Random Walks on Dense Graphs and Graphons

Journal of Theoretical Probability - J THEOR PROBABILITY, 2001

We consider random walks with small fixed steps inside of edges of a graph <img src="/fulltext-image.asp?format=htmlnonpaginated&src=R47027122T5P5217_html\10959_2004_Article_300626_TeX2GIFIE1.gif" border="0" alt=" $${\mathcal{G}}$$ " />, prescribing a natural rule of probabilities of jumps over a vertex. We show that after an appropriate rescaling such random walks weakly converge to the natural Brownian motion on <img src="/fulltext-image.asp?format=htmlnonpaginated&src=R47027122T5P5217_html\10959_2004_Article_300626_TeX2GIFIE2.gif" border="0" alt=" $${\mathcal{G}}$$ " /> constructed in Ref. 1.

Mechanics Research Communications, 2012

Preliminary but interesting and definite results are given on the application of graph theory concepts (random walk on graphs) to the double diffusivity theory proposed by Aifantis in the late 70s to model transport in media with high diffusivity paths such as metal polycrystals with a continuous distribution of grain boundaries possessing much higher diffusivity than the bulk, as well as in nanopolycrystals for which it has been shown recently that the double diffusivity model fits experimental observations. The new information provided by employing the graph theory tool is concerned with certain restrictions and relations that the phenomenological coefficients, entering in the coupled partial differential equations of double diffusivity, should satisfy depending on the topology and related details of the graph model adopted. © 2012 Elsevier Ltd. All rights reserved.

Physical Review E, 2019

We study the rare fluctuations or large deviations of time-integrated functionals or observables of an unbiased random walk evolving on Erdös-Rényi random graphs, and construct a modified, biased random walk that explains how these fluctuations arise in the long-time limit. Two observables are considered: the sum of the degrees visited by the random walk and the sum of their logarithm, related to the trajectory entropy. The modified random walk is used for both quantities to explain how sudden changes in degree fluctuations, similar to dynamical phase transitions, are related to localization transitions. For the second quantity, we also establish links between the large deviations of the trajectory entropy and the maximum entropy random walk.

Journal of Mathematical Analysis and Applications, 2022

Several variants of the graph Laplacian have been introduced to model non-local diffusion processes, which allow a random walker to "jump" to non-neighborhood nodes, most notably the transformed path graph Laplacians and the fractional graph Laplacian. From a rigorous point of view, this new dynamics is made possible by having replaced the original graph G with a weighted complete graph G on the same node-set, that depends on G and wherein the presence of new edges allows a direct passage between nodes that were not neighbors in G. We show that, in general, the graph G is not compatible with the dynamics characterizing the original model graph G: the random walks on G subjected to move on the edges of G are not stochastically equivalent, in the wide sense, to the random walks on G. From a purely analytical point of view, the incompatibility of G with G means that the normalized graph Ĝ can not be embedded into the normalized graph Ĝ . Eventually, we provide a regularization method to guarantee such compatibility and preserving at the same time all the nice properties granted by G .

Bulletin of the Brazilian …, 2006

We study two versions of random walks systems on complete graphs. In the first one, the random walks have geometrically distributed lifetimes so we define and identify a non-trivial critical parameter related to the proportion of visited vertices before the process dies out. In the second version, the lifetimes depend on the past of the process in a non-Markovian setup. For that version, we present results obtained from computational analysis, simulations and a mean field approximation. These three approaches match.

Geometric And Functional Analysis, 1998

Annales Henri Poincaré, 2012

We show that fast diffusions on finite graphs with semi permeable membranes on vertices may be approximated by finite-state Markov chains provided the related permeability coefficients are appropriately small. The convergence theorem involves a singular perturbation with singularity in both operator and boundary/transmission conditions, and the related semigroups of operators converge in an irregular manner. The result is motivated by recent models of synaptic depression.

Journal of Statistical Mechanics: Theory and Experiment, 2018

In this paper, we explore different Markovian random walk strategies on networks with transition probabilities between nodes defined in terms of functions of the Laplacian matrix. We generalize random walk strategies with local information in the Laplacian matrix, that describes the connections of a network, to a dynamics determined by functions of this matrix. The resulting processes are non-local allowing transitions of the random walker from one node to nodes beyond its nearest neighbors. We find that only two types of Laplacian functions are admissible with distinct behaviors for long-range steps in the infinite network limit: type (i) functions generate Brownian motions, type (ii) functions Lévy flights. For this asymptotic long-range step behavior only the lowest non-vanishing order of the Laplacian function is relevant, namely first order for type (i), and fractional order for type (ii) functions. In the first part, we discuss spectral properties of the Laplacian matrix and a series of relations that are maintained by a particular type of functions that allow to define random walks on any type of undirected connected networks. Once described general properties, we explore characteristics of random walk strategies that emerge from particular cases with functions defined in terms of exponentials, logarithms and powers of the Laplacian as well as relations of these dynamics with non-local strategies like Lévy flights and fractional transport. Finally, we analyze the global capacity of these random walk strategies to explore networks like lattices and trees and different types of random and complex networks.

Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, 2009

The aim of this article is to discuss some of the notions and applications of random walks on finite graphs, especially as they apply to random graphs. In this section we give some basic definitions, in Section 2 we review applications of random walks in computer science, and in Section 3 we focus on walks in random graphs. Given a graph G = (V, E), let d G (v) denote the degree of vertex v for all v ∈ V. The random walk W v = (W v (t), t = 0, 1,. . .) is defined as follows: W v (0) = v and given x = W v (t), W v (t + 1) is a randomly chosen neighbour of x. When one thinks of a random walk, one often thinks of Polya's Classical result for a walk on the d-dimensional lattice Z d , d ≥ 1. In this graph two vertices x = (x 1 , x 2 ,. .. , x d) and y = (y 1 , y 2 ,. .. , y d) are adjacent iff there is an index i such that (i) x j = y j for j = i and (ii) |x i − y i | = 1. Polya [33] showed that if d ≤ 2 then a walk starting at the origin returns to the origin with probability 1 and that if d ≥ 3 then it returns with probability p(d) < 1. See also Doyle and Snell [22]. A random walk on a graph G defines a Markov chain on the vertices V. If G is a finite, connected and non-bipartite graph, then this chain has a stationary distribution π given by π v = d G (v)/(2|E|). Thus if P (t) v (w) = Pr(W v (t) = w), then lim t→∞ P (t) v (w) = π w , independent of the starting vertex v. In this paper we only consider finite graphs, and we will focus on two aspects of a random walk: The Mixing Time and the Cover Time.

2002

We study diffusions, variational principles and associated boundary value problems on directed graphs with natural weightings. Using random walks and exit times, we associate to certain subgraphs (domains) a pair of sequences, each of which is invariant under the action of the automorphism group of the underlying graph. We prove that these invariants differ by an explicit combinatorial factor given