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Robustness of randomized rumour spreading

Part of: Discrete mathematics in relation to computer science Graph theory

Published online by Cambridge University Press:  12 August 2020

Rami Daknama ,
Konstantinos Panagiotou  and
Simon Reisser
Rami Daknama
Affiliation:
Institute for Mathematics, Ludwig-Maximilians-Universität München, 80333Munich, Germany
Konstantinos Panagiotou*
Affiliation:
Institute for Mathematics, Ludwig-Maximilians-Universität München, 80333Munich, Germany
Simon Reisser
Affiliation:
Institute for Mathematics, Ludwig-Maximilians-Universität München, 80333Munich, Germany
*
*Corresponding author. Email: kpanagio@math.lmu.de
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Abstract

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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.

MSC classification

Primary: 05C85: Graph algorithms
Secondary: 68R10: Graph theory (including graph drawing)
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 1 , January 2021 , pp. 37 - 78
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Footnotes

1

An extended abstract of this paper was published in the Proceedings of the European Symposium on Algorithms 2019 (ESA ’19).

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