Phase transitions in a complex network

We study a mean field model of a complex network, focusing on edge and triangle densities. Our first result is the derivation of a variational characterization of the entropy density, compatible with the infinite node limit. We then determine the optimizing graphs for small triangle density and a range of edge density, though we can only prove they are local, not global, maxima of the entropy density. With this assumption we then prove that the resulting entropy density must lose its analyticity in various regimes. In particular this implies the existence of a phase transition between distinct heterogeneous multipartite phases at low triangle density, and a phase transition between these phases and the disordered phase at high triangle density.