Related papers: Random Walks on Small World Networks

We explore some of the connections between the local picture left by the trace of simple random walk on a discrete cylinder with base a d-dimensional torus, d at least 2, of side-length N running for times of order N^{2d} and the model of…

Random walks find applications in many areas of science and are the heart of essential network analytic tools. When defined on temporal networks, even basic random walk models may exhibit a rich spectrum of behaviours, due to the…

Let $X$ be a lazy random walk on a graph $G$. If $G$ is undirected, then the mixing time is upper bounded by the maximum hitting time of the graph. This fails for directed chains, as the biased random walk on the cycle $\mathbb{Z}_n$ shows.…

Random walks on discrete lattices are fundamental models that form the basis for our understanding of transport and diffusion processes. For a single random walker on complex networks, many properties such as the mean first passage time and…

Random walks constitute a fundamental mechanism for a large set of dynamics taking place on networks. In this article, we study random walks on weighted networks with an arbitrary degree distribution, where the weight of an edge between two…

For $d\ge 3$ we construct a new coupling of the trace left by a random walk on a large $d$-dimensional discrete torus with the random interlacements on $\mathbb Z^d$. This coupling has the advantage of working up to macroscopic subsets of…

In this study we have carried out computer simulations of random walks on Watts-Strogatz-type small world networks and measured the mean number of visited sites and the return probabilities. These quantities were found to obey scaling…

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 propose an approximation for the first return time distribution of random walks on undirected networks. We combine a message-passing solution with a mean-field approximation, to account for the short- and long-term behaviours…

We apply the power-of-two-choices paradigm to a random walk on a graph: rather than moving to a uniform random neighbour at each step, a controller is allowed to choose from two independent uniform random neighbours. We prove that this…

In this paper, we propose a new Markov chain which generalizes random-to-random shuffling on permutations to random-to-random shuffling on linear extensions of a finite poset of size $n$. We conjecture that the second largest eigenvalue of…

For a random walk $S_n, n\geq 0$ in $\mathbb{Z}^d$, let $l(n,x)$ be its local time at the site $x\in \mathbb{Z}^d$. Define the $\alpha$-fold self intersection local time $L_n(\alpha) := \sum_{x} l(n,x)^{\alpha}$, and let…

Random walks are basic diffusion processes on networks and have applications in, for example, searching, navigation, ranking, and community detection. Recent recognition of the importance of temporal aspects on networks spurred studies of…

The one-dimensional elephant random walk is a typical model of discrete-time random walk with step-reinforcement, and is introduced by Sch\"{u}tz and Trimper (2004). It has a parameter $\alpha \in (-1,1)$: The case $\alpha=0$ corresponds to…

We study the entropy of the set traced by an $n$-step random walk on $\Z^d$. We show that for $d \geq 3$, the entropy is of order $n$. For $d = 2$, the entropy is of order $n/\log^2 n$. These values are essentially governed by the size of…

This work focuses on the quantum mixing time, which is crucial for efficient quantum sampling and algorithm performance. We extend Richter's previous analysis of continuous time quantum walks on the periodic lattice $\mathbb{Z}_{n_1}\times…

In this paper we study random walks on dynamical random environments in $1 + 1$ dimensions. Assuming that the environment is invariant under space-time shifts and fulfills a mild mixing hypothesis, we establish a law of large numbers and a…

A significant generalization of the Erd\"os-R\'enyi random graph model is an `inhomogeneous' random graph where the edge probabilities vary according to vertex types. We identify the threshold value for this random graph with a finite…

The rotor-router model is a deterministic process analogous to a simple random walk on a graph. This paper is concerned with a generalized model, functional-router model, which imitates a Markov chain possibly containing irrational…

We study a general class of random walks driven by a uniquely ergodic Markovian environment. Under a coupling condition on the environment we obtain strong ergodicity properties for the environment as seen from the position of the walker,…