Condensation of degrees emerging through a first-order phase transition in classical random graphs

Analytic solution of the two-star model with correlated degrees

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.

Generalized optimal paths and weight distributions revealed through the large deviations of random walks on networks

Numerous problems of both theoretical and practical interest are related to finding shortest (or otherwise optimal) paths in networks, frequently in the presence of some obstacles or constraints. A somewhat related class of problems focuses on finding optimal distributions of weights which, for a given connection topology, maximize some kind of flow or minimize a given cost function. We show that both sets of problems can be approached through an analysis of the large-deviation functions of random walks. Specifically, a study of ensembles of trajectories allows us to find optimal paths, or design optimal weighted networks, by means of an auxiliary stochastic process (the generalized Doob transform). The paths are not limited to shortest paths, and the weights must not necessarily optimize a given function. Paths and weights can in fact be tailored to a given statistics of a time-integrated observable, which may be an activity or current, or local functions marking the passing of the random walker through a given node or link. We illustrate this idea with an exploration of optimal paths in the presence of obstacles, and networks that optimize flows under constraints on local observables.