Discordant Voting Processes on Finite Graphs

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. Indeed the main purpose of the paper is to illustrate the lack of consistency obtained from the push and pull variants of discordant voting, and to contract this with the extremely consistent behavior of the oblivious protocol. For oblivious voting on connected n-vertex graphs and starting from an initial half red, half blue colouring, the expected time to consensus is n2/4 independent of the underlying graph. For the push and pull protocols and the same initial colouring we have the following. For the complete graph Kn, the push protocol has ET = Θ(n log n), whereas the pull protocol has ET = Θ(2n). For the cycle Cn all three protocols have ET = Θ(n 2). For the star graph however, the pull protocol has ET = O(n2), whereas the push protocol ∗This work was supported by EPSRC grant EP/M005038/1, “Randomized algorithms for computer networks”, NSF grant DMS0753472 and Becas CHILE. †Department of Informatics, King’s College London, UK. [email protected] ‡School of Computing,University of Leeds, Leeds, UK. [email protected] §Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA 15213, USA. [email protected] ¶Department of Informatics, King’s College London, UK. [email protected] 1 ar X iv :1 60 4. 06 88 4v 2 [ cs .D M ] 8 J un 2 01 6 is slower with ET = Θ(n2 log n). For the double star (two stars joined at the central vertices), for push we have ET = Ω(2n/5) and for pull ET = O(n4). Finally, for the barbell graph (two cliques of equal size joined by an edge) both push and pull have ET = Ω(2n/10). 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(n3) on any connected graph, and ET = O(n) on many classes of expanders.