Random walk on graphs with regular resistance and volume growth

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.

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. .

Proceedings of the London Mathematical Society, 1992

We introduce an (r, /?)-net (0 < 2r < R) of a metric space M as a maximal graph whose vertices are elements in M of pairwise distance at least r such that any two vertices of distance at most R are adjacent. We show that, for a large class of metric spaces, including many Riemannian manifolds, the property of transience of a net and the property of the net carrying a non-constant harmonic function of bounded energy is independent of the choice of the net. We give a new necessary and sufficient condition for a graph with bounded degrees and satisfying an isoperimetric inequality to have no non-constant harmonic functions. For this purpose we develop equivalent analytic conditions for graphs satisfying an isoperimetric inequality. Some of these results have been discovered recently by others in more general settings, but our treatment here is specific and self-contained. We use graph transience to prove that Scherk's surface is hyperbolic, a problem posed by Osserman in 1965.

Transactions of the American Mathematical Society, 1984

The difference Laplacian on a square lattice in Rn has been studied by many authors. In this paper an analogous difference operator is studied for an arbitrary graph. It is shown that many properties of the Laplacian in the continuous setting (e.g. the maximum principle, the Harnack inequality, and Cheeger's bound for the lowest eigenvalue) hold for this difference operator. The difference Laplacian governs the random walk on a graph, just as the Laplace operator governs the Brownian motion. As an application of the theory of the difference Laplacian, it is shown that the random walk on a class of graphs is transient.

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 &lt; 1 is the spectral radius, then for any Floyd function f that satisfies ∑∞ n=1 nf(n) &lt; ∞, the Dirichlet problem with respect to the Floyd boundary is solvable.

2013

We study a family of random geometric graphs on hyperbolic spaces. In this setting, N points are chosen randomly on the hyperbolic plane of curvature −ζ 2 and any two of them are joined by an edge with probability that depends on their hyperbolic distance, independently of every other pair. In particular, when the positions of the points have been fixed, the distribution over the set of graphs on these points is the Boltzmann distribution, where the Hamiltonian is given by the sum of weighted indicator functions for each pair of points. The weight is proportional to a real parameter β > 0 (interpreted as the inverse temperature) as well as to the hyperbolic distance between the corresponding points. The N points are distributed according to a quasi-uniform distribution which is a distorted version of the uniform distribution and depends on a parameter α > 0-when α = ζ it coincides with the uniform distribution. This class of random graphs was proposed by Krioukov et al. [17] as a model for complex networks. We provide a rigorous analysis of aspects of this model for all values of the ratio ζ/α, focusing on its dependence on the parameter β. We show that a phase transition occurs around β = 1. More specifically, let us assume that 0 < ζ/α < 2. In this case, we show that when β > 1 the degree of a typical vertex is bounded in probability (in fact it follows a distribution which for large values exhibits a power-law tail whose exponent depends only on ζ and α), whereas for β < 1 the degree is a random variable whose expected value grows polynomially in N. When β = 1, we establish logarithmic growth. For the case β > 1, we establish a connection with a class of inhomogeneous random graphs known as the Chung-Lu model. We show that the probability that two given points are adjacent is expressed through the kernel of this inhomogeneous random graph. We also consider the case ζ/α ≥ 2.

Monatshefte f?r Mathematik, 2002

Consider the tesselation of the hyperbolic plane by m-gons, ℓ per vertex. In its 1-skeleton, we compute the growth series of vertices, geodesics, tuples of geodesics with common extremities. We also introduce and enumerate holly trees, a family of reduced loops in these graphs. We then apply Grigorchuk's result relating cogrowth and random walks to obtain lower estimates on the spectral radius of the Markov operator associated with a symmetric random walk on these graphs.

Electronic Journal of Probability, 2001

This paper proves upper and lower off-diagonal, sub-Gaussian transition probabilities estimates for strongly recurrent random walks under sufficient and necessary conditions. Several equivalent conditions are given showing their particular role influence on the connection between the sub-Gaussian estimates, parabolic and elliptic Harnack inequality.

In this work, we study a family of random geometric graphs on hyperbolic spaces. In this setting, N points are chosen randomly on a hyperbolic space and any two of them are joined by an edge with probability that depends on their hyperbolic distance, independently of every other pair. In particular, when the positions of the points have been fixed, the distribution over the set of graphs on these points is the Boltzmann distribution, where the Hamiltonian is given by the sum of weighted indicator functions for each pair of points, with the weight being proportional to a real parameter \beta&gt;0 (interpreted as the inverse temperature) as well as to the hyperbolic distance between the corresponding points. This class of random graphs was introduced by Krioukov et al. We provide a rigorous analysis of aspects of this model and its dependence on the parameter \beta, verifying some of their observations. We show that a phase transition occurs around \beta =1. More specifically, we show...

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 ...

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...

Performance Evaluation Review, 2012

In this paper we study the behavior of a continuous time random walk (CTRW) on a time varying dynamic graph. We establish conditions under which the CTRW is a stationary and ergodic process. In general, the stationary distribution of the walker depends on the walker rate and is difficult to characterize. However, we characterize the stationary distribution in the following cases: i) the walker rate is significantly larger or smaller than the rate in which the graph changes (time-scale separation), ii) the walker rate is proportional to the degree of the node that it resides on (coupled dynamics), and iii) the degrees of vertices belonging to the same connected component are identical (structural constraints). We provide numerical results of examples that illustrate our theoretical findings and other peculiarities, as well as two applications.

Physical Review E, 2009

We study the random walk problem on a class of deterministic Scale-Free networks displaying a degree sequence for hubs scaling as a power law with an exponent γ = log 3/ log 2. We find exact results concerning different first-passage phenomena and, in particular, we calculate the probability of first return to the main hub. These results allow to derive the exact analytic expression for the mean time to first reach the main hub, whose leading behavior is given by τ ∼ V 1−1/γ , where V denotes the size of the structure, and the mean is over a set of starting points distributed uniformly over all the other sites of the graph. Interestingly, the process turns out to be particularly efficient. We also discuss the thermodynamic limit of the structure and some local topological properties.

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.

DOKTORSAVHANDLINGAR-CHALMERS TEKNISKA …, 2001