Planted clique detection and recovery from the hypergraph adjacency matrix

TL;DR

This paper investigates the planted clique problem using only the hypergraph adjacency matrix, providing spectral detection and recovery guarantees under various regimes.

Contribution

It introduces spectral methods for detection and recovery of planted cliques from hypergraph adjacency matrices, with explicit probabilistic bounds and analysis.

Findings

Spectral norm test is asymptotically powerful at the √n scale.

Spectral method achieves exact recovery at the √n scale.

Results extend to sparse hypergraph regimes with dependent probabilities.

Abstract

Hypergraph data are often projected onto a weighted graph by constructing an adjacency matrix whose $(i,j)$ entry counts the number of hyperedges containing both nodes $i$ and $j$. This reduction is computationally convenient, but it can lose information: distinct hypergraphs may induce the same matrix, and the matrix entries are generally dependent because each hyperedge contributes to multiple pairs. We study the planted clique problem under this matrix-only observation model. For detection, we show that a spectral norm test is asymptotically powerful at the $\sqrt{n}$ scale, with explicit dependence on the background hyperedge probability $p$. For recovery, we analyze a polynomial-time spectral method based on the leading eigenvector and prove exact recovery at the canonical $\sqrt{n}$ scale, again with explicit dependence on $p$. We also extend both results to sparse regimes in which the hyperedge probability may depend on \(n\). Our analysis adapts a leave--one--out eigenvector framework to this setting. These results provide rigorous detection and recovery guarantees when only the adjacency matrix is observed.