Coalescing-Branching Random Walks on Graphs

Abstract

We study a distributed randomized information propagation mechanism in networks we call the coalescingbranching 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 lightweight 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. CCS Concepts: r Mathematics of computing → Graph algorithms; r Theory of computation → Distributed algorithms; Random walks and Markov chains; r Computing methodologies → Selforganization;

Better Bounds for Coalescing-Branching Random Walks

“…• We show that the cover time for the 2-cobra walk for a d-regular graph with conductance φG is O(φ −2 G log 2 n). This generalizes a similar result in [13] for expander graphs with sufficiently high expansion. Our new result holds for any d-regular graph, and expresses the bound as a function of the conductance.…”

“…We show that the cover time for the d-dimensional grid using a 2-cobra walk [0, n] d is O(n), where the order notation hides constant factors and other terms that depend on d; indeed, we show all vertices are covered in O(n) steps with high probability. Previous work has shown that the cover time is O(npolylog(n)) [13]. 1 Our result is clearly tight in its dependence on n. Moreover, it shows that in some circumstances one can avoid using tools such as Matthews' Theorem, which had been used previously in this setting [13], and necessarily adds in an additional logarithmic factor in the number of vertices over the hitting time.…”

“…Cobra walks bear the closest resemblance to push-based gossip models. Indeed, [17] show that the push process completes in every undirected graph in O(n log n) steps with high probability, and this bound has been conjectured to hold for cobra walks [13]. However, the similarity between the two is in many ways superficial.…”

“…We make use of an extension of Matthews' Theorem, which relates the cover time of a random walk to the maximum hitting time. The following theorem was proven in [13]:…”

“…Random walks provide a fundamental mathematical model for many basic network processes. In disease models, transmission of a virus can be modeled by the virus moving according to a random walk on a graph representing a human contact network; computer viruses can be modeled similarly [18,28,13]. Variants of random walks can also be used for information dissemination, using message-passing protocols where a message is passed from neighbor to neighbor via a random walk [9,17].…”