Related papers: High-ordered Random Walks and Generalized Laplacia…
The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs and natural Laplacians acting on it and…
We present an elementary way to transform an expander graph into a simplicial complex where all high order random walks have a constant spectral gap, i.e., they converge rapidly to the stationary distribution. As an upshot, we obtain new…
In this paper we address the problem of understanding the success of algorithms that organize patches according to graph-based metrics. Algorithms that analyze patches extracted from images or time series have led to state-of-the art…
Analyzing the mixing time of random walks is a well-studied problem with applications in random sampling and more recently in graph partitioning. In this work, we present new analysis of random walks and evolving sets using more…
In this paper, we introduce random walks with absorbing states on simplicial complexes. Given a simplicial complex of dimension $d$, a random walk with an absorbing state is defined which relates to the spectrum of the $k$-dimensional…
We introduce and study Laplacians on a finite metric graph endowed with generalized densities, that is, measures of finite mass. One important motivation is that this setting provides a common framework for several interesting classes of…
Graph Neural Networks (GNNs) are a popular approach for predicting graph structured data. As GNNs tightly entangle the input graph into the neural network structure, common explainable AI approaches are not applicable. To a large extent,…
Determining and analyzing the spectra of graphs is an important and exciting research topic in theoretical computer science. The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on…
Transformers were originally proposed as a sequence-to-sequence model for text but have become vital for a wide range of modalities, including images, audio, video, and undirected graphs. However, transformers for directed graphs are a…
This paper deals with spectral graph theory issues related to questions of monotonicity and comparison of eigenvalues. We consider finite directed graphs with non symmetric edge weights and we introduce a special self-adjoint operator as…
The motivation of this work is to extend the techniques of higher order random walks on simplicial complexes to analyze mixing times of Markov chains for combinatorial problems. Our main result is a sharp upper bound on the second…
Graph convolutional networks(GCNs) have become the most popular approaches for graph data in these days because of their powerful ability to extract features from graph. GCNs approaches are divided into two categories, spectral-based and…
Graph disaggregation is a technique used to address the high cost of computation for power law graphs on parallel processors. The few high-degree vertices are broken into multiple small-degree vertices, in order to allow for more efficient…
Node-level random walk has been widely used to improve Graph Neural Networks. However, there is limited attention to random walk on edge and, more generally, on $k$-simplices. This paper systematically analyzes how random walk on different…
Network topology is a flourishing interdisciplinary subject that is relevant for different disciplines including quantum gravity and brain research. The discrete topological objects that are investigated in network topology are simplicial…
We establish and generalise several bounds for various random walk quantities including the mixing time and the maximum hitting time. Unlike previous analyses, our derivations are based on rather intuitive notions of local expansion…
Consider the following iterated process on a hypergraph $H$. Each vertex $v$ has an initial vertex weight. At each step, we uniformly at random select an edge $F$ in $H$, and for each vertex $v$ in $F$ we replace the weight of $v$ by the…
In this paper, we analyze the eigenfunctions of the edge-based Laplacian on a graph and the relationship of these functions to random walks on the graph. We commence by discussing the set of eigenfunctions supported at the vertices, and…
Many regular graphs admit a natural partition of their edge set into cliques of the same order such that each vertex is contained in the same number of cliques. In this paper, we study the mixing rate of certain random walks on such graphs…
Graph Laplacians as well as related spectral inequalities and (co-)homology provide a foray into discrete analogues of Riemannian manifolds, providing a rich interplay between combinatorics, geometry and theoretical physics. We apply some…