# Brief Announcement: Flooding in Dynamic Graphs with Arbitrary Degree Sequence

## Abstract

**1. 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** = *w*_{1}, …, *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 *G*_{n,p} Erdös-Rényi random graph.