Statistics for the Triangle Density in ERGM and Its Mean-Field Approximation

We consider the edge-triangle model (also known as the Strauss model) and its mean-field approximation, within the region of parameters called replica symmetric regime. While our motivation stems from analyzing the asymptotic behavior of the triangle density in the edge-triangle model, a significant part of our work is devoted to studying an approximation of this observable in the mean-field setting, where explicit computations are possible. More specifically, for the first model, we prove that the triangle density concentrates with high probability in a neighborhood of its typical values. For the second model we can go further and prove, for the approximated triangle density, a standard and non-standard central limit theorem at the critical point, still not known for the edge-triangle model. Additionally, we obtain many concentration results derived via large deviations and statistical mechanics techniques. Although a rigorous comparison between these two models is still lacking, we believe that they are asymptotically equivalent in many respects. To support this conjectured behavior, we complement the analysis with simulations related to the central limit theorem for the edge-triangle model.