Brief Announcement: Flooding in Dynamic Graphs with Arbitrary Degree Sequence

Abstract

Abstract1. Introduction. The simplest communication mechanism that implements the broadcast operation is the flooding protocol, according to which the source node is initially informed, and, when a not informed node has an informed neighbor, then it becomes informed at the next time step. In this paper we study the flooding completion time in the case of dynamic graphs with arbitrary degree sequence, which are a special case of random evolving graphs. A random evolving graph is a sequence of graphs (G t )t ≥ 0 with the same set of nodes, in which, at each time step t, the graph G t is chosen randomly according to a probability distribution over a specified family of graphs. A special case of random evolving graph is the edge-Markovian model (see the definition below), for which tight upper bounds on the flooding completion time have been obtained by using a so-called reduction lemma, which intuitively shows that the flooding completion time of an edge-Markovian evolving graph is equal to the diameter of a suitably defined weighted random graph. In this paper, we show that this technique can be applied to the analysis of the flooding completion time in the case of a random evolving graph based on the following generative model. Given a sequence w = w1, …, w n of non-negative numbers, the graph G w is a random graph with n nodes in which each edge (i,j) exists with probability \(p_{i,j}=\frac{w_iw_j}{\sum_{k=1}^nw_k}\) (independently of the other edges). It is easy to see that the expected degree of node i is w i : hence, if we choose w to be a sequence satisfying a power law, then G w is a power-law graph, while if we choose w i = pn, then G w is the Gn,p Erdös-Rényi random graph.KeywordsCompletion TimeSource NodeRandom GraphWeighted GraphDegree SequenceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.