Abstract
We develop a simple and generic method to analyze randomized rumor spreading processes in fully connected networks. In contrast to all previous works, which heavily exploit the precise definition of the process under investigation, we only need to understand the probability and the covariance of the events that uninformed nodes become informed. This universality allows us to easily analyze the classic push, pull, and pushpull protocols both in their pure version and in several variations such as messages failing with constant probability or nodes calling a random number of others each round. Some dynamic models can be analyzed as well, e.g., when the network is a G(n,p) random graph sampled independently each round [Clementi et al. (ESA 2013)].
Despite this generality, our method determines the expected rumor spreading time precisely apart from additive constants, which is more precise than almost all previous works. We also prove tail bounds showing that a deviation from the expectation by more than an additive number of r rounds occurs with probability at most exp(Omega(r)).
We further use our method to discuss the common assumption that nodes can answer any number of incoming calls. We observe that the restriction that only one call can be answered leads to a significant increase of the runtime of the pushpull protocol. In particular, the double logarithmic end phase of the process now takes logarithmic time. This also increases the message complexity from the asymptotically optimal Theta(n*log(log(n))) [Karp, Shenker, Schindelhauer, VĂ¶cking (FOCS 2000)] to Theta(n*log(n)). We propose a simple variation of the pushpull protocol that reverts back to the double logarithmic end phase and thus to the $\Theta(n*log(log(n))) message complexity.
BibTeX  Entry
@InProceedings{doerr_et_al:LIPIcs:2017:7379,
author = {Benjamin Doerr and Anatolii Kostrygin},
title = {{Randomized Rumor Spreading Revisited}},
booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
pages = {138:1138:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770415},
ISSN = {18688969},
year = {2017},
volume = {80},
editor = {Ioannis Chatzigiannakis and Piotr Indyk and Fabian Kuhn and Anca Muscholl},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7379},
URN = {urn:nbn:de:0030drops73798},
doi = {10.4230/LIPIcs.ICALP.2017.138},
annote = {Keywords: Epidemic algorithm, rumor spreading, dynamic graph}
}
Keywords: 

Epidemic algorithm, rumor spreading, dynamic graph 
Collection: 

44th International Colloquium on Automata, Languages, and Programming (ICALP 2017) 
Issue Date: 

2017 
Date of publication: 

07.07.2017 