Remarks on random walks on graphs and the Floyd boundary (original) (raw)
On the asymptotic spectrum of random walks on infinite families of graphs
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 ...
Random walks on graphs: ideas, techniques and results
Journal of Physics A: Mathematical and General, 2005
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 new ideas have been introduced, which can be fruitfully extended to different areas and disciplines. Here we aim at giving a brief but comprehensive perspective of these progresses, with a particular emphasis on physical aspects. Contents 1 Introduction 2 Mathematical description of graphs 3 The random walk problem 4 The generating functions 5 Random walks on finite graphs 6 Infinite graphs 7 Random walks on infinite graphs 8 Recurrence and transience: the type problem 9 The local spectral dimension 10 Averages on infinite graphs 11 The type problem on the average 1 12 The average spectral dimension 21 13 A survey of analytical results on specific networks 23 13.1 Renormalization techniques. .
A note on recurrent random walks on graphs
Journal of Statistical Physics, 1990
We consider random walks on polynomially growing graphs for which the resistances are also polynomially growing. In this setting we can show the same relation that was found earlier but that needed more complex conditions. The diffusion speed is determined by the geometric and resistance properties of the graph.
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.
Random walk on graphs with regular resistance and volume growth
Annales de l'Institut Henri Poincare (B) Probability and Statistics, 2008
In this paper characterizations of graphs satisfying heat kernel estimates for a wide class of space-time scaling functions are given. The equivalence of the two-sided heat kernel estimate and the parabolic Harnack inequality is also shown via the equivalence of the upper (lower) heat kernel estimate to the parabolic mean value (and super mean value) inequality.
Random Walks in a Dirichlet Environment
Electronic Journal of Probability, 2006
This paper states a law of large numbers for a random walk in a random iid environment on Z d , where the environment follows some Dirichlet distribution. Moreover, we give explicit bounds for the asymptotic velocity of the process and also an asymptotic expansion of this velocity at low disorder
Random walks on the finite components of
2005
The expected n-step return-probability EµP o [ ˆ Xn = o] of a random walk ˆ Xn with symmetric transition probabilities on a random partial graph of a regular graph G of degree δ with transitive automorphism group Aut(G) is considered. The law µ of the random edge-set is assumed to be stationary with respect to some transitive, unimodular subgroup Γ of Aut(G). By the spectral theory of finite random walks, using interlacing techniques, bounds in terms of functionals of the cluster size are obtained:
Boundaries and harmonic functions for random walks with random transition probabilities
2004
The usual random walk on a group (homogeneous both in time and in space) is determined by a probability measure on the group. In a random walk with random transition probabilities this single measure is replaced with a stationary sequence of measures, so that the resulting (random) Markov chains are still space homogeneous, but no longer time homogeneous. We study various notions of measure theoretical boundaries associated with this model and establish an analogue of the Poisson formula for (random) bounded harmonic functions. Under natural conditions on transition probabilities we identify these boundaries for several classes of groups with hyperbolic properties and prove the boundary triviality (i.e., the absence of non-constant random bounded harmonic functions) for groups of subexponential growth, in particular, for nilpotent groups.
The Poisson boundary of lamplighter random walks on trees
Geometriae Dedicata, 2007
Let T q be the homogeneous tree with degree q + 1 ≥ 3 and G a finitely generated group whose Cayley graph is T q . The associated lamplighter group is the wreath product Z r ≀ G, where Z r is the cyclic group of order r. For a large class of random walks on this group, we prove almost sure convergence to a natural geometric boundary. If the probability law governing the random walk has finite first moment, then the probability space formed by this geometric boundary together with the limit distribution of the random walk is proved to be maximal, that is, the Poisson boundary. We also prove that the Dirichlet problem at infinity is solvable for continuous functions on the active part of the boundary, if the lamplighter "operates at bounded range".
Random Walks Systems on Complete Graphs
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.
Two Results on Asymptotic Behaviour of Random Walks in Random Environment
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...
Random Walks on Dense Graphs and Graphons
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.
Random walks on the finite components of random partial
2005
The expected n-step return-probability of a random walk ˆ Xn with symmetric transition probabilities on a random partial graph of a regular graph G of degree δ with transitive automorphism group Aut(G) is considered. The law µ of the random edge-set is assumed to be stationary with respect to some transitive subgroup Γ of Aut(G). By the spectral theory of finite random walks, bounds in terms of the expected number of open clusters per vertex and moments of the cluster size are obtained:
High Dimensional Spectral Graph Theory and Non- backtracking Random Walks on Graphs
2015
This thesis has two primary areas of focus. First we study connection graphs, which are weighted graphs in which each edge is associated with a d-dimensional rotation matrix for some fixed dimension d, in addition to a scalar weight. Second, we study non-backtracking random walks on graphs, which are random walks with the additional constraint that they cannot return to the immediately previous state at any given step. Our work in connection graphs is centered on the notion of consistency, that is, the product of rotations moving from one vertex to another is independent of the path taken, and a generalization called epsilon-consistency. We present higher dimensional versions of the combinatorial Laplacian matrix and normalized Laplacian matrix from spectral graph theory, and give results characterizing the consistency of a connection graph in terms of the spectra of these matrices. We generalize several tools from classical spectral graph theory, such as PageRank and effective resi...
Asymptotics of the transition probabilities of the simple random walk on self-similar graphs
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.