Stochastic Analysis of Rumor Spreading with Multiple Pull Operations

Abstract

We propose and analyze a new asynchronous rumor spreading protocol to deliver a rumor to all the nodes of a large-scale distributed network. This spreading protocol relies on what we call a k-pull operation, with \(k \ge 2\). Specifically a k-pull operation consists, for an uninformed node s, in contacting \(k-1\) other nodes at random in the network, and if at least one of them knows the rumor, then node s learns it. We perform a thorough study of the total number \(T_{k,n}\) of k-pull operations needed for all the n nodes to learn the rumor. We compute the expected value and the variance of \(T_{k,n}\), together with their limiting values when n tends to infinity. We also analyze the limiting distribution of \((T_{k,n} - {E}(T_{k,n}))/n\) and prove that it has a double exponential distribution when n tends to infinity. Finally, we show that when \(k > 2\), our new protocol requires less operations than the traditional 2-push-pull and 2-push protocols by using stochastic dominance arguments. All these results generalize the standard case \(k=2\).