
Semidefinite Programs for Exact Recovery of a Hidden Community
We study a semidefinite programming (SDP) relaxation of the maximum like...
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Recovering a Single Community with Side Information
We study the effect of the quality and quantity of side information on t...
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Consistent recovery threshold of hidden nearest neighbor graphs
Motivated by applications such as discovering strong ties in social netw...
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Community Detection: Exact Recovery in Weighted Graphs
In community detection, the exact recovery of communities (clusters) has...
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Submatrix localization via message passing
The principal submatrix localization problem deals with recovering a K× ...
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Information Limits for Detecting a Subhypergraph
We consider the problem of recovering a subhypergraph based on an observ...
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The Overlap Gap Property in Principal Submatrix Recovery
We study support recovery for a k × k principal submatrix with elevated ...
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Information Limits for Recovering a Hidden Community
We study the problem of recovering a hidden community of cardinality K from an n × n symmetric data matrix A, where for distinct indices i,j, A_ij∼ P if i, j both belong to the community and A_ij∼ Q otherwise, for two known probability distributions P and Q depending on n. If P= Bern(p) and Q= Bern(q) with p>q, it reduces to the problem of finding a denselyconnected Ksubgraph planted in a large ErdösRényi graph; if P=N(μ,1) and Q=N(0,1) with μ>0, it corresponds to the problem of locating a K × K principal submatrix of elevated means in a large Gaussian random matrix. We focus on two types of asymptotic recovery guarantees as n →∞: (1) weak recovery: expected number of classification errors is o(K); (2) exact recovery: probability of classifying all indices correctly converges to one. Under mild assumptions on P and Q, and allowing the community size to scale sublinearly with n, we derive a set of sufficient conditions and a set of necessary conditions for recovery, which are asymptotically tight with sharp constants. The results hold in particular for the Gaussian case, and for the case of bounded log likelihood ratio, including the Bernoulli case whenever p/q and 1p/1q are bounded away from zero and infinity. An important algorithmic implication is that, whenever exact recovery is information theoretically possible, any algorithm that provides weak recovery when the community size is concentrated near K can be upgraded to achieve exact recovery in linear additional time by a simple voting procedure.
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