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 two-star with degree-degree correlations in sparse regime. The constraints average correlation between degrees adjacent nodes (nearest neighbors) and at end-points two-stars (next nearest neighbors). We compute exactly network free energy show that this undergoes a first-order transition condensed phase. For non-negative degree next neighbors, distribution inside phase has single peak largest degree, while for negative neighbors is characterized by bimodal distribution. calculate assortativities they non-monotonic functions parameters, discontinuous behavior transition. critical line terminates second-order point, whose location diagram can be accurately determined. Our results help develop more detailed models networks correlated degrees.