Finite-size effects in exponential random graphs

In this article, we show numerically the strong finite-size effects in exponential random graphs. Particularly, for the two-star model above the critical value of the chemical potential for triplets a ground state is a star-like graph with the finite set of hubs at network density $p<0.5$ or as the single cluster at $p>0.5$. We find that there exists the critical value of number of nodes $N^{*}(p)$ when the ground state undergoes clear-cut crossover. At $N>N^{*}(p),$ the network flows via a cluster evaporation to the state involving the small star in the Erdős–Rényi environment. The similar evaporation of the cluster takes place at $N>N^{*}(p)$ in the Strauss model. We suggest that the entropic trap mechanism is relevant for microscopic mechanism behind the crossover regime.