Random Walks on the finite Components of random Partial Graphs of Transitive Graphs (original) (raw)
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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:
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:
Random walks on edge-transitive graphs (II
Statistics & Probability Letters, 1999
We give formulas, in terms of the number of pure k-cycles, for the expected hitting times between vertices at distances greater than 1 for random walks on edge-transitive graphs, extending our prior results for neighboring vertices and also extending results of Devroye-Sbihi and Biggs concerning distance-regular graphs. We apply these formulas to a class of Cayley graphs and give explicit values for the expected hitting times.
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 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. .
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.
Random Walks on Graphs: A Survey
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.
Component structure of the vacant set induced by a random walk on a random graph
Random Structures & Algorithms, 2013
We consider random walks on two classes of random graphs and explore the likely structure of the vacant set viz. the set of unvisited vertices. Let Γ(t) be the subgraph induced by the vacant set. We show that for random graphs G n,p (above the connectivity threshold) and for random regular graphs G r , r ≥ 3 there is a phase transition in the sense of the well-known Erdős-Renyi phase transition. Thus for t ≤ (1 − ǫ)t * we have a unique giant plus logarithmic size components and for t ≥ (1 + ǫ)t * we only have logarithmic sized components.
Journal of Theoretical Probability, 2010
A comparison technique for random walks on finite graphs is introduced, using the well-known interlacing method. It yields improved return probability bounds. A key feature is the incorporation of parts of the spectrum of the transition matrix other than just the principal eigenvalue. As an application, an upper bound of the expected return probability of a random walk with symmetric transition probabilities is found. In this case, the state space is a random partial graph of a regular graph of bounded geometry and transitive automorphism group. The law of the random edge-set is assumed to be stationary with respect to some transitive subgroup of the automorphism group ('invariant percolation'). Given that this subgroup is unimodular, it is shown that stationarity strengthens the upper bound of the expected return probability, compared with standard bounds derived from the Cheeger inequality.
On the stability of the behavior of random walks on groups
Journal of Geometric Analysis, 2000
We show that, for random walks on Cayley graphs, the long time behavior of the probability of return after 2n steps is invariant by quasi-isometry. 1. Introduction Let G be a finitely generated group. For any finite generating set S satisfying S = S-1 , consider the Cayley graph (G, S) with vertex set G and an edge from x to y if and only if y = xs for some s ~ S. Thus, edges are oriented but this is merely a convention since (x, y) is an edge if and only if (y, x) is an edge. We allow the identity element id to be in S in which case our graph has a loop at each vertex. Clearly the graph (G, S) is invariant under the left action of G. Denote by Ixl the distance from the neutral element id to x in the Cayley graph (G, S), that is, ]xl is the minimal number k of elements of S needed to write x as x = sis2 .. 9 sk, si ~ S. The volume growth function of (G, S) is defined by V(n) =#Ix ~ a : Ixl _< n]. This paper focuses on the probability of return after 2n steps of the simple random walk on (G, S). For a survey of this topic, see [36]. The simple random walk on (G, S) is the Markov process (Xi)~ c with values in G which evolves as follows: If the current state is x, the next state is a neighbor of x chosen uniformly at random. This implicitly defines a probability measure Ps on G r~ such that Ps (Xn = y/ Xo = x) = Iz (sn' (x-ly) where 1 Ixs(g) = ~-~ls(g) and/1 ~n) is the n-fold convolution power of/x. Following usual notation we will also write P~(.) = Ps('/Xo = x) for the law of the walk based on S and started at x e G. To avoid parity problems, we consider only the probability of return at even times and set 4)s(n) = P~ (XEn = id) =/x(s2n)(id) 9