Limit theorems for walks and triangles on Erd\"os-R\'enyi random graphs with large interaction radius

Abstract

We study cumulants of numbers of $q$-step walks on Erd\H os-R\'enyi random graphs with distance-dependent edge probability in the limit when the number of vertices $N$, concentration $c$, and interaction radius $R$ tend to infinity. These cumulants can be associated with a formal cumulant expansion of the free energy of matrix models of exponential random graphs widely known in mathematical and theoretical physics. We show that in three different asymptotic regimes, the limiting values of $k$-th cumulants ${\cal F}_k^{(q)}$ exist and can be associated with one or another family of tree-type diagrams, in dependence on the asymptotic behavior of parameters $cR/N$ for $q$-step non-closed walks and $c^2R/N^2$ for 3-step closed walks, respectively. In some cases, we obtain explicit expressions for ${\cal F}_k^{(q)}$ with the help of a modified Pr\"ufer codification. These results allow us to prove Limit Theorems for the number of non-closed walks and for the number of triangles in corresponding ensembles of large random graphs. We indicate an asymptotic threshold that separates the normal probability distribution and the Poisson one for the number of triangles in random graphs. We show that in the random graph ensemble that we consider the average vertex degree can be bounded from above while the total number of triangles infinitely increases, thus rigorously solving a graph collapse problem known in applications.