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)

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