Coalescing-branching random walks on graphs

Abstract

We study a distributed randomized information propagation mechanism in networks we call the coalescing-branching random walk (cobra walk, for short). A cobra walk is a generalization of the well-studied "standard" random walk, and is useful in modeling and understanding the SusceptibleInfected-Susceptible (SIS)-type of epidemic processes in networks. It can also be helpful in performing light-weight information dissemination in resource-constrained networks. A cobra walk is parameterized by a branching factor k. The process starts from an arbitrary node, which is labeled active for step 1. (For instance, this could be a node that has a piece of data, rumor, or a virus.) In each step of a cobra walk, each active node chooses k random neighbors to become active for the next step ("branching"). A node is active for step t + 1 only if it is chosen by an active node in step t ("coalescing"). This results in a stochastic process in the underlying network with properties that are quite different from both the standard random walk (which is equivalent * Chinmoy ...$15.00.to the cobra walk with branching factor 1) as well as other gossip-based rumor spreading mechanisms.We focus on the cover time of the cobra walk, which is the number of steps for the walk to reach all the nodes, and derive almost-tight bounds for various graph classes. Our main technical result is an O(log 2 n) high probability bound for the cover time of cobra walks on expanders, if either the expansion factor or the branching factor is sufficiently large; we also obtain an O(log n) high probability bound for the partial cover time, which is the number of steps needed for the walk to reach at least a constant fraction of the nodes. We show that the cobra walk takes O(n log n) steps on any n-node tree for k ≥ 2, andÕ(n 1/d ) steps on a d-dimensional grid for k ≥ 2, with high probability.

Improved Cover Time Bounds for the Coalescing-Branching Random Walk on Graphs

“…Dutta et al [5,6] studied the following coalescing-branching random walk process for propagating information on a connected graph with n vertices and m edges. At the start of a round each vertex containing information 'pushes' this information to b randomly selected neighbours.…”

“…The w.h.p. 1 cover time results for the COBRA process obtained in [5,6] for the case b = 2 include the following. (i) For the complete graph K n all vertices are visited in O(log n) rounds.…”

“…Previous work on multiple random walks includes [1][2][3]7], where cover times were analysed for various classes of graphs, assuming that the random walks are independent. The analyses of the COBRA process given in Dutta et al [5,6] and Mitzenmacher et al [8] use a number of tools from multiple random walks, but applicability of those tools turns out to be limited because the random walks in COBRA are highly dependent. An alternative approach was proposed by Cooper et al [4], who introduced and analysed a related epidemic process BIPS and showed that it is a dual (in some sense) of the COBRA process under time reversal.…”

Tight Bounds for Coalescing-Branching Random Walks on Regular Graphs

“…A Coalescing-Branching Random Walk with branching factor k, also called k-CoBra walk, is a generalization of the simple random walk on a graph and was introduced by Dutta et al [12,13]. The process starts with one particle on an arbitrary node of a connected graph.…”

“…Dutta et al [12,13] showed an O log 2 n bound that holds w.h.p. (with high probability 1 ) for regular constantdegree graphs with suciently high expansion parameters.…”