Spectral Algorithms Optimally Recover (Censored) Planted Dense Subgraphs

• URL: http://arxiv.org/abs/2203.11847v1
• Date: Tue, 22 Mar 2022 16:23:43 GMT
• Title: Spectral Algorithms Optimally Recover (Censored) Planted Dense Subgraphs
• Authors: Souvik Dhara, Julia Gaudio, Elchanan Mossel, and Colin Sandon
• Abstract summary: We study spectral algorithms for the planted dense subgraph problem (PDS) and a censored variant (CPDS) of PDS, where the edge statuses are missing at random. We show that a simple spectral algorithm based on the top two eigenvectors of the adjacency matrix can recover $Sstar$ up to the information theoretic threshold. For the CDPS model, we obtain the information theoretic limit for the recovery problem, and further show that a spectral algorithm based on a special matrix called the signed adjacency matrix recovers $Sstar$ up to the information theoretic threshold
• Score: 6.760971938794954
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study spectral algorithms for the planted dense subgraph problem (PDS), as well as for a censored variant (CPDS) of PDS, where the edge statuses are missing at random. More precisely, in the PDS model, we consider $n$ vertices and a random subset of vertices $S^{\star}$ of size $\Omega(n)$, such that two vertices share an edge with probability $p$ if both of them are in $S^{\star}$, and all other edges are present with probability $q$, independently. The goal is to recover $S^{\star}$ from one observation of the network. In the CPDS model, edge statuses are revealed with probability $\frac{t \log n}{n}$. For the PDS model, we show that a simple spectral algorithm based on the top two eigenvectors of the adjacency matrix can recover $S^{\star}$ up to the information theoretic threshold. Prior work by Hajek, Wu and Xu required a less efficient SDP based algorithm to recover $S^{\star}$ up to the information theoretic threshold. For the CDPS model, we obtain the information theoretic limit for the recovery problem, and further show that a spectral algorithm based on a special matrix called the signed adjacency matrix recovers $S^{\star}$ up to the information theoretic threshold.

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