No 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;

Coalescing-branching random walks, or cobra walks for short, are a natural variant of random walks on graphs that can model the spread of disease through contacts or the spread of information in networks. In a k-cobra walk, at each time step a subset of the vertices are active; each active vertex chooses k random neighbors (sampled indpendently and uniformly with replacement) that become active at the next step, and these are the only active vertices at the next step. A natural quantity to study for cobra walks is the cover time, which corresponds to the expected time when all nodes have become infected or received the disseminated information.In this work, we extend previous results for cobra walks in multiple ways. We show that the cover time for the 2-cobra walk on [0, n] d is O(n) (where the order notation hides constant factors that depend on d); previous work had shown the cover time was O(n·polylog(n)). We show that the cover time for a 2-cobra walk on an n-vertex d-regular graph with conductance φG is O(φ −2 G log 2 n), significantly generalizing a previous result that held only for expander graphs with sufficiently high expansion. And finally we show that the cover time for a 2-cobra walk on a graph with n vertices is always O(n 11/4 log n); this is the first result showing that the bound of Θ(n 3 ) for the worst-case cover time for random walks can be beaten using 2-cobra walks.

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.