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 Susceptible-Infected- 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 vertex, which is labeled active for step 1. In each step of a cobra walk, each active vertex chooses k random neighbors to become active for the next step (“branching”). A vertex is active for step t + 1 only if it is chosen by an active vertex 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 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 vertices, and derive almost-tight bounds for various graph classes. We show 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 vertices. We also show that the cover time of the cobra walk is, with high probability, O ( n log  n ) on any n -vertex tree for k ≥ 2, Õ ( n 1/d ) on a d -dimensional grid for k ≥ 2, and O (log  n ) on the complete graph.