neighbors) and between the degrees at the end-points of two-stars (next nearest neighbors). We compute exactly the network free energy and show that this model undergoes a first-order transition to a condensed phase. For non-negative degree correlations between next nearest neighbors, the degree distribution inside the condensed phase has a single peak at the largest degree, while for negative degree correlations between next nearest neighbors the condensed phase is characterized by a bimodal degree distribution. We calculate the degree assortativities and show they are nonmonotonic functions of the model parameters, with a discontinuous behavior at the first-order transition. The first-order critical line terminates at a second-order critical point, whose location in the phase diagram can be accurately determined. Our results can help to develop more detailed models of complex networks with correlated degrees.
We explore the cooperative behaviour and phase transitions of interacting networks by studying a simplified model consisting of Ising spins placed on the nodes of two coupled Erdös-Rényi random graphs. We derive analytical expressions for the free-energy of the system and the magnetization of each graph, from which the phase diagrams, the stability of the different states, and the nature of the transitions among them, are clearly characterized. We show that a metastable state appears discontinuously by varying the model parameters, yielding a region in the phase diagram where two solutions coexist. By performing Monte-Carlo simulations, we confirm the exactness of our main theoretical results and show that the typical time the system needs to escape from a metastable state grows exponentially fast as a function of the temperature, characterizing ergodicity breaking in the thermodynamic limit.
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