Phase transitions in a complex network

Abstract

We study a mean field model of a complex network, focusing on edge and\ntriangle densities. Our first result is the derivation of a variational\ncharacterization of the entropy density, compatible with the infinite node\nlimit. We then determine the optimizing graphs for small triangle density and a\nrange of edge density, though we can only prove they are local, not global,\nmaxima of the entropy density. With this assumption we then prove that the\nresulting entropy density must lose its analyticity in various regimes. In\nparticular this implies the existence of a phase transition between distinct\nheterogeneous multipartite phases at low triangle density, and a phase\ntransition between these phases and the disordered phase at high triangle\ndensity.\n