Jian Ding

Jian Ding, Yihong Wu, Jiaming Xu et al.

We study the problem of reconstructing a perfect matching $M^*$ hidden in a randomly weighted $n\times n$ bipartite graph. The edge set includes every node pair in $M^*$ and each of the $n(n-1)$ node pairs not in $M^*$ independently with probability $d/n$. The weight of each edge $e$ is independently drawn from the distribution $\mathcal{P}$ if $e \in M^*$ and from $\mathcal{Q}$ if $e \notin M^*$. We show that if $\sqrt{d} B(\mathcal{P},\mathcal{Q}) \le 1$, where $B(\mathcal{P},\mathcal{Q})$ stands for the Bhattacharyya coefficient, the reconstruction error (average fraction of misclassified edges) of the maximum likelihood estimator of $M^*$ converges to $0$ as $n\to \infty$. Conversely, if $\sqrt{d} B(\mathcal{P},\mathcal{Q}) \ge 1+ε$ for an arbitrarily small constant $ε>0$, the reconstruction error for any estimator is shown to be bounded away from $0$ under both the sparse and dense model, resolving the conjecture in [Moharrami et al. 2019, Semerjian et al. 2020]. Furthermore, in the special case of complete exponentially weighted graph with $d=n$, $\mathcal{P}=\exp(λ)$, and $\mathcal{Q}=\exp(1/n)$, for which the sharp threshold simplifies to $λ=4$, we prove that when $λ\le 4-ε$, the optimal reconstruction error is $\exp\left( - Θ(1/\sqrtε) \right)$, confirming the conjectured infinite-order phase transition in [Semerjian et al. 2020].

Jian Ding, Yihong Wu, Jiaming Xu et al.

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{2α_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 $α_n \triangleq -2 \log \int \sqrt{d P_n d Q_n}$ is the Rényi 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.

Jian Ding, Zongming Ma, Yihong Wu et al.

Random graph matching refers to recovering the underlying vertex correspondence between two random graphs with correlated edges; a prominent example is when the two random graphs are given by Erdős-Rényi graphs $G(n,\frac{d}{n})$. This can be viewed as an average-case and noisy version of the graph isomorphism problem. Under this model, the maximum likelihood estimator is equivalent to solving the intractable quadratic assignment problem. This work develops an $\tilde{O}(n d^2+n^2)$-time algorithm which perfectly recovers the true vertex correspondence with high probability, provided that the average degree is at least $d = Ω(\log^2 n)$ and the two graphs differ by at most $δ= O( \log^{-2}(n) )$ fraction of edges. For dense graphs and sparse graphs, this can be improved to $δ= O( \log^{-2/3}(n) )$ and $δ= O( \log^{-2}(d) )$ respectively, both in polynomial time. The methodology is based on appropriately chosen distance statistics of the degree profiles (empirical distribution of the degrees of neighbors). Before this work, the best known result achieves $δ=O(1)$ and $n^{o(1)} \leq d \leq n^c$ for some constant $c$ with an $n^{O(\log n)}$-time algorithm \cite{barak2018nearly} and $δ=\tilde O((d/n)^4)$ and $d = \tildeΩ(n^{4/5})$ with a polynomial-time algorithm \cite{dai2018performance}.

Vivek Bagaria, Jian Ding, David Tse et al.

We introduce the problem of hidden Hamiltonian cycle recovery, where there is an unknown Hamiltonian cycle in an $n$-vertex complete graph that needs to be inferred from noisy edge measurements. The measurements are independent and distributed according to $\calP_n$ for edges in the cycle and $\calQ_n$ otherwise. This formulation is motivated by a problem in genome assembly, where the goal is to order a set of contigs (genome subsequences) according to their positions on the genome using long-range linking measurements between the contigs. Computing the maximum likelihood estimate in this model reduces to a Traveling Salesman Problem (TSP). Despite the NP-hardness of TSP, we show that a simple linear programming (LP) relaxation, namely the fractional $2$-factor (F2F) LP, recovers the hidden Hamiltonian cycle with high probability as $n \to \infty$ provided that $α_n - \log n \to \infty$, where $α_n \triangleq -2 \log \int \sqrt{d P_n d Q_n}$ is the Rényi divergence of order $\frac{1}{2}$. This condition is information-theoretically optimal in the sense that, under mild distributional assumptions, $α_n \geq (1+o(1)) \log n$ is necessary for any algorithm to succeed regardless of the computational cost. Departing from the usual proof techniques based on dual witness construction, the analysis relies on the combinatorial characterization (in particular, the half-integrality) of the extreme points of the F2F polytope. Represented as bicolored multi-graphs, these extreme points are further decomposed into simpler "blossom-type" structures for the large deviation analysis and counting arguments. Evaluation of the algorithm on real data shows improvements over existing approaches.