Rumor Spreading on Random Regular Graphs and Expanders

TL;DR

This paper analyzes the efficiency of the push rumor spreading model on random regular graphs and expanders, providing precise asymptotic broadcast times and the influence of node degree.

Contribution

It offers a detailed characterization of broadcast times in random regular graphs and expanders, revealing the impact of node degree on spreading efficiency.

Findings

Broadcasts complete in approximately C_d ln n rounds for degree d

Broadcast time approaches C ln n as degree d increases

Precise formulas for broadcast times based on node degree

Abstract

Broadcasting algorithms are important building blocks of distributed systems. In this work we investigate the typical 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, which is at least 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 within (1 + o(1))C_d ln n rounds, where C_d = 1/ ln(2(1-1/d)) - 1/(d ln(1 - 1/d)). In particular, we can characterize 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 is very large. There we show that the broadcast time is (1+o(1))C ln n with probability 1 - o(1), where C= 1/ ln 2 + 1, is the limit of C_d as d grows.