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 Pn for edges in the hidden 2k-NN graph and Qn 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 ≤ no(1) if liminf frac2αnlog n>1; (2) almost exact recovery for 1 ≤ k ≤ oleft( fraclog nlog log n right) if liminf frackD(Pn||Qn)log n>1, where αn triangleq -2 log int √d Pn d Qn is the R'enyi divergence of order frac12 and D(Pn||Qn) 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.