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In this paper, we develop a novel weighted Laplacian method, which is partially inspired by the theory of graph Laplacian, to study recent popular graph problems, such as multilevel graph partitioning and balanced minimum cut problem, in a…
In this work, we present an alternative formulation of the higher eigenvalue problem associated to the infinity Laplacian, which opens the door for numerical approximation of eigenfunctions. A rigorous analysis is performed to show the…
In graph signal processing, data samples are associated to vertices on a graph, while edge weights represent similarities between those samples. We propose a convex optimization problem to learn sparse well connected graphs from data. We…
The Laplacian matrix and its pseudo-inverse for a strongly connected directed graph is fundamental in computing many properties of a directed graph. Examples include random-walk centrality and betweenness measures, average hitting and…
We define a graph-based rate optimization problem and consider its computation, which provides a unified approach to the computation of various theoretical limits, including the (conditional) graph entropy, rate-distortion functions and…
We consider the Cauchy problem for the evolutive discrete p-Laplacian in infinite graphs, with initial data decaying at infinity. We prove optimal sup and gradient bounds for nonnegative solutions, when the initial data has finite mass, and…
This thesis studies the graph alignment problem, the noisy version of the graph isomorphism problem, which aims to find a matching between the nodes of two graphs which preserves most of the edges. Focusing on the planted version where the…
In this article we study the well-posedness (uniqueness and existence of solutions) of nonlinear elliptic Partial Differential Equations (PDEs) on a finite graph. These results are obtained using the discrete comparison principle and…
Our primary motivation is existence and uniqueness for the obstacle problem on graphs. That is, we look for unique solutions to the problem $Lu = \chi_{\{u>0\}}$, where $L$ is the Laplacian matrix associated to a graph, and $u$ is a…
We apply eigenvalue interlacing techniques for obtaining lower and upper bounds for the sums of Laplacian eigenvalues of graphs, and characterize equality. This leads to generalizations of, and variations on theorems by Grone, and Grone and…
We investigate a generalized poly-Laplacian system with a parameter on weighted finite graph, a generalized poly-Laplacian system with a parameter and Dirichlet boundary value on weighted locally finite graphs, and a $(p,q)$-Laplacian…
Scalings in which the graph Laplacian approaches a differential operator in the large graph limit are used to develop understanding of a number of algorithms for semi-supervised learning; in particular the extension, to this graph setting,…
In this paper, we study a nonlocal variational problem which consists of minimizing in $L^2$ the sum of a quadratic data fidelity and a regularization term corresponding to the $L^p$-norm of the nonlocal gradient. In particular, we study…
Graph sparsification underlies a large number of algorithms, ranging from approximation algorithms for cut problems to solvers for linear systems in the graph Laplacian. In its strongest form, "spectral sparsification" reduces the number of…
In classical potential theory, one can solve the Dirichlet problem on unbounded domains such as the upper half plane. These domains have two types of boundary points; the usual finite boundary points and another point at infinity. W. Woess…
This paper studies infinite graphs produced from a natural unfolding operation applied to finite graphs. Graphs produced via such operations are of finite degree and automatic over the unary alphabet (that is, they can be described by…
Several invariants of polarized metrized graphs and their applications in Arithmetic Geometry are studied recently. In this paper, we give fast algorithms to compute these invariants by expressing them in terms of the discrete Laplacian…
We give a combinatorial condition for the existence of efficient, LP-based FPT algorithms for a broad class of graph-theoretical optimisation problems. Our condition is based on the notion of biased graphs known from matroid theory.…
We design improved approximation algorithms for NP-hard graph problems by incorporating predictions (e.g., learned from past data). Our prediction model builds upon and extends the $\varepsilon$-prediction framework by Cohen-Addad, d'Orsi,…
We consider the problem of learning a sparse graph under the Laplacian constrained Gaussian graphical models. This problem can be formulated as a penalized maximum likelihood estimation of the Laplacian constrained precision matrix. Like in…