Given a simple graph on $n$ vertices and a parameter $k$, the triangle-densest-$k$-subgraph problem is known to be computationally hard in the worst case. To circumvent the computational hardness, we study an average-case model where a triangle-dense subgraph on $k$ vertices is planted in an Erdős–Rényi random graph on $n$ vertices. For the recovery of the planted subgraph, we propose a simple spectral algorithm and a semidefinite program, both of which use a graph matrix whose entries are local signed triangle counts. Theoretical guarantees for these algorithms are established through spectral analysis of the graph matrix. Finally, we provide evidence showing a statistical-to-computational gap analogous to that for the planted clique problem. The computational threshold in terms of the subgraph size $k$ is at least $\sqrt{n}$ in the framework of low-degree polynomial algorithms, while the information-theoretic threshold is at most logarithmic in $n$.

We study a random graph model for small-world networks which are ubiquitous in social and biological sciences. In this model, a dense cycle of expected bandwidth$n \tau $, representing the hidden one-dimensional geometry of vertices, is planted in an ambient random graph on n vertices. For both detection and recovery of the planted dense cycle, we characterize the information-theoretic thresholds in terms of n,$\tau $, and an edge-wise signal-to-noise ratio$\lambda $. In particular, the information-theoretic thresholds differ from the computational thresholds established in a recent work for low-degree polynomial algorithms, thereby justifying the existence of statistical-to-computational gaps for this problem.

We consider a symmetric mixture of linear regressions with random samples from the pairwise comparison design, which can be seen as a noisy version of a type of Euclidean distance geometry problem. We analyze the expectation-maximization (EM) algorithm locally around the ground truth and establish that the sequence converges linearly, providing an $\ell_\infty$-norm guarantee on the estimation error of the iterates. Furthermore, we show that the limit of the EM sequence achieves the sharp rate of estimation in the $\ell_2$-norm, matching the information-theoretically optimal constant. We also argue through simulation that convergence from a random initialization is much more delicate in this setting, and does not appear to occur in general. Our results show that the EM algorithm can exhibit several unique behaviors when the covariate distribution is suitably structured.

Planted dense cycles are a type of latent structure that appears in many applications, such as small-world networks in social sciences and sequence assembly in computational biology. We consider a model where a dense cycle with expected bandwidth $n \tau$ and edge density $p$ is planted in an Erdős–Rényi graph $G(n,q)$. We characterize the computational thresholds for the associated detection and recovery problems for the class of low-degree polynomial algorithms. In particular, a gap exists between the two thresholds in a certain regime of parameters. For example, if $n^{-3/4} \ll \tau \ll n^{-1/2}$ and $p = C q = \Theta(1)$ for a constant $C>1$, the detection problem is computationally easy while the recovery problem is hard for low-degree algorithms.

We propose an efficient algorithm for graph matching based on similarity scores constructed from counting a certain family of weighted trees rooted at each vertex. For two Erdős–Rényi graphs G(n,q) whose edges are correlated through a latent vertex correspondence, we show that this algorithm correctly matches all but a vanishing fraction of the vertices with high probability, provided that nq→∞ and the edge correlation coefficient ρ satisfies ρ2>α ≈ 0.338, where α is Otter’s tree-counting constant. Moreover, this almost exact matching can be made exact under an extra condition that is information-theoretically necessary. This is the first polynomial-time graph matching algorithm that succeeds at an explicit constant correlation and applies to both sparse and dense graphs. In comparison, previous methods either require ρ=1−o(1) or are restricted to sparse graphs.

Graph matching, also known as network alignment, refers to finding a bijection between the vertex sets of two given graphs so as to maximally align their edges. This fundamental computational problem arises frequently in multiple fields such as computer vision and biology. Recently, there has been a plethora of work studying efficient algorithms for graph matching under probabilistic models. In this work, we propose a new algorithm for graph matching: Our algorithm associates each vertex with a signature vector using a multistage procedure and then matches a pair of vertices from the two graphs if their signature vectors are close to each other. We show that, for two Erdős–Rényi graphs with edge correlation $1-\alpha$, our algorithm recovers the underlying matching exactly with high probability when $\alpha \le 1 / (\log \log n)^C$, where $n$ is the number of vertices in each graph and $C$ denotes a positive universal constant. This improves the condition $\alpha \le 1 / (\log n)^C$ achieved in previous work.

Graph matching, also known as network alignment, aims at recovering the latent vertex correspondence between two unlabeled, edge-correlated weighted graphs. To tackle this task, we propose a spectral method, GRAph Matching by Pairwise eigen-Alignments (GRAMPA), which first constructs a similarity matrix as a weighted sum of outer products between all pairs of eigenvectors of the two graphs, and then outputs a matching by a simple rounding procedure. For a universality class of correlated Wigner models, GRAMPA achieves exact recovery of the latent matching between two graphs with edge correlation $1 - 1/\mathrm{polylog}(n)$ and average degree at least $\mathrm{polylog}(n)$. This matches the state-of-the-art guarantees for polynomial-time algorithms established for correlated Erdős-Rényi graphs, and significantly improves over existing spectral methods. The superiority of GRAMPA is also demonstrated on a variety of synthetic and real datasets, in terms of both statistical accuracy and computational efficiency.

There has been a recent surge of interest in studying permutation-based models for ranking from pairwise comparison data. Despite being structurally richer and more robust than parametric ranking models, permutation-based models are less well understood statistically and generally lack efficient learning algorithms. In this work, we study a prototype of permutation-based ranking models, namely, the noisy sorting model. We establish the optimal rates of learning the model under two sampling procedures. Furthermore, we provide a fast algorithm to achieve near-optimal rates if the observations are sampled independently. Along the way, we discover properties of the symmetric group which are of theoretical interest.

Many applications, including rank aggregation and crowd-labeling, can be modeled in terms of a bivariate isotonic matrix with unknown permutations acting on its rows and columns. We consider the problem of estimating such a matrix based on noisy observations of a subset of its entries, and design and analyze a polynomial-time algorithm that improves upon the state of the art. In particular, our results imply that any such $n \times n$ matrix can be estimated efficiently in the normalized Frobenius norm at rate $\widetilde{\mathcal O}(n^{-3/4})$, thus narrowing the gap between $\widetilde{\mathcal O}(n^{-1})$ and $\widetilde{\mathcal O}(n^{-1/2})$, which were hitherto the rates of the most statistically and computationally efficient methods, respectively.