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.
Highlights
The process of reaching consensus in a graph by means of local interactions is known as voting
The performance of randomized voting processes is usually measured by the consensus time and the probability a given opinion wins
This paper considers a different asynchronous voting process, discordant voting
Summary
The process of reaching consensus in a graph by means of local interactions is known as voting. For discordant voting using the oblivious protocol, the expected time to consensus is the same for any connected n-vertex graph It is independent of graph structure and of the number of edges, and depends only on the initial number of vertices of each color (red, blue). Let T be the time to consensus of the asynchronous discordant voting process starting from any initial coloring with an equal number of red and blue vertices, i.e., R = B = n/2. At this point little remains of the possibility of a meta-theorem except a vague hope that at least one of the push and pull protocols always has polynomial time to consensus This is disproved by the example of the barbell graph, which consists of two cliques of size n/2 joined by a single edge. We use the term ordinary to refer to the standard asynchronous voting model in which the protocol makes no distinction between discordant and nondiscordant neighbors
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