A note on recurrent random walks on graphs

2005

In this part we shall explore the tight relation between (simple) random walks and electric networks. This will provide us with an exact criteria for recurrency and with the monotonicity law which states that a subgraph of a recurrent graph is recurrent. This part is based on Doyle and Snell [3], which is a much longer, more detailed, but easy, read.

Springer Series in Synergetics, 2011

The present work aims to contribute to the study of networks by mapping the temporal evolution of the degree to a random walk in degree space. We analyzed how and when the degree approximates a pre-established value through a parallel with the first-passage problem of random walks. The mean time for the first-passage was calculated for the dynamical versions the Watts-Strogatz and Erdős-Rényi models. We also analyzed the degree variance for the random recursive tree and Barabási-Albert models.

Geometric And Functional Analysis, 1998

Revista Matemática Iberoamericana, 2000

This paper studies the on-and off-diagonal upper estimate and the two-sided transition probability estimate of random walks on weighted graphs.

2021

We show that for a uniformly irreducible random walk on a graph, with bounded range, there is a Floyd function for which the random walk converges to its corresponding Floyd boundary. Moreover if we add the assumptions, p(n)(v,w) ≤ Cρ, where ρ < 1 is the spectral radius, then for any Floyd function f that satisfies ∑∞ n=1 nf(n) < ∞, the Dirichlet problem with respect to the Floyd boundary is solvable.

Random walks and discrete potential …, 1999

Abstract. We observe that the spectral measure of the Markov operator depends continuously on the graph in the space of graphs with uniformly bounded degree. We investigate the behaviour of the largest eigenvalue and the density of eigenvalues for ...

The Annals of Probability, 2007

We show that the edges crossed by a random walk in a network form a recurrent graph a.s. In fact, the same is true when those edges are weighted by the number of crossings.

arXiv (Cornell University), 2002

It is shown explicitly how self-similar graphs can be obtained as 'blow-up' constructions of finite cell graphsĈ. This yields a larger family of graphs than the graphs obtained by discretising continuous self-similar fractals. For a class of symmetrically self-similar graphs we study the simple random walk on a cell graphĈ, starting in a vertex v of the boundary ofĈ. It is proved that the expected number of returns to v before hitting another vertex in the boundary coincides with the resistance scaling factor. Using techniques from complex rational iteration and singularity analysis for Green functions we compute the asymptotic behaviour of the n-step transition probabilities of the simple random walk on the whole graph. The results of Grabner and Woess for the Sierpiński graph are generalised to the class of symmetrically self-similar graphs and at the same time the error term of the asymptotic expression is improved. Finally we present a criterion for the occurrence of oscillating phenomena of the n-step transition probabilities.

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.

2005

A geometric random graph, G d (n, r), is formed as follows: place n nodes uniformly at random onto the surface of the d-dimensional unit torus and connect nodes which are within a distance r of each other. The G d (n, r) has been of great interest due to its success as a model for ad-hoc wireless networks. It is well known that the connectivity of G d (n, r) exhibits a threshold property: there exists a constant α d such that for any > 0, for r d < α d (1 −) log n/n the G d (n, r) is not connected with high probability 1 and for r d > α d (1 +) log n/n the G d (n, r) is connected w.h.p.. In this paper, we study mixing properties of random walks on G d (n, r) for r d (n) = ω(log n/n). Specifically, we study the scaling of mixing times of the fastest-mixing reversible random walk, and the natural random walk. We find that the mixing time of both of these random walks have the same scaling laws and scale proportional to r −2 (for all d). These results hold for G d (n, r) when distance is defined using any L p norm. Though the results of this paper are not so surprising, they are nontrivial and require new methods.

SIAM Journal on Applied Mathematics, 2021

Graph-limit theory focuses on the convergence of sequences of increasingly large graphs, providing a framework for the study of dynamical systems on massive graphs, where classical methods would become computationally intractable. Through an approximation procedure, the standard ordinary differential equations are replaced by nonlocal evolution equations on the unit interval. In this work, we adopt this methodology to prove the validity of the continuum limit of random walks, a largely studied model for diffusion on graphs. We focus on two classes of processes on dense weighted graphs, in discrete and in continuous time, whose dynamics are encoded in the transition matrix of the associated Markov chain, or in the random-walk Laplacian. We further show that previous works on the discrete heat equation, associated to the combinatorial Laplacian, fall within the scope of our approach. Finally, we characterize the relaxation time of the process in the continuum limit.

2016

Two results on Asymptotic Behaviour of Random Walks in Random Environment Jeremy Voltz Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2016 In the first chapter of this thesis, we consider a model of directed polymer in 1 + 1 dimensions in a product-type random environment ω(t, x) = b(t)F (x), where the fields F and b are i.i.d., with F (x) continuous, symmetric and bounded, and b(t) = ±1 with probabilty 1/2. Thus ω can be viewed as the field F oscillating in time. We consider directed last-passage percolation through this random field; namely, we investigate the behavior of the length n polymer path with maximal action, where the action of a path is simply the sum of the environment variables it moves through. We prove a law of large numbers for the maximal action of the path from the origin to a fixed endpoint (n, bαnc), and investigate the limiting shape function a(α). We prove that this shape function is non-linear, and has a corner at α = 0, thus i...

2000

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.