Modular Statistics for Subgraph Counts in Sparse Random Graphs

Abstract

Answering a question of Kolaitis and Kopparty, we show that, for given integer $q>1$ and pairwise nonisomorphic connected graphs $G_1,\dots, G_k$, if $p=p(n) $ is such that $\Pr(G_{n,p}\supseteq G_i)\rightarrow 1$ $\forall i$, then, with $\xi_i$ the number of copies of $G_i$ in $G_{n,p}$, $(\xi_1,\dots, \xi_k)$ is asymptotically uniformly distributed on ${\bf Z}_q^k$.