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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 <inline-formula> <tex-math notation="LaTeX">$2k$ </tex-math></inline-formula>-nearest neighbor (NN) graph in an <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-vertex complete graph, whose edge weights are independent and distributed according to <inline-formula> <tex-math notation="LaTeX">$P_{n}$ </tex-math></inline-formula> for edges in the hidden <inline-formula> <tex-math notation="LaTeX">$2k$ </tex-math></inline-formula>-NN graph and <inline-formula> <tex-math notation="LaTeX">$Q_{n}$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$n\to \infty $ </tex-math></inline-formula>: (1) exact recovery: all edges are classified correctly with probability tending to one; (2) almost exact recovery: the expected number of misclassified edges is <inline-formula> <tex-math notation="LaTeX">$o(nk)$ </tex-math></inline-formula>. We show that the maximum likelihood estimator achieves (1) exact recovery for <inline-formula> <tex-math notation="LaTeX">$2 \le k \le n^{o(1)}$ </tex-math></inline-formula> if <inline-formula> <tex-math notation="LaTeX">$\liminf \frac {2\alpha _{n}}{\log n}&gt;1$ </tex-math></inline-formula>; (2) almost exact recovery for <inline-formula> <tex-math notation="LaTeX">$1 \le k \le o\left ({\frac {\log n}{\log \log n} }\right)$ </tex-math></inline-formula> if <inline-formula> <tex-math notation="LaTeX">$\liminf \frac {kD(P_{n}||Q_{n})}{\log n}&gt;1$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$\alpha _{n} \triangleq -2 \log \int \sqrt {d P_{n} d Q_{n}} ^{^{^{^{}}}}$ </tex-math></inline-formula> is the R&#x00E9;nyi divergence of order <inline-formula> <tex-math notation="LaTeX">$\frac {1}{2}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$D(P_{n}||Q_{n})$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$2k$ </tex-math></inline-formula>-NN graphs that differ from the hidden one by a given number of edges. We also analyze several computationally efficient algorithms and provide sufficient conditions under which they achieve exact/almost exact recovery. In particular, we develop a polynomial-time algorithm that attains the threshold for exact recovery under the small-world model.

Consistent Recovery Threshold of Hidden Nearest Neighbor Graphs