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

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 $K_n$

Quasi-Majority Functional Voting on Expander Graphs

Consider a distributed graph where each vertex holds one of two distinct opinions. In this paper, we are interested in synchronous voting processes where each vertex updates its opinion according to a predefined common local updating rule. For example, each vertex adopts the majority opinion among 1) itself and two randomly picked neighbors in best-of-two or 2) three randomly picked neighbors in best-of-three. Previous works intensively studied specific rules including best-of-two and best-of-three individually. In this paper, we generalize and extend previous works of best-of-two and best-of-three on expander graphs by proposing a new model, quasi-majority functional voting. This new model contains best-of-two and best-of-three as special cases. We show that, on expander graphs with sufficiently large initial bias, any quasi-majority functional voting reaches consensus within $O(\log n)$ steps with high probability. Moreover, we show that, for any initial opinion configuration, any quasi-majority functional voting on expander graphs with higher expansion (e.g., Erd\H{o}s-R\'enyi graph $G(n,p)$ with $p=\Omega(1/\sqrt{n})$) reaches consensus within $O(\log n)$ with high probability. Furthermore, we show that the consensus time is $O(\log n/\log k)$ of best-of-$(2k+1)$ for $k=o(n/\log n)$

Paecilomyces lilacinus-induced Scleritis Following Bleb-associated Endophthalmitis after Trabeculectomy

Paecilomyces lilacinus (P. lilacinus) is a rare cause of fungal scleritis. We herein report a case of P. lilacinus-induced scleritis following bleb-associated endophthalmitis after trabeculectomy that was successfully treated with surgical excision of the affected sclera in combination with antifungal medication. An 85-year-old female underwent trabeculectomy of the left eye. A dellen formed in the corneal periphery due to limbal elevation of the filtering bleb and progressed to an infectious corneal ulcer, leading to blebitis. Eight days after the onset of blebitis, the patient was diagnosed with endophthalmitis, which resolved after vitrectomy. The growth of P. lilacinus was identified on swabs of the conjunctiva and the corneal specimen. Scleritis developed after the resolution of the endophthalmitis, and an early excision of the affected sclera, in addition to antifungal medication, resolved it completely. However, the scleritis recurred in a different region of the left eye. After 7 months of antifungal medication, the left eye showed no residual infection. When treating P. lilacinus-induced scleritis, surgical excision of the affected sclera has been shown to be an effective treatment strategy. Nevertheless, it is possible that the infection may recur in another part of the eyeball after the complete resolution of the primary lesion