Coalescing-branching random walks on graphs

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 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 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(log2 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, and Õ(n1/d) steps on a d-dimensional grid for k ≥ 2, with high probability.