Detecting weighted hidden cliques

TL;DR

This paper extends the hidden clique detection problem to weighted graphs, establishing statistical limits and efficient spectral tests for hypothesis testing under various knowledge scenarios about edge weight distributions.

Contribution

It introduces a weighted hidden clique model, derives statistical thresholds for detectability, and proposes spectral algorithms effective when the clique size is at least proportional to the square root of the number of vertices.

Findings

Statistical limits for hypothesis distinguishability in weighted graphs.

Spectral tests effective for clique sizes $k= ext{Omega}(\sqrt{n})$.

Analysis of cases with partial and full knowledge of distributions.

Abstract

We study a generalization of the classical hidden clique problem to graphs with real-valued edge weights. Formally, we define a hypothesis testing problem. Under the null hypothesis, edges of a complete graph on $n$ vertices are associated with independent and identically distributed edge weights from a distribution $P$. Under the alternate hypothesis, $k$ vertices are chosen at random and the edge weights between them are drawn from a distribution $Q$, while the remaining are sampled from $P$. The goal is to decide, upon observing the edge weights, which of the two hypotheses they were generated from. We investigate the problem under two different scenarios: (1) when $P$ and $Q$ are completely known, and (2) when there is only partial information of $P$ and $Q$. In the first scenario, we obtain statistical limits on $k$ when the two hypotheses are distinguishable, and when they are not. Additionally, in each of the scenarios, we provide bounds on the minimal risk of the hypothesis testing problem when $Q$ is not absolutely continuous with respect to $P$. We also provide computationally efficient spectral tests that can distinguish the two hypotheses as long as $k=\Omega(\sqrt{n})$ in both the scenarios.