Rumor spreading on random regular graphs and expanders

Abstract

Broadcasting algorithms are important building blocks of distributed systems. In this work we investigate further the performance of the classical and well-studied push model. Assume that initially one node in a given network holds some piece of information. In each round, every one of the informed nodes chooses independently a neighbor uniformly at random and transmits the message to it. In this paper, we consider random networks where each vertex has degree d ≥ 3, i.e., the underlying graph is drawn uniformly at random from the set of all d-regular graphs with n vertices. We show that with probability 1-o(1) the push model broadcasts the message to all nodes in Cd ln n+ξ rounds, where |ξ|=O((ln ln n)2) and Cd=1/ln(2(1-1/d))- 1/dln(1-1/d). In particular, we determine precisely the effect of the node degree to the typical broadcast time of the push model. Moreover, we consider pseudo-random regular networks, where we assume that the degree of each node depends on n. There we show that the broadcast time is (1+o(1))Cln n with probability 1-o(1), where C-limd→∞Cd=1/ln2+1. © 2010 Springer-Verlag.