Multiple cover time

Motivated by applications in Markov estimation and distributed computing, we define the blanket time of an undirected graph G to be the expected time for a random walk to hit every vertex of G within a constant factor of the number of times predicted by the stationary distribution. Thus the blanket time is, essentially, the number of steps required of a random walk in order that the observed distribution reflect the stationary distribution. We provide substantial evidence for the following conjecture: that the blanket time of a graph never exceeds the cover time by more than a constant factor. In other words, at the cost of a multiplicative constant one can hit every vertex often instead of merely once. We prove the conjecture in the case where the cover time and maximum hitting time differ by a logarithmic factor. This case includes almost all graphs, as well as most “natural” graphs: the hypercube, k-dimensional lattices for k ≥ 2, balanced k-ary trees, and expanders. We further prove the conjecture for perhaps the most natural graphs not falling in the above case: paths and cycles. Finally, we prove the conjecture in the case of independent stochastic processes. © 1996 John Wiley & Sons, Inc. Random Struct. Alg., 9 , 403–411 (1996)     

Component structure of the vacant set induced by a random walk on a random graph

We consider random walks on several classes of graphs and explore the likely structure of the vacant set, i.e. the set of unvisited vertices. Let Γ(t) be the subgraph induced by the vacant set of the walk at step t. We show that for random graphs G_n,p (above the connectivity threshold) and for random regular graphs G_r,r ≥ 3, the graph Γ(t) undergoes a phase transition in the sense of the well‐known ErdJW‐RSAT1100590x.png ‐Renyi phase transition. Thus for t ≤ (1 ‐ ε)t^*, there is a unique giant component, plus components of size O(log n), and for t ≥ (1 + ε)t^* all components are of size O(log n). For G_n,p and G_r we give the value of t^*, and the size of Γ(t). For G_r, we also give the degree sequence of Γ(t), the size of the giant component (if any) of Γ(t) and the number of tree components of Γ(t) of a given size k = O(log n). We also show that for random digraphs D_n,p above the strong connectivity threshold, there is a similar directed phase transition. Thus for t ≤ (1 ‐ ε)t^*, there is a unique strongly connected giant component, plus strongly connected components of size O(log n), and for t ≥ (1 + ε)t^* all strongly connected components are of size O(log n). © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Highly connected random geometric graphs

Let P be a Poisson process of intensity 1 in a square S_n 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(logn), 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.     

The cover time of the giant component of a random graph

We study the cover time of a random walk on the largest component of the random graph G_n,p. We determine its value up to a factor 1 + o(1) whenever np = c > 1, c = O(lnn). In particular, we show that the cover time is not monotone for c = Θ(lnn). We also determine the cover time of the k‐cores, k ≥ 2. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

The mixing time of the giant component of a random graph

We show that the total variation mixing time of the simple random walk on the giant component of supercritical and is . This statement was proved, independently, by Fountoulakis and Reed. Our proof follows from a structure result for these graphs which is interesting in its own right. We show that these graphs are “decorated expanders” — an expander glued to graphs whose size has constant expectation and exponential tail, and such that each vertex in the expander is glued to no more than a constant number of decorations. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 383–407, 2014     

The rank of random graphs

We show that almost surely the rank of the adjacency matrix of the Erd?s‐Rényi random graph G(n,p) equals the number of nonisolated vertices for any c ln n/n ≤ p ≤ 1/2, where c is an arbitrary positive constant larger than 1/2. In particular, the adjacency matrix of the giant component (a.s.) has full rank in this range. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Random walks which prefer unvisited edges: Exploring high girth even degree expanders in linear time

Let be a connected graph with vertices. A simple random walk on the vertex set of G is a process, which at each step moves from its current vertex position to a neighbouring vertex chosen uniformly at random. We consider a modified walk which, whenever possible, chooses an unvisited edge for the next transition; and makes a simple random walk otherwise. We call such a walk an edge‐process (or E‐process). The rule used to choose among unvisited edges at any step has no effect on our analysis. One possible method is to choose an unvisited edge uniformly at random, but we impose no such restriction. For the class of connected even degree graphs of constant maximum degree, we bound the vertex cover time of the E‐process in terms of the edge expansion rate of the graph G, as measured by eigenvalue gap of the transition matrix of a simple random walk on G. A vertex v is ?‐good, if any even degree subgraph containing all edges incident with v contains at least ? vertices. A graph G is ?‐good, if every vertex has the ?‐good property. Let G be an even degree ?‐good expander of bounded maximum degree. Any E‐process on G has vertex cover time This is to be compared with the lower bound on the cover time of any connected graph by a weighted random walk. Our result is independent of the rule used to select the order of the unvisited edges, which could, for example, be chosen on‐line by an adversary. As no walk based process can cover an n vertex graph in less than n – 1 steps, the cover time of the E‐process is of optimal order when . With high probability random r‐regular graphs, even, have . Thus the vertex cover time of the E‐process on such graphs is . © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 36–54, 2015     

Critical window for the vacant set left by random walk on random regular graphs

We consider the simple random walk on a random d ‐regular graph with n vertices, and investigate percolative properties of the set of vertices not visited by the walk until time \begin{align*}\left\lfloor un \right\rfloor\end{align*}, where u > 0 is a fixed positive parameter. It was shown in ?erný et al., (Ann Inst Henri Poincaré Probab Stat 47 (2011) 929–968) that this so‐called vacant set exhibits a phase transition at u = u_?: there is a giant component if u < u_? and only small components when u > u_?. In this paper we show the existence of a critical window of size n^‐1/3 around u_?. In this window the size of the largest cluster is of order n^2/3. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

On k‐connectivity for a geometric random graph

For n points uniformly randomly distributed on the unit cube in d dimensions, with d≥2, let ρ_n (respectively, σ_n) denote the minimum r at which the graph, obtained by adding an edge between each pair of points distant at most r apart, is k‐connected (respectively, has minimum degree k). Then P[ρ_n=σ_n]→1 as n→∞. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 15, 145–164, 1999     

Cover time and mixing time of random walks on dynamic graphs

The application of simple random walks on graphs is a powerful tool that is useful in many algorithmic settings such as network exploration, sampling, information spreading, and distributed computing. This is due to the reliance of a simple random walk on only local data, its negligible memory requirements, and its distributed nature. It is well known that for static graphs the cover time, that is, the expected time to visit every node of the graph, and the mixing time, that is, the time to sample a node according to the stationary distribution, are at most polynomial relative to the size of the graph. Motivated by real world networks, such as peer‐to‐peer and wireless networks, the conference version of this paper was the first to study random walks on arbitrary dynamic networks. We study the most general model in which an oblivious adversary is permitted to change the graph after every step of the random walk. In contrast to static graphs, and somewhat counter‐intuitively, we show that there are adversary strategies that force the expected cover time and the mixing time of the simple random walk on dynamic graphs to be exponentially long, even when at each time step the network is well connected and rapidly mixing. To resolve this, we propose a simple strategy, the lazy random walk, which guarantees, under minor conditions, polynomial cover time and polynomial mixing time regardless of the changes made by the adversary.     

Firefighting on a random geometric graph

LetG_λ be the graph whose vertices are points of a planar Poisson process of density λ, with vertices adjacent if they are within distance 1. A “fire” begins at some vertex and spreads to all neighbors in discrete steps; in the meantime f vertices can be deleted at each time‐step. Let f_λ be the least f such that, with probability 1, any fire on G_λ can be stopped in finite time. We show that f_λ is bounded between two linear functions of λ. The lower bound makes use of a new result concerning oriented percolation in the plane; the constant factor in the upper bound is is tight, provided a certain conjecture, for which we offer supporting evidence, is correct. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 466–477, 2015     

Some properties for the largest component of random geometric graphs with applications in sensor networks

In this paper we consider the standard Poisson Boolean model of random geometric graphs G（Hλ,s; 1） in Rd and study the properties of the order of the largest component L1 （G（Hλ,s; 1）） . We prove that ElL1 （G（Hλ,s; 1））] is smooth with respect to A, and is derivable with respect to s. Also, we give the expression of these derivatives. These studies provide some new methods for the theory of the largest component of finite random geometric graphs （not asymptotic graphs as s - co） in the high dimensional space （d 〉 2）. Moreover, we investigate the convergence rate of E[L1（G（Hλ,s; 1））]. These results have significance for theory development of random geometric graphs and its practical application. Using our theories, we construct and solve a new optimal energy-efficient topology control model of wireless sensor networks, which has the significance of theoretical foundation and guidance for the design of network layout.     

On MAXCUT in strictly supercritical random graphs,and coloring of random graphs and random tournaments

We use a theorem by Ding, Lubetzky, and Peres describing the structure of the giant component of random graphs in the strictly supercritical regime, in order to determine the typical size of MAXCUT of in terms of ɛ. We then apply this result to prove the following conjecture by Frieze and Pegden. For every , there exists such that w.h.p. is not homomorphic to the cycle on vertices. We also consider the coloring properties of biased random tournaments. A p‐random tournament on n vertices is obtained from the transitive tournament by reversing each edge independently with probability p. We show that for the chromatic number of a p‐random tournament behaves similarly to that of a random graph with the same edge probability. To treat the case we use the aforementioned result on MAXCUT and show that in fact w.h.p. one needs to reverse edges to make it 2‐colorable.     

Expected cover times of random walks on symmetric graphs

We give very simple proofs for an (N–1)H _N–1 lower bound and anN ² upper bound for the expected cover time of symmetric graphs.     

Disjoint Hamilton cycles in the random geometric graph

We consider the standard random geometric graph process in which n vertices are placed at random on the unit square and edges are sequentially added in increasing order of edge‐length. For fixed k?1, weprove that the first edge in the process that creates a k‐connected graph coincides a.a.s. with the first edge that causes the graph to contain k/2 pairwise edge‐disjoint Hamilton cycles (for even k), or (k?1)/2 Hamilton cycles plus one perfect matching, all of them pairwise edge‐disjoint (for odd k). This proves and extends a conjecture of Krivelevich and M ler. In the special case when k = 2, our result says that the first edge that makes the random geometric graph Hamiltonian is a.a.s. exactly the same one that gives 2‐connectivity, which answers a question of Penrose. (This result appeared in three independent preprints, one of which was a precursor to this article.) We prove our results with lengths measured using the ?_p norm for any p>1, and we also extend our result to higher dimensions. © 2011 Wiley Periodicals, Inc. J Graph Theory 68:299‐322, 2011     

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 (n logn) for the expected cover time for trees. We present examples showing all our bounds to be tight.Mike Saks was supported by NSF-DMS87-03541 and by AFOSR-0271. Jeff Kahn was supported by MCS-83-01867 and by AFOSR-0271.     

Lower bounds for covering times for reversible Markov chains and random walks on graphs

For simple random walk on aN-vertex graph, the mean time to cover all vertices is at leastcN log(N), wherec>0 is an absolute constant. This is deduced from a more general result about stationary finite-state reversible Markov chains. Under weak conditions, the covering time for such processes is at leastc times the covering time for the corresponding i.i.d. process.     

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

We 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–1H_N-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.