  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.
