Discrete Incremental Voting on Expanders

TL;DR

The paper studies discrete incremental voting (DIV) on graphs, where opinions are integers in $\{1,\dots,k\}$ and a random vertex adjusts by $\pm 1$ toward a neighbour's opinion. On expanders with small spectral parameter, DIV rapidly collapses the opinion spectrum to two adjacent values, after which standard two-opinion pull voting governs the final consensus. Under the conditions $\lambda k = o(1)$, $k = o(n/\log n)$, and $\pi_{\min}=\Theta(1/n)$, the final opinion is the initial weighted average $c$ rounded to $\lfloor c\rfloor$ or $\lceil c\rceil$ with probabilities determined by the position of $c$ within its unit interval; the paper also provides explicit bounds on the expected completion time $\mathbb{E}[T]$. The approach combines expander mixing, martingale concentration (Azuma), and a coupling to two-opinion pull voting to establish the main theorem, with applications to graphs including complete graphs and random regular graphs. Overall, the work demonstrates that DIV can robustly compute the initial integer average in many expanders using only local, asynchronous updates.

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 $λ$ 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 $c=\sum_{v \in V} π_v X_v$, where $X_v$ is the initial opinion of vertex $v$, and $π_v$ is the stationary distribution of the vertex. Then provided $λk=o(1)$ 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 $\lceil c\rceil$.

Theorems & Definitions (21)

• theorem 1
• theorem 2
• Remark 1
• Lemma 3
• Lemma 4: The Azuma-Hoeffding inequality
• Lemma 5
• Lemma 6
• Corollary 7
• theorem 8: Restatement of \ref{['thm:expander_gen']}
• Lemma 9