Spectral Algorithms Optimally Recover (Censored) Planted Dense Subgraphs
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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^⋆ of size Ω(n), such that two vertices share an edge with probability p if both of them are in S^⋆, and all other edges are present with probability q, independently. The goal is to recover S^⋆ from one observation of the network. In the CPDS model, edge statuses are revealed with probability 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^⋆ up to the information theoretic threshold. Prior work by Hajek, Wu and Xu required a less efficient SDP based algorithm to recover S^⋆ 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^⋆ up to the information theoretic threshold.
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