The Erdős-Rényi Random Graph Conditioned on Every Component Being a Clique

Martijn Gösgens, Lukas Lüchtrath, Elena Magnanini, Marc Noy, Élie de Panafieu

Published 2024-05-22Version 1

We consider an Erd\H{o}s-R\'enyi random graph, conditioned on the rare event that all connected components are fully connected. Such graphs can be considered as partitions of vertices into cliques. Hence, this conditional distribution defines a distribution over partitions. Using tools from analytic combinatorics, we prove limit theorems for several graph observables: the number of cliques; the number of edges; and the degree distribution. We consider several regimes of the connection probability $p$ as the number of vertices $n$ diverges. For $p=\frac{1}{2}$, the conditioning yields the uniform distribution over set partitions, which is well-studied, but has not been studied as a graph distribution before. For $p < \frac{1}{2}$, we show that the number of cliques is of the order $n/\sqrt{\log n}$, while for $p > \frac{1}{2}$, we prove that the graph consists of a single clique with high probability. This shows that there is a phase transition at $p=\frac{1}{2}$. We additionally study the near-critical regime $p_n\downarrow\frac{1}{2}$, as well as the sparse regime $p_n\downarrow0$.