Self-assembly of geometric structures in exponential random graphs

The exponential random graph model (ERGM) is a family of probabilistic graph models described by Boltzmann-like distributions whose Hamiltonians encode the network statistics. The standard ERGMs based on counting subgraphs of specific shapes exhibit a variety of phases. However, conventional phases are phenomenologically undesirable as they consist of nearly-empty or nearly-complete graphs, as in the much-studied Strauss model based on the triangle count. We introduce simple modifications to the standard ERGMs that successfully produce graph ensembles with finite mean degree and macroscopic numbers of triangles or squares, providing a qualitative improvement of the Strauss model.