Planted Bipartite Graph Detection

• URL: http://arxiv.org/abs/2302.03658v2
• Date: Wed, 6 Mar 2024 08:30:48 GMT
• Title: Planted Bipartite Graph Detection
• Authors: Asaf Rotenberg and Wasim Huleihel and Ofer Shayevitz
• Abstract summary: We consider the task of detecting a hidden bipartite subgraph in a given random graph. Under the null hypothesis, the graph is a realization of an ErdHosR'enyi random graph over $n$ with edge density $q$. Under the alternative, there exists a planted $k_mathsfR times k_mathsfL$ bipartite subgraph with edge density $p>q$.
• Score: 13.95780443241133
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider the task of detecting a hidden bipartite subgraph in a given random graph. This is formulated as a hypothesis testing problem, under the null hypothesis, the graph is a realization of an Erd\H{o}s-R\'{e}nyi random graph over $n$ vertices with edge density $q$. Under the alternative, there exists a planted $k_{\mathsf{R}} \times k_{\mathsf{L}}$ bipartite subgraph with edge density $p>q$. We characterize the statistical and computational barriers for this problem. Specifically, we derive information-theoretic lower bounds, and design and analyze optimal algorithms matching those bounds, in both the dense regime, where $p,q = \Theta\left(1\right)$, and the sparse regime where $p,q = \Theta\left(n^{-\alpha}\right), \alpha \in \left(0,2\right]$. We also consider the problem of testing in polynomial-time. As is customary in similar structured high-dimensional problems, our model undergoes an "easy-hard-impossible" phase transition and computational constraints penalize the statistical performance. To provide an evidence for this statistical computational gap, we prove computational lower bounds based on the low-degree conjecture, and show that the class of low-degree polynomials algorithms fail in the conjecturally hard region.

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