The Erd\H{o}s-R\'enyi Random Graph Conditioned on Every Component Being a Clique

TL;DR

This paper studies a conditioned Erdős-Rényi graph where all components are cliques, revealing phase transitions and implications for community detection through combinatorial and probabilistic analysis.

Contribution

It introduces a new conditioned Erdős-Rényi model related to community detection and analyzes its properties across different regimes of connection probability.

Findings

Phase transition at p=1/2 between multiple cliques and a single clique.

Number of cliques scales as n/√log n for p<1/2.

Graph becomes a single clique with high probability for p>1/2.

Abstract

Motivated by an application in community detection, we consider an \ER 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. We show that a popular community detection method is equivalent to Bayesian inference with this distribution as prior over the community partitions. Using tools from analytic combinatorics, we prove limit theorems for several graph observables in this conditional distribution: 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 $p=\tfrac{1}{2}$, the conditioning yields the uniform distribution over set partitions, which is well-studied, but has not been studied as a graph distribution before. For $p<\tfrac{1}{2}$, we show that the number of cliques is of the order $n/\sqrt{\log n}$, while for $p>\tfrac{1}{2}$, we prove that the graph consists of a single clique with high probability. This shows that there is a phase transition at $p=\tfrac{1}{2}$. We additionally study the near-critical regime $p_n\downarrow\tfrac{1}{2}$, as well as the sparse regime $p_n\downarrow0$. Finally, we discuss the implications of these results for community detection.