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

Let G = (V,E) be a connected graph with |V | = n 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 1 -λmax of the transition matrix of a simple random walk on G. A vertex v is ℓ -good, if any even degree subgraph containing all edges incident with v contains at least ℓ vertices. A graph G is ℓ -good, if every vertex has the ℓ -good property. Let G be an even degree ℓ -good expander of bounded maximum degree. Any E -process on G has vertex cover time equation image This is to be compared with the Ω(nlog n) 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. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 00, 000–000, 2013 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 ℓ =Θ (log n). With high probability random r -regular graphs, r ≥ 4 even, have ℓ =Ω (log n). Thus the vertex cover time of the E -process on such graphs is Θ(n)

Vacant sets and vacant nets: Component structures induced by a random walk

Given a discrete random walk on a finite graph $G$, the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step $t$.%These sets induce subgraphs of the underlying graph. Let $\Gamma(t)$ be the subgraph of $G$ induced by the vacant set of the walk at step $t$. Similarly, let $\widehat \Gamma(t)$ be the subgraph of $G$ induced by the edges of the vacant net. For random $r$-regular graphs $G_r$, it was previously established that for a simple random walk, the graph $\Gamma(t)$ of the vacant set undergoes a phase transition in the sense of the phase transition on Erd\H{os}-Renyi graphs $G_{n,p}$. Thus, for $r \ge 3$ there is an explicit value $t^*=t^*(r)$ of the walk, such that for $t\leq (1-\epsilon)t^*$, $\Gamma(t)$ has a unique giant component, plus components of size $O(\log n)$, whereas for $t\geq (1+\epsilon)t^*$ all the components of $\Gamma(t)$ are of size $O(\log n)$. We establish the threshold value $\widehat t$ for a phase transition in the graph $\widehat \Gamma(t)$ of the vacant net of a simple random walk on a random $r$-regular graph. We obtain the corresponding threshold results for the vacant set and vacant net of two modified random walks. These are a non-backtracking random walk, and, for $r$ even, a random walk which chooses unvisited edges whenever available. This allows a direct comparison of thresholds between simple and modified walks on random $r$-regular graphs. The main findings are the following: As $r$ increases the threshold for the vacant set converges to $n \log r$ in all three walks. For the vacant net, the threshold converges to $rn/2 \; \log n$ for both the simple random walk and non-backtracking random walk. When $r\ge 4$ is even, the threshold for the vacant net of the unvisited edge process converges to $rn/2$, which is also the vertex cover time of the process.Comment: Added results pertaining to modified walk

Randomized Search of Graphs in Log Space and Probabilistic Computation

Reingold has shown that L = SL, that s-t connectivity in a poly-mixing digraph is complete for promise-RL, and that s-t connectivity for a poly-mixing out-regular digraph with known stationary distribution is in L. Several properties that bound the mixing times of random walks on digraphs have been identified, including the digraph conductance and the digraph spectral expansion. However, rapidly mixing digraphs can still have exponential cover time, thus it is important to specifically identify structural properties of digraphs that effect cover times. We examine the complexity of random walks on a basic parameterized family of unbalanced digraphs called Strong Chains (which model weakly symmetric logspace computations), and a special family of Strong Chains called Harps. We show that the worst case hitting times of Strong Chain families vary smoothly with the number of asymmetric vertices and identify the necessary condition for non-polynomial cover time. This analysis also yields bounds on the cover times of general digraphs. Next we relate random walks on graphs to the random walks that arise in Monte Carlo methods applied to optimization problems. We introduce the notion of the asymmetric states of Markov chains and use this definition to obtain some results about Markov chains. We also obtain some results on the mixing times for Markov Chain Monte Carlo Methods. Finally, we consider the question of whether a single long random walk or many short walks is a better strategy for exploration. These are walks which reset to the start after a fixed number of steps. We exhibit digraph families for which a few short walks are far superior to a single long walk. We introduce an iterative deepening random search. We use this strategy estimate the cover time for poly-mixing subgraphs. Finally we discuss complexity theoretic implications and future work

Sampling from the random cluster model on random regular graphs at all temperatures via Glauber dynamics

We consider the performance of Glauber dynamics for the random cluster model with real parameter $q>1$ and temperature $\beta>0$. Recent work by Helmuth, Jenssen and Perkins detailed the ordered/disordered transition of the model on random $\Delta$-regular graphs for all sufficiently large $q$ and obtained an efficient sampling algorithm for all temperatures $\beta$ using cluster expansion methods. Despite this major progress, the performance of natural Markov chains, including Glauber dynamics, is not yet well understood on the random regular graph, partly because of the non-local nature of the model (especially at low temperatures) and partly because of severe bottleneck phenomena that emerge in a window around the ordered/disordered transition. Nevertheless, it is widely conjectured that the bottleneck phenomena that impede mixing from worst-case starting configurations can be avoided by initialising the chain more judiciously. Our main result establishes this conjecture for all sufficiently large $q$ (with respect to $\Delta$). Specifically, we consider the mixing time of Glauber dynamics initialised from the two extreme configurations, the all-in and all-out, and obtain a pair of fast mixing bounds which cover all temperatures $\beta$, including in particular the bottleneck window. Our result is inspired by the recent approach of Gheissari and Sinclair for the Ising model who obtained a similar-flavoured mixing-time bound on the random regular graph for sufficiently low temperatures. To cover all temperatures in the RC model, we refine appropriately the structural results of Helmuth, Jenssen and Perkins about the ordered/disordered transition and show spatial mixing properties "within the phase", which are then related to the evolution of the chain