Analytic solution of the two-star model with correlated degrees

TL;DR

This paper provides an exact solution for the two-star exponential random graph model with degree correlations, revealing a first-order phase transition and detailed degree distribution behaviors in the condensed phase.

Contribution

It offers an analytical solution to the correlated two-star model, including phase transition analysis and degree distribution characterization in the sparse regime.

Findings

First-order transition to a condensed phase identified.

Degree distribution peaks at maximum degree or bimodal depending on correlations.

Degree assortativity varies non-monotonically with model parameters.

Abstract

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 non-monotonic 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.