Mean-field solution of equilibrium ensembles of undirected networks with 3-edge interactions

We study the equilibrium statistical mechanics of ensembles of undirected networks with triangle- and 3-star-type interaction among bi-valued edges. We present analytical expressions for the statistics' averages in mean-field approximation. We find this model's phase diagram to be identical after parameter substitution to that of a reduced model with triangle interactions only. For sufficiently large networks, both triangle- and 3-star-type interactions affect the network's topology very similarly. We find the mean-field solution to agree excellently with Markov Chain Monte Carlo simulations in an important part of parameter space. Implications for the analysis of network topologies are being discussed.