Non-isomorphic subgraphs in random graphs
Michael Krivelevich, Maksim Zhukovskii
TL;DR
The authors investigate $\mu(G)$, the count of non-isomorphic induced subgraphs in $G(n,p)$, across almost the entire range of $p$. They establish precise threshold behavior: the count becomes exponential once $p$ passes $\frac{1}{n}$, approaches the maximum base $2^n$ once $p$ grows beyond $\frac{2\ln n}{n}$, and provide first- and second-order asymptotics for $p\gg \frac{\ln n}{n}$. A key novelty is connecting the growth of $\mu(G(n,p))$ to the structure of the giant component via Galton–Watson tree analysis and the contiguous giant-component model, enabling lower bounds through abundant non-isomorphic subtrees. The results extend to random $d$-regular graphs, showing exponential diversity whenever $d$ is large enough, and to all $d\ge3$ via expander-based arguments, highlighting the fundamental role of underlying component structure and automorphisms in subgraph diversity. Overall, the paper sharpens the understanding of how randomness and phase transitions shape the diversity of induced subgraphs in random graph models with concrete implications for reconstruction, labeling schemes, and graph entropy concepts.
Abstract
We establish the asymptotic behaviour of $μ(G(n,p))$, the number of unlabelled induced subgraphs in the binomial random graph $G(n,p)$, for almost the entire range of the probability parameter $p=p(n)\in[0,1]$. In particular, we show that typically the number of subgraphs becomes exponential when $p$ passes $1/n$, reaches maximum possible base of exponent (asymptotically) when $p\gg 1/n$, and reaches the asymptotic value $2^n$ when $p$ passes $2\ln n/n$. For $p\gg \ln n/n$, we get the first order term and asymptotics of the second order term of $μ(G(n,p))$. We also prove that random regular graphs $G_{n,d}$ typically have $μ(G_{n,d})\geq 2^{c_d n}$ for all $d\geq 3$ and some positive constant $c_d$ such that $c_d\to 1$ as $d\to\infty$.