On Randomized Broadcasting in Power Law Networks

Abstract

Broadcasting algorithms have various range of applications in different fields of computer science. In this paper we consider randomized broadcasting algorithms in power law graphs which are often used to model large scale real world networks such as the Internet. We prove that for certain (truncated) power law networks there exists a time efficient randomized broadcasting algorithm whose communication complexity is bounded by an asymptotically optimal value.

In order to describe these power law graphs, we first consider the generalized random graph model G(d) = (V,E), where d = (d ₁, ..., d _n ) is a given sequence of expected degrees, and two nodes v _i ,v _j ∈V share an edge in G(d) with probability p _i,j = d _i d _j /∑$_{k=1}^{n}$ d _k , independently [7]. We show for these graphs that if the expected minimal degree d _min is larger than log^δ n, δ>2, and the number of nodes with expected degree d _i is proportional to (d _i – d _min+1)^− β, where β>2 is a constant, then a simple randomized broadcasting protocol exists, which spreads any information r to all nodes of a graph G(d) within O(logn) steps by using at most O(n max{ loglogn, logn/logd _min}) transmissions. Furthermore, we discuss the applicability of our methods in more general power law graph models. Please note that our results hold with probability 1– 1/n ^Ω(1), even if n and d are completely unknown to the nodes of the graph.

The algorithm we present in this paper uses a very simple communication rule, and can efficiently handle restricted node failures or dynamical changes in the size of the network. In addition, our methods might be useful for further research in this field.