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An algorithm observes the trajectories of random walks over an unknown graph $G$, starting from the same vertex $x$, as well as the degrees along the trajectories. For all finite connected graphs, one can estimate the number of edges $m$ up…
We give the first polynomial-time, differentially node-private, and robust algorithm for estimating the edge density of Erd\H{o}s-R\'enyi random graphs and their generalization, inhomogeneous random graphs. We further prove…
I report on the development of a novel statistical mechanical formalism for the analysis of random graphs with many short loops, and processes on such graphs. The graphs are defined via maximum entropy ensembles, in which both the degrees…
We consider the problem of approximating a maximum weighted matching, when the edges of an underlying weighted graph $G(V,E)$ are revealed in a streaming fashion. We analyze a variant of the previously best-known…
Graph matching is a challenging problem with very important applications in a wide range of fields, from image and video analysis to biological and biomedical problems. We propose a robust graph matching algorithm inspired in…
We propose a method for solving statistical mechanics problems defined on sparse graphs. It extracts a small Feedback Vertex Set (FVS) from the sparse graph, converting the sparse system to a much smaller system with many-body and dense…
We study the graphs formed from instances of the stable matching problem by connecting pairs of elements with an edge when there exists a stable matching in which they are matched. Our results include the NP-completeness of recognizing…
We consider the task of detecting a hidden bipartite subgraph in a given random graph. This is formulated as a hypothesis testing problem, under the null hypothesis, the graph is a realization of an Erd\H{o}s-R\'{e}nyi random graph over $n$…
To characterize the location (mean, median) of a set of graphs, one needs a notion of centrality that is adapted to metric spaces, since graph sets are not Euclidean spaces. A standard approach is to consider the Frechet mean. In this work,…
In this paper, we study the asymptotic behavior of the extreme eigenvalues and eigenvectors of the high dimensional spiked sample covariance matrices, in the supercritical case when a reliable detection of spikes is possible. Especially, we…
We investigate the noise sensitivity of the top eigenvector of a sparse random symmetric matrix. Let $v$ be the top eigenvector of an $N\times N$ sparse random symmetric matrix with an average of $d$ non-zero centered entries per row. We…
Graphs arising in statistical problems, signal processing, large networks, combinatorial optimization, and data analysis are often dense, which causes both computational and storage bottlenecks. One way of \textit{sparsifying} a…
An \emph{additive $+\beta$ spanner} of a graph $G$ is a subgraph which preserves distances up to an additive $+\beta$ error. Additive spanners are well-studied in unweighted graphs but have only recently received attention in weighted…
Finding optimal matchings in dense graphs is of general interest and of particular importance in social, transportation and biological networks. While developing optimal solutions for various matching problems is important, the running…
We present deterministic distributed algorithms for computing approximate maximum cardinality matchings and approximate maximum weight matchings. Our algorithm for the unweighted case computes a matching whose size is at least $(1-\eps)$…
We consider a class of sparse random matrices which includes the adjacency matrix of the Erd\H{o}s-R\'enyi graph $\mathcal{G}(N,p)$. We show that if $N^{\varepsilon} \leq Np \leq N^{1/3-\varepsilon}$ then all nontrivial eigenvalues away…
Wishart correlation matrices are the standard model for the statistical analysis of time series. The ensemble averaged eigenvalue density is of considerable practical and theoretical interest. For complex time series and correlation…
In this paper we give a method to associate a graph with an arbitrary density matrix referred to a standard orthonormal basis in the Hilbert space of a finite dimensional quantum system. We study the related issues like classification of…
The eigenvalue statistics of quantum ideal gases with single particle energies $e_n=n^\alpha$ are studied. A recursion relation for the partition function allows to calculate the mean density of states from the asymptotic expansion for the…
We give an upper bound on the maximal eigenvalue of the adjacency matrix of a connected graph in terms of its maximum degree, diameter and order. This bound is best possible up to a constant factor and improves prevoius results of…