Related papers: About the distance between random walkers on some …
The number of walks from one vertex to another in a finite graph can be counted by the adjacency matrix. In this paper, we prove two theorems that connect the graph Laplacian with two types of walks in a graph. By defining two types of…
Random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology. In recent years, relevant mathematical results have been obtained in this field, and…
This paper concerns a random walk that moves on the integer lattice and has zero mean and a finite variance. We obtain first an asymptotic estimate of the transition probability of the walk absorbed at the origin, and then, using the…
Motivated by a problem arising from pharmaceutical science [B. Baeumer et al., Discr. Contin. Dyn. Sys. B 12], we study random walks on the contact graph of a bidisperse random sphere packing. For a random walk on the unweighted graph that…
Random walks on regular bounded degree expander graphs have numerous applications. A key property of these walks is that they converge rapidly to the uniform distribution on the vertices. The recent study of expansion of high dimensional…
Quantum walks have emerged as an interesting alternative to the usual circuit model for quantum computing. While still universal for quantum computing, the quantum walk model has very different physical requirements, which lends itself more…
We survey recent results on some one- and two-dimensional patterns generated by random permutations of natural numbers. In the first part, we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time…
In this note, we try to analyze and clarify the intriguing interplay between some counting problems related to specific thermalized weighted graphs and random walks consistent with such graphs.
We address the question of symmetries of an important type of quantum walks. We introduce all the necessary definitions and provide a rigorous formulation of the problem. Using a thorough analysis, we reach the complete answer by presenting…
The range, local times, and periodicity of symmetric, weakly asymmetric and asymmetric random walks at the time of exit from a strip with $N$ locations are considered. Several results on asymptotic distributions are obtained.
We study graph-theoretic properties of the trace of a random walk on a random graph. We show that for any $\varepsilon>0$ there exists $C>1$ such that the trace of the simple random walk of length $(1+\varepsilon)n\ln{n}$ on the random…
We study the properties of discrete-time random walks on networks formed by randomly interconnected cliques, namely, random networks of cliques. Our purpose is to derive the parameters that define the network structure -- specifically, the…
Several interesting approaches have been reported in the literature on complex networks, random walks, and hierarchy of graphs. While many of these works perform random walks on stable, fixed networks, in the present work we address the…
By assigning a probability measure via the spectrum of the normalized Laplacian to each graph and using L^p Wasserstein distances between probability measures, we define the corresponding spectral distances d_p on the set of all graphs.…
We consider two notions describing how one finite graph may be larger than another. Using them, we prove several theorems for such pairs that compare the number of spanning trees, the return probabilities of random walks, and the number of…
We establish recurrence criteria for sums of independent random variables which take values in Euclidean lattices of varying dimension. In particular, we describe transient inhomogenous random walks in the plane which interlace two…
We study numerically the behavior of continuous-time quantum walks over networks which are topologically equivalent to square lattices. On short time scales, when placing the initial excitation at a corner of the network, we observe a fast,…
In this article, we develop a theory for understanding the traces left by a random walk in the vicinity of a randomly chosen reference vertex. The analysis is related to interlacements but goes beyond previous research by showing weak limit…
We consider transient nearest neighbor random walks on the positive part of the real line. We give criteria for the finiteness of the number of cutpoints and strong cutpoints. Examples and open problems are presented.
We introduce the concept of distance ideals of graphs, which can be regarded as a generalization of the Smith normal form and the spectra of the distance matrix of a graph. We obtain a classification of the graphs with at most one trivial…