Cooper, C, Dyer, M, Frieze, A et al. (1 more author) (2016) Discordant voting processes on finite graphs. In: Leibniz International Proceedings in Informatics, LIPIcs. 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016), 11-15 Jul 2016, Rome, Italy. Schloss Dagstuhl - Leibniz-Zentrum für Informatik ISBN 9783959770132
Abstract
We consider an asynchronous voting process on graphs which we call discordant voting, and which can be described as follows. Initially each vertex holds one of two opinions, red or blue say. Neighbouring vertices with different opinions interact pairwise. After an interaction both vertices have the same colour. The quantity of interest is T, the time to reach consensus, i.e. the number of interactions needed for all vertices have the same colour. An edge whose endpoint colours differ (i.e. one vertex is coloured red and the other one blue) is said to be discordant. A vertex is discordant if its is incident with a discordant edge. In discordant voting, all interactions are based on discordant edges. Because the voting process is asynchronous there are several ways to update the colours of the interacting vertices. Push: Pick a random discordant vertex and push its colour to a random discordant neighbour. Pull: Pick a random discordant vertex and pull the colour of a random discordant neighbour. Oblivious: Pick a random endpoint of a random discordant edge and push the colour to the other end point. We show that ET, the expected time to reach consensus, depends strongly on the underlying graph and the update rule. For connected graphs on n vertices, and an initial half red, half blue colouring the following hold. For oblivious voting, ET = n2/4 independent of the underlying graph. For the complete graph Kn, the push protocol has ET = =(n log n), whereas the pull protocol has ET = =(2n). For the cycle Cn all three protocols have ET = =(n2). For the star graph however, the pull protocol has ET = O(n2), whereas the push protocol is slower with ET = =(n2 log n). The wide variation in ET for the pull protocol is to be contrasted with the well known model of synchronous pull voting, for which ET = O(n) on many classes of expanders.
Metadata
Item Type: | Proceedings Paper |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © Colin Cooper, Martin Dyer, Alan Frieze, and Nicolás Rivera. This is an open access article licensed under Creative Commons Attribution License CC-BY. |
Keywords: | Distributed consensus, Voter model, Interacting particles, Randomized algorithm |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 03 Apr 2017 15:38 |
Last Modified: | 05 Oct 2017 15:38 |
Published Version: | https://doi.org/10.4230/LIPIcs.ICALP.2016.145 |
Status: | Published |
Publisher: | Schloss Dagstuhl - Leibniz-Zentrum für Informatik |
Identification Number: | 10.4230/LIPIcs.ICALP.2016.145 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:114300 |