Computational Lower Bounds for Community Detection on Random Graphs

TL;DR

This paper establishes computational lower bounds for detecting small dense communities in large random graphs, revealing a phase transition at a critical sparsity level and linking the problem's hardness to the planted clique problem.

Contribution

It identifies a phase transition in the computational complexity of community detection based on graph sparsity and connects this to the hardness of the planted clique problem.

Findings

Existence of a critical sparsity threshold at alpha=2/3

Below threshold, detection is computationally hard for small communities

Above threshold, linear-time detection is statistically optimal

Abstract

This paper studies the problem of detecting the presence of a small dense community planted in a large Erd\H{o}s-R\'enyi random graph $\mathcal{G}(N,q)$, where the edge probability within the community exceeds $q$ by a constant factor. Assuming the hardness of the planted clique detection problem, we show that the computational complexity of detecting the community exhibits the following phase transition phenomenon: As the graph size $N$ grows and the graph becomes sparser according to $q=N^{-\alpha}$, there exists a critical value of $\alpha = \frac{2}{3}$, below which there exists a computationally intensive procedure that can detect far smaller communities than any computationally efficient procedure, and above which a linear-time procedure is statistically optimal. The results also lead to the average-case hardness results for recovering the dense community and approximating the densest $K$-subgraph.