Bounds on the Voter Model in Dynamic Networks

In the voter model, each node of a graph has an opinion, and in every round each node chooses independently a random neighbour and adopts its opinion. We are interested in the consensus time, which is the first point in time where all nodes have the same opinion. We consider dynamic graphs in which the edges are rewired in every round (by an adversary) giving rise to the graph sequence $G_1, G_2, \dots$, where we assume that $G_i$ has conductance at least $\phi_i$. We assume that the degrees of nodes don't change over time as one can show that the consensus time can become super-exponential otherwise. In the case of a sequence of $d$-regular graphs, we obtain asymptotically tight results. Even for some static graphs, such as the cycle, our results improve the state of the art. Here we show that the expected number of rounds until all nodes have the same opinion is bounded by $O(m/(d_{min} \cdot \phi))$, for any graph with $m$ edges, conductance $\phi$, and degrees at least $d_{min}$. In addition, we consider a biased dynamic voter model, where each opinion $i$ is associated with a probability $P_i$, and when a node chooses a neighbour with that opinion, it adopts opinion $i$ with probability $P_i$ (otherwise the node keeps its current opinion). We show for any regular dynamic graph, that if there is an $\epsilon>0$ difference between the highest and second highest opinion probabilities, and at least $\Omega(\log n)$ nodes have initially the opinion with the highest probability, then all nodes adopt w.h.p. that opinion. We obtain a bound on the convergences time, which becomes $O(\log n/\phi)$ for static graphs

Fast plurality consensus in regular expanders

Pull voting is a classic method to reach consensus among $n$ vertices with differing opinions in a distributed network: each vertex at each step takes on the opinion of a random neighbour. This method, however, suffers from two drawbacks. Even if there are only two opposing opinions, the time taken for a single opinion to emerge can be slow and the final opinion is not necessarily the initially held majority. We refer to a protocol where 2 neighbours are contacted at each step as a 2-sample voting protocol. In the two-sample protocol a vertex updates its opinion only if both sampled opinions are the same. Not much was known about the performance of two-sample voting on general expanders in the case of three or more opinions. In this paper we show that the following performance can be achieved on a $d$-regular expander using two-sample voting. We suppose there are $k \ge 3$ opinions, and that the initial size of the largest and second largest opinions is $A_1, A_2$ respectively. We prove that, if $A_1 - A_2 \ge C n \max\{\sqrt{(\log n)/A_1}, \lambda\}$, where $\lambda$ is the absolute second eigenvalue of matrix $P=Adj(G)/d$ and $C$ is a suitable constant, then the largest opinion wins in $O((n \log n)/A_1)$ steps with high probability. For almost all $d$-regular graphs, we have $\lambda=c/\sqrt{d}$ for some constant $c>0$. This means that as $d$ increases we can separate an opinion whose majority is $o(n)$, whereas $\Theta(n)$ majority is required for $d$ constant. This work generalizes the results of Becchetti et. al (SPAA 2014) for the complete graph $K_n$

On the convergence of densities of finite voter models to the Wright-Fisher diffusion

We study voter models defined on large sets. Through a perspective emphasizing the martingale property of voter density processes, we prove that in general, their convergence to the Wright-Fisher diffusion only involves certain averages of the voter models over a small number of spatial locations. This enables us to identify suitable mixing conditions on the underlying voting kernels, one of which may just depend on their eigenvalues in some contexts, to obtain the convergence of density processes. Our examples show that these conditions are satisfied by a large class of voter models on growing finite graphs

Phase Transitions of Best-of-Two and Best-of-Three on Stochastic Block Models

This paper is concerned with voting processes on graphs where each vertex holds one of two different opinions. In particular, we study the \emph{Best-of-two} and the \emph{Best-of-three}. Here at each synchronous and discrete time step, each vertex updates its opinion to match the majority among the opinions of two random neighbors and itself (the Best-of-two) or the opinions of three random neighbors (the Best-of-three). Previous studies have explored these processes on complete graphs and expander graphs, but we understand significantly less about their properties on graphs with more complicated structures. In this paper, we study the Best-of-two and the Best-of-three on the stochastic block model $G(2n,p,q)$, which is a random graph consisting of two distinct Erd\H{o}s-R\'enyi graphs $G(n,p)$ joined by random edges with density $q\leq p$. We obtain two main results. First, if $p=\omega(\log n/n)$ and $r=q/p$ is a constant, we show that there is a phase transition in $r$ with threshold $r^*$ (specifically, $r^*=\sqrt{5}-2$ for the Best-of-two, and $r^*=1/7$ for the Best-of-three). If $r>r^*$, the process reaches consensus within $O(\log \log n+\log n/\log (np))$ steps for any initial opinion configuration with a bias of $\Omega(n)$. By contrast, if $r<r^*$, then there exists an initial opinion configuration with a bias of $\Omega(n)$ from which the process requires at least $2^{\Omega(n)}$ steps to reach consensus. Second, if $p$ is a constant and $r>r^*$, we show that, for any initial opinion configuration, the process reaches consensus within $O(\log n)$ steps. To the best of our knowledge, this is the first result concerning multiple-choice voting for arbitrary initial opinion configurations on non-complete graphs

The Power of Two Choices in Distributed Voting

Distributed voting is a fundamental topic in distributed computing. In pull voting, in each step every vertex chooses a neighbour uniformly at random, and adopts its opinion. The voting is completed when all vertices hold the same opinion. On many graph classes including regular graphs, pull voting requires $\Theta(n)$ expected steps to complete, even if initially there are only two distinct opinions. In this paper we consider a related process which we call two-sample voting: every vertex chooses two random neighbours in each step. If the opinions of these neighbours coincide, then the vertex revises its opinion according to the chosen sample. Otherwise, it keeps its own opinion. We consider the performance of this process in the case where two different opinions reside on vertices of some (arbitrary) sets $A$ and $B$, respectively. Here, $|A| + |B| = n$ is the number of vertices of the graph. We show that there is a constant $K$ such that if the initial imbalance between the two opinions is ?$\nu_0 = (|A| - |B|)/n \geq K \sqrt{(1/d) + (d/n)}$, then with high probability two sample voting completes in a random $d$ regular graph in $O(\log n)$ steps and the initial majority opinion wins. We also show the same performance for any regular graph, if $\nu_0 \geq K \lambda_2$ where $\lambda_2$ is the second largest eigenvalue of the transition matrix. In the graphs we consider, standard pull voting requires $\Omega(n)$ steps, and the minority can still win with probability $|B|/n$.Comment: 22 page