Large subgraphs in pseudo-random graphs

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 $(np - Cn^δ,np+Cn^δ)$ and $(np^2- C n^δ, np^2 +C n^δ)$ respectively, for some absolute constant $C$, and $p, δ\in (0,1)$. 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 Erdős-Réyni random graph with edge connectivity probability $p$ as long as $s \le (((1-δ)\wedge \frac{1}{2})-o(1))\log n/\log (1/p)$, 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, Erdős-Réyni 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.

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