Random walks on graphs with volume and time doubling

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. .

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.

Various aspects of the theory of random walks on graphs are surveyed. In particular, estimates on the important parameters of access time, commute time, cover time and mixing time are discussed. Connections with the eigenvalues of graphs and with electrical networks, and the use of these connections in the study of random walks is described. We also sketch recent algorithmic applications of random walks, in particular to the problem of sampling.

2012

A simple random walk on a graph is a sequence of movements from one vertex to another where at each step an edge is chosen uniformly at random from the set of edges incident on the current vertex, and then transitioned to next vertex. Central to this thesis is the cover time of the walk, that is, the expectation of the number of steps required to visit every vertex, maximised over all starting vertices. In our first contribution, we establish a relation between the cover times of a pair of graphs, and the cover time of their Cartesian product. This extends previous work on special cases of the Cartesian product, in particular, the square of a graph. We show that when one of the factors is in some sense larger than the other, its cover time dominates, and can become within a logarithmic factor of the cover time of the product as a whole. Our main theorem effectively gives conditions for when this holds. The techniques and lemmas we introduce may be of independent interest. In our sec...

Potential Analysis, 2003

In this paper some isoperimetric problems are studied, particularly the extremal property of the mean exit time of the random walk from finite sets. This isoperimetric problem is inserted into the set of equivalent conditions of the diagonal upper estimate of transition probability 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.

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.

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.

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 ...

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.

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.

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:

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.

Open Journal of Discrete Mathematics, 2016

We study the mixing rate of non-backtracking random walks on graphs by looking at non-backtracking walks as walks on the directed edges of a graph. A result known as Ihara's Theorem relates the adjacency matrix of a graph to a matrix related to non-backtracking walks on the directed edges. We prove a weighted version of Ihara's Theorem which relates the transition probability matrix of a non-backtracking walk to the transition matrix for the usual random walk. This allows us to determine the spectrum of the transition probability matrix of a non-backtracking random walk in the case of regular graphs and biregular graphs. As a corollary, we obtain a result of Alon et. al. in [1] that in most cases, a nonbacktracking random walk on a regular graph has a faster mixing rate than the usual random walk. In addition, we obtain an analogous result for biregular 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.