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

TL;DR

This paper establishes limit theorems for walks and triangles in Erdős-Rényi random graphs with large interaction radius, revealing asymptotic behaviors and phase transitions in graph structures.

Contribution

It introduces new asymptotic regimes for cumulants of walks and triangles, connecting them to tree-type diagrams and solving a graph collapse problem.

Findings

Limit theorems for non-closed walks and triangles established.

Explicit forms of cumulants obtained in certain regimes.

Identifies conditions for bounded degree with infinitely increasing triangles.

Abstract

We study cumulants of numbers of $q$-step walks on Erd\"os-R\'enyi-type random graphs of long-range percolation radius model in the limit when the number of vertices $N$, concentration $c$, and the 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 of 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 certain cases, we obtain ${\cal F}_k^{(q)}$ in explicit form. 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. As a consequence, we indicate an asymptotic regime when in random graphs that we consider, the average vertex degree remains bounded while the total number of triangles infinitely increases, thus rigorously solving a graph collapse problem known in applications.