Discordant Voting Processes on Finite Graphs

Abstract

We consider an asynchronous voting process on graphs called discordant voting, which can be described as follows. Initially each vertex holds one of two opinions, red or blue. Neighbouring vertices with different opinions interact pairwise along an edge. After an interaction both vertices have the same colour. The quantity of interest is the time to reach consensus, i.e. the number of steps needed for all vertices have the same colour. We show that for a given initial colouring of the vertices, the expected time to reach consensus, depends strongly on the underlying graph and the update rule (push, pull, oblivious).

Discordant Voting Protocols for Cyclically Linked Agents

“…So usually many idle rounds go by before the opinion of some vertex is altered. This example demonstrates the advantage of discordant (oblivious, push, pull) voting protocols, defined in [3]. An edge uv is discordant if u and v have different opinion, and a vertex is discordant if it is in a discordant edge.…”

“…In particular, the expected times to absorption from each transient state as initial state are the coordinates of the vector N 1; see [13] for further details. The following lemma is the basic observation of the elementary method we use to improve the upper estimations for the expected time to absorption presented in [3]. We can think about x[t] as a "guesstimate" of the expected value of the sum of the entries of u during a random walk with initial state t before reaching an absorbing state.…”

“…The topic of the present paper is the expected time to reach consensus with the discordant push, pull and oblivious processes on the n-cycle. It was proven in [3] that all three processes have a quadratic runtime at worst. In particular, push voting is expected to terminate in at most 33n 2 steps regardless of the initial state, and from some initial state it is indeed expected to take at least n 2 /4 + O(n) time to reach a unanimous vote [3,Section 4].…”

“…It was proven in [3] that all three processes have a quadratic runtime at worst. In particular, push voting is expected to terminate in at most 33n 2 steps regardless of the initial state, and from some initial state it is indeed expected to take at least n 2 /4 + O(n) time to reach a unanimous vote [3,Section 4]. We improve the bounds and obtain that the precise asymptotical behavior of the worst expected runtime of all three discordant protocols on the n-cycle is n 2 /4 + O(n 3/2 ), and an effective constant can be given in the error term for each of these protocols.…”

Threshold behaviour of discordant voting on the complete graph

“…In stark contrast to the oblivious protocol, the push and pull protocols can exhibit very different expected times to consensus, which depend strongly on the structure of underlying graph in question (see [3] for more detail). For example, on the complete graph K n , starting from an initial colouring where half the vertices are red and half blue, then ET (Push) = Θ(n log n), whereas ET (Pull) = Θ(2 n ).…”