The Cover Time of Random Walks on Graphs

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 second contribution, we determine the precise asymptotic value of the cover time of a random graph with given degree sequence. This is a graph picked uniformly at random from all simple graphs with that degree sequence. We also show that with high probability, a structural property of the graph called conductance, is bounded below by a constant. This is of independent interest. Finally, we explore random walks with weighted random edge choices. We present a weighting scheme that has a smaller worst case cover time than a simple random walk. We give an upper bound for a random graph of given degree sequence weighted according to our scheme. We demonstrate that the speed-up (that is, the ratio of cover times) over a simple random walk can be unboundedComment: 179 pages, PhD thesi

Cutoff for non-backtracking random walks on sparse random graphs

A finite ergodic Markov chain is said to exhibit cutoff if its distance to stationarity remains close to 1 over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Discovered in the context of card shuffling (Aldous-Diaconis, 1986), this phenomenon is now believed to be rather typical among fast mixing Markov chains. Yet, establishing it rigorously often requires a challengingly detailed understanding of the underlying chain. Here we consider non-backtracking random walks on random graphs with a given degree sequence. Under a general sparsity condition, we establish the cutoff phenomenon, determine its precise window, and prove that the (suitably rescaled) cutoff profile approaches a remarkably simple, universal shape

On the cover time of the emerging giant

Let $p=\frac{1+\varepsilon}{n}$. It is known that if $N=\varepsilon^3n\to\infty$ then w.h.p. $G_{n,p}$ has a unique giant largest component. We show that if in addition, $\varepsilon=\varepsilon(n)\to 0$ then w.h.p. the cover time of $G_{n,p}$ is asymptotic to $n\log^2N$; previously Barlow, Ding, Nachmias and Peres had shown this up to constant multiplicative factors

Cover time of a random graph with a degree sequence II: Allowing vertices of degree two

We study the cover time of a random graph chosen uniformly at random from the set of graphs with vertex set $[n]$ and degree sequence $\mathbf{d}=(d_i)_{i=1}^n$. In a previous work, the asymptotic cover time was obtained under a number of assumptions on $\mathbf{d}$, the most significant being that $d_i\geq 3$ for all $i$. Here we replace this assumption by $d_i\geq 2$. As a corollary, we establish the asymptotic cover time for the 2-core of the emerging giant component of $\mathcal{G}(n,p)$.Comment: 48 page

Viral processes by random walks on random regular graphs

We study the SIR epidemic model with infections carried by $k$ particles making independent random walks on a random regular graph. Here we assume $k\leq n^{\epsilon}$, where $n$ is the number of vertices in the random graph, and $\epsilon$ is some sufficiently small constant. We give an edge-weighted graph reduction of the dynamics of the process that allows us to apply standard results of Erd\H{o}s-R\'{e}nyi random graphs on the particle set. In particular, we show how the parameters of the model give two thresholds: In the subcritical regime, $O(\ln k)$ particles are infected. In the supercritical regime, for a constant $\beta\in(0,1)$ determined by the parameters of the model, $\beta k$ get infected with probability $\beta$, and $O(\ln k)$ get infected with probability $(1-\beta)$. Finally, there is a regime in which all $k$ particles are infected. Furthermore, the edge weights give information about when a particle becomes infected. We exploit this to give a completion time of the process for the SI case.Comment: Published in at http://dx.doi.org/10.1214/13-AAP1000 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Stationary distribution and cover time of sparse directed configuration models

We consider sparse digraphs generated by the configuration model with given in-degree and out-degree sequences. We establish that with high probability the cover time is linear up to a poly-logarithmic correction. For a large class of degree sequences we determine the exponent \u3b3 651 of the logarithm and show that the cover time grows as nlog\u3b3(n), where n is the number of vertices. The results are obtained by analysing the extremal values of the stationary distribution of the digraph. In particular, we show that the stationary distribution \u3c0 is uniform up to a poly-logarithmic factor, and that for a large class of degree sequences the minimal values of \u3c0 have the form 1nlog1 12\u3b3(n), while the maximal values of \u3c0 behave as 1nlog1 12\u3ba(n) for some other exponent \u3ba 08[0,1]. In passing, we prove tight bounds on the diameter of the digraphs and show that the latter coincides with the typical distance between two vertices