相关论文: Sharp thresholds in inference of planted subgraphs

Sourav Chatterjee, Persi Diaconis, Allan Sly and Lingfu Zhang, prompted by a question of Ramis Movassagh, renewed the study of a process proposed in the early 1980s by Jean Bourgain. A state vector $v \in \mathbb R^n$, labeled with the…

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…

We consider statistical inference for network-linked regression problems, where covariates may include network summary statistics computed for each node. In settings involving network data, it is often natural to posit that latent variables…

We study the large-deviation properties of minimum spanning trees for two ensembles of random graphs with $N$ nodes. First, we consider complete graphs. Second, we study Erd\H{o}s-R\'{e}nyi (ER) random graphs with edge probability $p=c/N$…

We improve the running time of the general algorithmic technique known as Baker's approach (1994) on H-minor-free graphs from O(n^{f(|H|)}) to O(f(|H|) n^{O(1)}). The numerous applications include e.g. a 2-approximation for coloring and…

Out-of-Distribution (OoD) inputs are examples that do not belong to the true underlying distribution of the dataset. Research has shown that deep neural nets make confident mispredictions on OoD inputs. Therefore, it is critical to identify…

There is a growing body of work on proving hardness results for average-case estimation problems by bounding the low-degree advantage (LDA) - a quantitative estimate of the closeness of low-degree moments - between a null distribution and a…

The theory of graphons comes with a natural sampling procedure, which results in an inhomogeneous variant of the Erd\H{o}s--R\'enyi random graph, called $W$-random graphs. We prove, via the method of moments, a limit theorem for the number…

A famous conjecture of Erd\H{o}s and S\'os states that every graph with average degree more than $k - 1$ contains all trees with $k$ edges as subgraphs. We prove that the Erd\H{o}s-S\'os conjecture holds approximately, if the size of the…

We consider Erd\H{o}s-R\'enyi graphs $G(n,p)$ for $0 < p < 1$ fixed and $n \rightarrow \infty$ and study the expected number of steps, $H_{wv}$, that a random walk started in $w$ needs to first arrive in $v$. A natural guess is that an…

NAC-colourings of graphs correspond to flexible quasi-injective realisations in $\mathbb {R} ^2$. A special class of NAC-colourings are those that arise from stable cuts. We give sharp thresholds for the random graph to have no stable cut…

We study computational limitations in \emph{multi-plant} average-case inference problems, in which $t$ disjoint planted structures of size $k$ are embedded in a random background on $n$ elements. A natural parameter in this setting is the…

We locate the critical threshold $p_c$ at which it becomes likely that the complete graph $K_n$ can be obtained from the Erd\H{o}s-R\'enyi graph ${\cal G}_{n,p}$ by iteratively completing copies of $K_4$ minus an edge. This refines work of…

The Erd\H{o}s--Ko--Rado (EKR) theorem and its generalizations can be viewed as classifications of maximum independent sets in appropriately defined families of graphs, such as the Kneser graph $K(n,k)$. In this paper, we investigate the…

We study a natural problem in graph sparsification, the Spanning Tree Congestion (\STC) problem. Informally, the \STC problem seeks a spanning tree with no tree-edge \emph{routing} too many of the original edges. The root of this problem…

We prove that the treewidth of an Erd\"{o}s-R\'{e}nyi random graph $\rg{n, m}$ is, with high probability, greater than $\beta n$ for some constant $\beta > 0$ if the edge/vertex ratio $\frac{m}{n}$ is greater than 1.073. Our lower bound…

Along with the fast development of network technology and the rapid growth of network equipment, the data throughput is sharply increasing. To handle the problem of backhaul bottleneck in cellular network and satisfy people's requirements…

Embedding graphs in a geographical or latent space, i.e.\ inferring locations for vertices in Euclidean space or on a smooth manifold or submanifold, is a common task in network analysis, statistical inference, and graph visualization. We…

Recently, Bollob\'as, Janson and Riordan introduced a very general family of random graph models, producing inhomogeneous random graphs with $\Theta(n)$ edges. Roughly speaking, there is one model for each {\em kernel}, i.e., each symmetric…

We consider the ensemble of adjacency matrices of Erd\H{o}s-R\'{e}nyi random graphs, that is, graphs on $N$ vertices where every edge is chosen independently and with probability $p\equiv p(N)$. We rescale the matrix so that its bulk…