Mixing times for random walks on geometric random 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.

Physica A-statistical Mechanics and Its Applications, 2008

Dynamical scalings for the end-to-end distance Ree and the number of distinct visited nodes Nv of random walks (RWs) on finite scale-free networks (SFNs) are studied numerically. 〈Ree〉 shows the dynamical scaling behavior 〈Ree(ℓ¯,t)〉=ℓ¯α(γ,N)g(t/ℓ¯z), where ℓ¯ is the average minimum distance between all possible pairs of nodes in the network, N is the number of nodes, γ is the degree

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. .

IEEE/ACM Transactions on Networking, 2000

In this paper we present a methodology employing statistical analysis and stochastic geometry to study geometric routing schemes in wireless ad-hoc networks. In particular, we analyze the network layer performance of one such scheme, the random 1 2 disk routing scheme, which is a localized geometric routing scheme in which each node chooses the next relay randomly among the nodes within its transmission range and in the general direction of the destination. The techniques developed in this paper enable us to establish the asymptotic connectivity and the convergence results for the mean and variance of the routing path lengths generated by geometric routing schemes in random wireless networks. In particular, we approximate the progress of the routing path towards the destination by a Markov process and determine the sufficient conditions that ensure the asymptotic connectivity for both dense and large-scale ad-hoc networks deploying the random 1 2 disk routing scheme. Furthermore, using this Markov characterization, we show that the expected length (hop-count) of the path generated by the random 1 2 disk routing scheme normalized by the length of the path generated by the ideal direct-line routing, converges to 3π/4 asymptotically. Moreover, we show that the variance-to-mean ratio of the routing path length converges to 9π 2 /64 − 1 asymptotically. Through simulation, we show that the aforementioned asymptotic statistics are in fact quite accurate even for finite granularity and size of the network.

Physical Review E, 2003

Using both numerical simulations and scaling arguments, we study the behavior of a random walker on a one-dimensional small-world network. For the properties we study, we find that the random walk obeys a characteristic scaling form. These properties include the average number of distinct sites visited by the random walker, the mean-square displacement of the walker, and the distribution of first-return times. The scaling form has three characteristic time regimes. At short times, the walker does not see the small-world shortcuts and effectively probes an ordinary Euclidean network in d dimensions. At intermediate times, the properties of the walker shows scaling behavior characteristic of an infinite small-world network. Finally, at long times, the finite size of the network becomes important, and many of the properties of the walker saturate. We propose general analytical forms for the scaling properties in all three regimes, and show that these analytical forms are consistent with our numerical simulations.

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.

2009

We study the cover time of random geometric graphs. Let I(d) = [0, 1] d denote the unit torus in d dimensions. Let D(x, r) denote the ball (disc) of radius r. Let Υ d be the volume of the unit ball D(0, 1) in d dimensions. A random geometric graph G = G(d, r, n) in d dimensions is defined as follows: Sample n points V independently and uniformly at random from I(d). For each point x draw a ball D(x, r) of radius r about x. The vertex set V (G) = V and the edge set E(G) = {{v, w} : w = v, w ∈ D(v, r)}. Let G(d, r, n), d ≥ 3 be a random geometric graph. Let c > 1 be constant, and let r = (c log n/(Υ d n))

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.

Complex Networks and Their Applications VII, 2018

The betweenness centrality of graphs using random walk paths instead of geodesics is studied. A scaling collapse with no adjustable parameters is obtained as the graph size N is varied; the scaling curve depends on the graph model. A normalized random betweenness, that counts each walk passing through a node only once, is also defined. It is argued to be more useful and seen to have simpler scaling behavior. In particular, the probability for a random walk on a preferential attachment graph to pass through the root node is found to tend to unity as N → ∞.

Discrete Applied Mathematics, 2009

Let P be a Poisson process of intensity one in a square Sn of area n. We construct a random geometric graph G n,k by joining each point of P to its k nearest neighbours. For many applications it is desirable that G n,k is highly connected, that is, it remains connected even after the removal of a small number of its vertices. In this paper we relate the study of the s-connectivity of G n,k to our previous work on the connectivity of G n,k . Roughly speaking, we show that for s = o(log n), the threshold (in k) for s-connectivity is asymptotically the same as that for connectivity, so that, as we increase k, G n,k becomes s-connected very shortly after it becomes connected.

Geometric And Functional Analysis, 1998

Proceeedings of the Second European Workshop on Wireless Sensor Networks, 2005., 2005

In (l), the authors proposed the partial cover of a random walk on a broadcast network to be used to gather information and supported their proposal with experimental results. In this paper, we demonstrate analytically that for sufficiently large broadcast radius T, the partial cover of a random walk on a random broadcast network is in fact efficient and generates a good distribution of the visited nodes. Our result is based on bounding the conductance, which intuitively measures the amount of bottlenecks in a graph. We show that the conductance of a random broadcast network in a unit square is O(T) with high probability, and this bound allows us to analyze properties of the random walk such as mixing time and Ioad balancing. We find that for the random walk to be &amp;amp;#x27;both efficient and have a high quality cover and partial cover (Le. rapid mixing), radius at least Trapid = O(l/poly(logn)) is sufficient and necessary. Experimental results on the random geometric graphs, namely graphs that represent broadcast networks, that resemble the conductance of the 3-dimensional grid indicate that the analytical bounds on efficiency, namely cover time and partial cover time, are not tight. In particular, T = Q(l/~z~/~) is sufficient radius to obtain optimal cover time and partial cover time.

2008

Consider n points (or nodes) distributed uniformly and independently on the unit interval [0, 1]. Two nodes are said to be adjacent if their distance is less than some given threshold value. For the underlying random graph we derive zero-one laws for the property of graph connectivity and give the asymptotics of the transition widths for the associated phase transition. These results all flow from a single convergence statement for the probability of graph connectivity under a particular class of scalings. Given the importance of this result, we give two separate proofs; one approach relies on results concerning maximal spacings, while the other one exploits a Poisson convergence result for the number of breakpoint users.

Periodica Mathematica Hungarica, 2017

We consider two or more simple symmetric walks on Z d and the 2-dimensional comb lattice, and investigate the properties of the distance among the walkers.