# Consistent recovery threshold of hidden nearest neighbor graphs

Motivated by applications such as discovering strong ties in social networks and assembling genome subsequences in biology, we study the problem of recovering a hidden 2k-nearest neighbor (NN) graph in an n-vertex complete graph, whose edge weights are independent and distributed according to P_n for edges in the hidden 2k-NN graph and Q_n otherwise. The special case of Bernoulli distributions corresponds to a variant of the Watts-Strogatz small-world graph. We focus on two types of asymptotic recovery guarantees as n→∞: (1) exact recovery: all edges are classified correctly with probability tending to one; (2) almost exact recovery: the expected number of misclassified edges is o(nk). We show that the maximum likelihood estimator achieves (1) exact recovery for 2 < k < n^o(1) if lim inf2α_n/log n>1; (2) almost exact recovery for 1 < k < o( log n/loglog n) if lim infkD(P_n||Q_n)/log n>1, where α_n -2 log∫√(d P_n d Q_n) is the Rényi divergence of order 1/2 and D(P_n||Q_n) is the Kullback-Leibler divergence. Under mild distributional assumptions, these conditions are shown to be information-theoretically necessary for any algorithm to succeed. A key challenge in the analysis is the enumeration of 2k-NN graphs that differ from the hidden one by a given number of edges.

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