Robustness of randomized rumour spreading

Abstract

In this work we consider three well-studied broadcast protocols: push, pull and push&pull. A key property of all these models, which is also an important reason for their popularity, is that they are presumed to be very robust, since they are simple, randomized and, crucially, do not utilize explicitly the global structure of the underlying graph. While sporadic results exist, there has been no systematic theoretical treatment quantifying the robustness of these models. Here we investigate this question with respect to two orthogonal aspects: (adversarial) modifications of the underlying graph and message transmission failures. We explore in particular the following notion of local resilience: beginning with a graph, we investigate up to which fraction of the edges an adversary may delete at each vertex, so that the protocols need significantly more rounds to broadcast the information. Our main findings establish a separation among the three models. On one hand, pull is robust with respect to all parameters that we consider. On the other hand, push may slow down significantly, even if the adversary may modify the degrees of the vertices by an arbitrarily small positive fraction only. Finally, push&pull is robust when no message transmission failures are considered, otherwise it may be slowed down. On the technical side, we develop two novel methods for the analysis of randomized rumour-spreading protocols. First, we exploit the notion of self-bounding functions to facilitate significantly the round-based analysis: we show that for any graph the variance of the growth of informed vertices is bounded by its expectation, so that concentration results follow immediately. Second, in order to control adversarial modifications of the graph we make use of a powerful tool from extremal graph theory, namely Szemerédi’s Regularity Lemma.

Asymptotics for push on the complete graph

“…Besides on complete graphs, push has been extensively studied on several other graph classes. For example, Erdős-Rényi random graphs [9,10], random regular graphs and expander graphs [11,18,5]. More general bounds that only depend on some graph parameter have also been derived, e.g.…”

Asymptotics for Pull on the Complete Graph

“…A key property that simplifies greatly the computations in this section is the following observation, in a similar form introduced in [5] and also applied in [6].…”

“…One direction of research describes the runtime of randomized rumour spreading protocols using only general graph parameters like conductance [2], diameter [11] or expansion [14]. A different direction is to obtain ever more precise bounds on the runtime on specific graph classes [12,3,5]. One major step in that direction was achieved in [8], where the authors studied the runtime of pull on the complete graph.…”