Triangle-3star-Correlations in the 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 discuss the analytical expressions for the statistics' averages in mean-field approximation as a function of these interaction parameters. We find these averages to be highly correlated for a large region of the parameter space. Quantifying these correlations we especially find triangle- and 3-star-motifs to correlate with one another. Markov Chain Monte Carlo simulations confirm the analytical mean-field results in an important part of parameter space. Implications for the analysis of network topologies are being discussed.