Large subgraphs in pseudo-random graphs

TL;DR

This paper demonstrates that pseudo-random graphs exhibit subgraph counts similar to Erdős-Rényi graphs for small motifs, extending the understanding of their structural similarities across various models.

Contribution

It establishes conditions under which pseudo-random graphs have subgraph counts comparable to Erdős-Rényi graphs, including both deterministic and random graph models.

Findings

Subgraph counts in pseudo-random graphs match Erdős-Rényi predictions for small motifs.

Results apply to diverse graph models including ERGMs, high-dimensional correlation networks, and regular graphs.

Optimal results are achieved for graphs derived from vector spaces over binary fields.

Abstract

We consider classes of pseudo-random graphs on $n$ vertices for which the degree of every vertex and the co-degree between every pair of vertices are in the intervals $(np - Cn^\delta,np+Cn^\delta)$ and $(np^2- C n^\delta, np^2 +C n^\delta)$ respectively, for some absolute constant $C$, and $p, \delta \in (0,1)$. We show that for such pseudo-random graphs the number of induced isomorphic copies of subgraphs of size $s$ are approximately same as that of an Erd\H{o}s-R\'{e}yni random graph with edge connectivity probability $p$ as long as $s \le (((1-\delta)\wedge \frac{1}{2})-o(1))\log n/\log (1/p)$, when $p \in (0,1/2]$. When $p \in (1/2,1)$ we obtain a similar result. Our result is applicable for a large class of random and deterministic graphs including exponential random graph models (ERGMs), thresholded graphs from high-dimensional correlation networks, Erd\H{o}s-R\'{e}yni random graphs conditioned on large cliques, random $d$-regular graphs and graphs obtained from vector spaces over binary fields. In the context of the last example, the results obtained are optimal. Straight-forward extensions using the proof techniques in this paper imply strengthening of the above results in the context of larger motifs if a model allows control over higher co-degree type functionals.