Discrete Incremental Voting on Expanders

TL;DR

This paper introduces discrete incremental voting on expanders, analyzing how opinions evolve in a graph where vertices update opinions gradually towards neighbors, and shows that under certain conditions the final opinion approximates a weighted average of initial opinions.

Contribution

It models a new form of pull voting with discrete opinion changes and proves that on expanders, the final consensus closely matches the initial weighted average under specific spectral and size conditions.

Findings

Final opinion approximates initial weighted average c.

Convergence occurs with high probability on expanders.

Results depend on spectral gap and number of opinions.

Abstract

Pull voting is a random process in which vertices of a connected graph have initial opinions chosen from a set of $k$ distinct opinions, and at each step a random vertex alters its opinion to that of a randomly chosen neighbour. If the system reaches a state where each vertex holds the same opinion, then this opinion will persist forthwith. In general the opinions are regarded as incommensurate, whereas in this paper we consider a type of pull voting suitable for integer opinions such as $\{1,2,\ldots,k\}$ which can be compared on a linear scale; for example, 1 ('disagree strongly'), 2 ('disagree'), $\ldots,$ 5 ('agree strongly'). On observing the opinion of a random neighbour, a vertex updates its opinion by a discrete change towards the value of the neighbour's opinion, if different. Discrete incremental voting is a pull voting process which mimics this behaviour. At each step a random vertex alters its opinion towards that of a randomly chosen neighbour; increasing its opinion by $+1$ if the opinion of the chosen neighbour is larger, or decreasing its opinion by $-1$, if the opinion of the neighbour is smaller. If initially there are only two adjacent integer opinions, for example $\{0,1\}$, incremental voting coincides with pull voting, but if initially there are more than two opinions this is not the case. For an $n$-vertex graph $G=(V,E)$, let $\lambda$ be the absolute second eigenvalue of the transition matrix $P$ of a simple random walk on $G$. Let the initial opinions of the vertices be chosen from $\{1,2,\ldots,k\}$. Let $c=\sum_{v \in V} \pi_v X_v$, where $X_v$ is the initial opinion of vertex $v$, and $\pi_v$ is the stationary distribution of the vertex. Then provided $\lambda k=o(1)$ and $k=o(n/\log n)$, with high probability the final opinion is the initial weighted average $c$ suitably rounded to $\lfloor c \rfloor$ or $\lceil c\rceil$.