Abstract:
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 $n\to \infty$ (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 $2 \le k \le n^{o(1)}$ if $ \liminf \frac{2\alpha_n}{\log n}>1$; (2) almost exact recovery for $ 1 \le k \le o\left( \frac{\log n}{\log \log n} \right)$ if $\liminf\frac{kD(P_n||Q_n)}{\log n}>1$, where $\alpha_n \triangleq -2 \log \int \sqrt{d P_n d Q_n}$ is the R\'enyi divergence of order $\frac{1}{2}$ 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. 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.
