Cover Time Of Random Walks Research Articles

Optimal Scale-Free Small-World Graphs with Minimum Scaling of Cover Time

The cover time of random walks on a graph has found wide practical applications in different fields of computer science, such as crawling and searching on the World Wide Web and query processing in sensor networks, with the application effects dependent on the behavior of the cover time: the smaller the cover time, the better the application performance. It was proved that over all graphs withNnodes, complete graphs have the minimum cover timeNlogN. However, complete graphs cannot mimic real-world networks with small average degree and scale-free small-world properties, for which the cover time has not been examined carefully, and its behavior is still not well understood. In this article, we first experimentally evaluate the cover time for various real-world networks with scale-free small-world properties, which scales asNlogN. To better understand the behavior of the cover time for real-world networks, we then study the cover time of three scale-free small-world model networks by using the connection between cover time and resistance diameter. For all the three networks, their cover time also behaves asNlogN. This work indicates that sparse networks with scale-free and small-world topology are favorable architectures with optimal scaling of cover time. Our results deepen understanding the behavior of cover time in real-world networks with scale-free small-world structure, and have potential implications in the design of efficient algorithms related to cover time.

Analytical results for the distribution of cover times of random walks on random regular graphs

We present analytical results for the distribution of cover times of random walks (RWs) on random regular graphs consisting of N nodes of degree c (c ⩾ 3). Starting from a random initial node at time t = 1, at each time step t ⩾ 2 an RW hops into a random neighbor of its previous node. In some of the time steps the RW may visit a new, yet-unvisited node, while in other time steps it may revisit a node that has already been visited before. The cover time T C is the number of time steps required for the RW to visit every single node in the network at least once. We derive a master equation for the distribution P t (S = s) of the number of distinct nodes s visited by an RW up to time t and solve it analytically. Inserting s = N we obtain the cumulative distribution of cover times, namely the probability P(T C ⩽ t) = P t (S = N) that up to time t an RW will visit all the N nodes in the network. Taking the large network limit, we show that P(T C ⩽ t) converges to a Gumbel distribution. We calculate the distribution of partial cover (PC) times P(T PC,k = t), which is the probability that at time t an RW will complete visiting k distinct nodes. We also calculate the distribution of random cover (RC) times P(T RC,k = t), which is the probability that at time t an RW will complete visiting all the nodes in a subgraph of k randomly pre-selected nodes at least once. The analytical results for the distributions of cover times are found to be in very good agreement with the results obtained from computer simulations.

Spectral Properties of Extended Sierpiński Graphs and Their Applications

The eigenvalues of a graph present a wide range of applications in structural and dynamical aspects of the graph. Determining and analyzing spectra of a graph has been an important and exciting research topic in recent years. In this paper, we study the spectra and their applications for extended Sierpinski graphs, which are closely related to WK-recursive networks that are widely used in the design and implementation of local area networks and parallel processing architectures. Moreover, a particular case of extended Sierpinski graphs is the dual of Apollonian network, which displays the prominent scale-free small-world characteristics as observed in various real networks. We derive recursive relations of the characteristic polynomials for extended Sierpinski graphs at two successive iterations, based on which we determine all the eigenvalues, their corresponding multiplicities and properties. We then use the obtained eigenvalues to evaluate the number of spanning trees, Kirchhoff index of extended Sierpinski graphs, as well as mean hitting time and cover time for random walks on the graphs.

Better Bounds for Coalescing-Branching Random Walks

Coalescing-branching random walks, or cobra walks for short, are a natural variant of random walks on graphs that can model the spread of disease through contacts or the spread of information in networks. In a k -cobra walk, at each timestep, a subset of the vertices are active; each active vertex chooses k random neighbors (sampled independently and uniformly with replacement) that become active at the next step, and these are the only active vertices at the next step. A natural quantity to study for cobra walks is the cover time, which corresponds to the expected time when all nodes have become infected or received the disseminated information. In this article, we extend previous results for cobra walks in multiple ways. We show that the cover time for the 2-cobra walk on [0, n ] d is O ( n ) (where the order notation hides constant factors that depend on d ); previous work had shown the cover time was O ( n ⋅polylog( n )). We show that the cover time for a 2-cobra walk on an n -vertex d -regular graph with conductance φ G is O ( d 4 φ −2 G log 2 n ), significantly generalizing a previous result that held only for expander graphs with sufficiently high expansion. And, finally, we show that the cover time for a 2-cobra walk on a graph with n vertices and m edges is always O ( mn 3/4 log n ; this is the first result showing that the bound of Θ( n 3 ) for the worst-case cover time for random walks can be beaten using 2-cobra walks.

Cover time for random walks on arbitrary complex networks.

We present an analytical method for computing the mean cover time of a discrete-time random walk process on arbitrary, complex networks. The cover time is defined as the time a random walker requires to visit every node in the network at least once. This quantity is particularly important for random search processes and target localization on network structures. Based on the global mean first-passage time of target nodes, we derive a method for computing the cumulative distribution function of the cover time based on first-passage time statistics. Our method is viable for networks on which random walks equilibrate quickly. We show that it can be applied successfully to various model and real-world networks. Our results reveal an intimate link between first-passage and cover time statistics and offer a computationally efficient way for estimating cover times in network-related applications.

The cover times of random walks on random uniform hypergraphs

Random walks in graphs have been applied to various network exploration and network maintenance problems. In some applications, however, it may be more natural, and more accurate, to model the underlying network not as a graph but as a hypergraph, and solutions based on random walks require a notion of random walks in hypergraphs. At each step, a random walk on a hypergraph moves from its current position v to a random vertex in a randomly selected hyperedge containing v. We consider two definitions of cover time of a hypergraph H. If the walk sees only the vertices it moves between, then the usual definition of cover time, C(H), applies. If the walk sees the complete edge during the transition, then an alternative definition of cover time, the inform time I(H) is used. The notion of inform time models passive listening which fits the following types of situations. The particle is a rumour passing between friends, which is overheard by other friends present in the group at the same time. The particle is a message transmitted randomly from location to location by a directional transmission in an ad-hoc network, but all receivers within the transmission range can hear.In this paper we give an expression for C(H) which is tractable for many classes of hypergraphs, and calculate C(H) and I(H) exactly for random r-regular, s-uniform hypergraphs. We find that for such hypergraphs, whp, C(H)/I(H)∼s(r−1)/r, if rs=O((loglogn)1−ϵ). For random r-regular, s-uniform multi-hypergraphs, constant r≥2, and 3≤s≤O(nϵ), we also prove that, whp, I(H)=O((n/s)logn), i.e. the inform time decreases directly with the edge size s.

Random Walks and Green's Function on Digraphs: A Framework for Estimating Wireless Transmission Costs

Various applications in wireless networks, such as routing and query processing, can be formulated as random walks on graphs. Many results have been obtained for such applications by utilizing the theory of random walks (or spectral graph theory), which is mostly developed for undirected graphs. However, this formalism neglects the fact that the underlying (wireless) networks in practice contain asymmetric links, which are best characterized by directed graphs (digraphs). Therefore, random walk on digraphs is a more appropriate model to consider for such networks. In this paper, by generalizing the random walk theory (or spectral graph theory) that has been primarily developed for undirected graphs to digraphs, we show how various transmission costs in wireless networks can be formulated in terms of hitting times and cover times of random walks on digraphs. Using these results, we develop a unified theoretical framework for estimating various transmission costs in wireless networks. Our framework can be applied to random walk query processing strategy and the three routing paradigms—best path routing, opportunistic routing, and stateless routing—to which nearly all existing routing protocols belong. Extensive simulations demonstrate that the proposed digraph-based analytical model can achieve more accurate transmission cost estimation over existing methods.

Probabilistic Self-Stabilization and Biased Random Walks on Dynamic Graphs

A distributed system is said to be probabilistic self-stabilizing, if it eventually converges to a legitimate computation with probability 1, starting from any global configuration. Like a self-stabilizing system, a probabilistic self-stabilizing system tolerates any number of transient failures and recovers a legitimate computation, but only probabilistically, unlike a self-stabilizing system, which recovers it deterministically even in the worst case. However, a self-stabilizing algorithm is in general difficult to design and even impossible for some problems. A probabilistic self-stabilizing, on the other hand, is easier to design. To see this, we discuss how a probabilistic self-stabilizing system is constructible from a given weak stabilizing system, which can recover a legitimate computation only in the best case. An execution of a probabilistic self-stabilizing system can be modeled by a random walk on a graph, and its performance can be evaluated in terms of some quantities, e.g., the hitting and the cover times, of the corresponding random walk. The hitting and the cover times of random walks have been studied extensively, but most of them consider standard (i.e., unbiased) random walks on static graphs. We discuss how to design biased random walks whose hitting and cover times are faster than standard random walks, to improve the performance of probabilistic self-stabilizing system. We also discuss random walks on dynamic graphs to analyze a probabilistic self-stabilizing system such that its communication network topology frequently changes.

Stationary distribution and cover time of random walks on random digraphs

We study properties of a simple random walk on the random digraph Dn,p when np=dlogn, d>1.We prove that whp the value πv of the stationary distribution at vertex v is asymptotic to deg−(v)/m where deg−(v) is the in-degree of v and m=n(n−1)p is the expected number of edges of Dn,p. If d=d(n)→∞ with n, the stationary distribution is asymptotically uniform whp.Using this result we prove that, for d>1, whp the cover time of Dn,p is asymptotic to dlog(d/(d−1))nlogn. If d=d(n)→∞ with n, then the cover time is asymptotic to nlogn.

Combinatorial Proof of the Identity for Cover Times on Finite Graphs

We give a combinatorial proof and an extension of the identity for the probability density of the cover time of a random walk on a finite graph obtained in [10] by considering it as the Mobius inversion formula. In addition, we obtain a similar identity for the cover time of multiple random walks.

Expander properties and the cover time of random intersection graphs

We investigate important combinatorial and algorithmic properties of G n , m , p random intersection graphs. In particular, we prove that with high probability (a) random intersection graphs are expanders, (b) random walks on such graphs are “rapidly mixing” (in particular they mix in logarithmic time) and (c) the cover time of random walks on such graphs is optimal (i.e. it is Θ ( n log n ) ). All results are proved for p very close to the connectivity threshold and for the interesting, non-trivial range where random intersection graphs differ from classical G n , p random graphs.

The hitting and cover times of random walks on finite graphs using local degree information

Standard random walks on finite graphs select the vertex visited next to the adjacent vertices at random with the same probability. Despite not using any global topological information, they guarantee O ( n 3 ) hitting and cover times for any graph, where n is the order of the graph. Motivated by network protocol applications, this paper investigates the impact of local topological information on designing “better” random walks. We first show that (a) for any transition probability matrix, the hitting (and hence the cover) time of a path graph is Ω ( n 2 ) . We next investigate for any graph G = ( V , E ) a transition probability matrix P = ( p ( u , v ) ) u , v ∈ V defined by p ( u , v ) = { deg − 1 / 2 ( v ) ∑ w ∈ N ( u ) deg − 1 / 2 ( w ) if v ∈ N ( u ) , 0 otherwise , where N ( u ) and deg ( u ) are respectively the set of adjacent vertices of u and the u ’s degree. Random walks obeying this transition probability matrix are shown to guarantee the following: For any graph, (b) the hitting time is O ( n 2 ) , and (c) the cover time is O ( n 2 log n ) . Facts (a) and (b) show that the degree information on the adjacent vertices is powerful enough for random walks to achieve the optimum hitting time.

On the Cover Time for Random Walks on Random Graphs

The cover time, C, for a simple random walk on a realization, GN, of G(N, p), the random graph on N vertices where each two vertices have an edge between them with probability p independently, is studied. The parameter p is allowed to decrease with N and p is written on the form f(N)/N, where it is assumed that f(N)≥c log N for some c>1 to asymptotically ensure connectedness of the graph. It is shown that if f(N) is of higher order than log N, then, with probability 1−o(1), (1−e)N log N≤E[C∣GN] ≤(1+e)N log N for any fixed e>0, whereas if f(N)=O(log N), there exists a constant a>0 such that, with probability 1−o(1), E[C∣GN] ≥(1+a)N log N. It is furthermore shown that if f(N) is of higher order than (log N)3 then Var(C∣GN)= o((N log N)2) with probability 1−o(1), so that with probability 1−o(1), the stronger statement that (1−e)N log N≤C≤(1+e)N log N holds.

Collecting coupons on trees, and the cover time of random walks

We consider the cover timeE u [G], the expected time it takes a random walk that starts at vertexu to visit alln vertices of a connected graphG. Aleliunas et al introduced the spanning tree argument: for any spanning treeT of the graphG, E u [G]≤W[T], whereW[T] is the sum of commute times along the edges ofT. By refining the spanning tree argument we obtain: $$E_u [G] \leqslant \frac{1}{2}(\mathop {\min }\limits_T [W[T]] + \mathop {\max }\limits_{\upsilon \in G} [H[u,\upsilon ] - H[\upsilon ,u]])$$

A tight lower bound on the cover time for random walks on graphs

AbstractWe prove that the expected time for a random walk to cover all n vertices of a graph is at least (1 + o(1))n In n.

A tight lower bound on the cover time for random walks on graphs

We prove that the expected time for a random walk to visit all n vertices of a connected graph is at most 4/27n3 + o(n3). 1995 Wiley Periodicals, Inc.

A tight upper bound on the cover time for random walks on graphs

AbstractWe prove that the expected time for a random walk to visit all n vertices of a connected graph is at most 4/27n3 + o(n3).

Expected hitting and cover times of random walks on some special graphs

AbstractWe give general bounds (and in some cases exact values) for the expected hitting and cover times of the simple random walk on some special undirected connected graphs using symmetry and properties of electrical networks. In particular we give easy proofs for an N–1HN‐1 lower bound and an N2 upper bound for the cover time of symmetric graphs and for the fact that the cover time of the unit cube is Φ(NlogN). We giver a counterexample to a conjecture of Freidland about a general bound for hitting times. Using the electric approach, we provide some genral upper and lower bounds for the expected cover times in terms of the diameter of the graph. These bounds are tight in many instances, particularly when the graph is a tree. © 1994 John Wiley & Sons, Inc.

Expected cover times of random walks on symmetric graphs

We give very simple proofs for an (N−1)HN−1 lower bound and anN2 upper bound for the expected cover time of symmetric graphs.

On the cover time of random walks on graphs

This article deals with random walks on arbitrary graphs. We consider the cover time of finite graphs. That is, we study the expected time needed for a random walk on a finite graph to visit every vertex at least once. We establish an upper bound ofO(n 2) for the expectation of the cover time for regular (or nearly regular) graphs. We prove a lower bound of Ω(n logn) for the expected cover time for trees. We present examples showing all our bounds to be tight.