Tight bounds for the cover time of multiple random walks

Abstract

a b s t r a c tWe study the cover time of multiple random walks on undirected graphs G = (V , E). We consider k parallel, independent random walks that start from the same vertex. The speedup is defined as the ratio of the cover time of a single random walk to the cover time of these k random walks. Recently, Alon et al. (2008) [5] derived several upper bounds on the cover time, which imply a speed-up of Ω(k) for several graphs; however, for many of them, k has to be bounded by O(log n). They also conjectured that, for any 1 ⩽ k ⩽ n, the speed-up is at most O(k) on any graph. We prove the following main results:• We present a new lower bound on the speed-up that depends on the mixing time. It gives a speed-up of Ω(k) on many graphs, even if k is as large as n. • We prove that the speed-up is O(k log n) on any graph. For a large class of graphs we can also improve this bound to O(k), matching the conjecture of Alon et al.• We determine the order of the speed-up for any value of 1 ⩽ k ⩽ n on hypercubes, random graphs and degree restricted expanders. For d-dimensional tori with d > 2, our bounds are tight up to logarithmic factors.• Our findings also reveal a surprisingly sharp threshold behaviour for certain graphs, e.g., the d-dimensional torus with d > 2 and hypercubes: there is a value T such that the speed-up is approximately min{T , k} for any 1 ⩽ k ⩽ n.

Cover time for random walks on arbitrary complex networks

“…For low-dimensional lattice networks with d ≤ 2, the relaxation time is large compared to a variety of complex networks (see Fig. 1 in [33]). Hence, we suspect that our method will not perform well for low-dimensional lattice networks and networks where nodes are embedded in low-dimensional space at position r u with short-range connection probability π vu ∝ r −ω vu with r vu = |r u − r v | and ω > d as those networks are comparable to low-dimensional lattices concerning search processes [34].…”

Network Sampling with Memory

“…In human populations, this is roughly comparable to starting random walks from multiple seeds. The use of multiple random walks has been found to reduce cover times (Alon, et al, 2008; Cooper, Frieze, and Radzik 2009; Elasser and Sauerwald 2010), however it is still possible for each individual random walk to get stuck in sub-graphs of the overall network. Ribeiro and Towsley (2010) propose an approach based upon multiple dependent random walks, where the multiple walks share the same sampling process, and show that this approach has a faster mixing time than multiple independent walks and single random walks in simulations on several large social-media networks 2 .…”

The multi-agent rotor-router on the ring

“…We compare our results with the so-called parallel random walk, achieved by deploying independent agents performing random walks in a graph independently and without any form of coordination. Recent work on the area of parallel random walks [2,10,9,13] contains a characterization of the improvement of the cover time due to the deployment of k independent random walkers with respect to the case with a single walker. It is shown in these works that the achieved speed-up depends on different parameters, such as the mixing time [10] and edge expansion [13] of the graph.…”

“…Recent work on the area of parallel random walks [2,10,9,13] contains a characterization of the improvement of the cover time due to the deployment of k independent random walkers with respect to the case with a single walker. It is shown in these works that the achieved speed-up depends on different parameters, such as the mixing time [10] and edge expansion [13] of the graph. The speed-up may sometimes be as low as Θ(log k) [2], and sometimes as high as exponential in terms of k [9].…”