Simple Dynamics for Plurality Consensus

We study a \emph{Plurality-Consensus} process in which each of $n$ anonymous agents of a communication network initially supports an opinion (a color chosen from a finite set $[k]$). Then, in every (synchronous) round, each agent can revise his color according to the opinions currently held by a random sample of his neighbors. It is assumed that the initial color configuration exhibits a sufficiently large \emph{bias} $s$ towards a fixed plurality color, that is, the number of nodes supporting the plurality color exceeds the number of nodes supporting any other color by $s$ additional nodes. The goal is having the process to converge to the \emph{stable} configuration in which all nodes support the initial plurality. We consider a basic model in which the network is a clique and the update rule (called here the \emph{3-majority dynamics}) of the process is the following: each agent looks at the colors of three random neighbors and then applies the majority rule (breaking ties uniformly). We prove that the process converges in time $\mathcal{O}( \min\{ k, (n/\log n)^{1/3} \} \, \log n )$ with high probability, provided that $s \geqslant c \sqrt{ \min\{ 2k, (n/\log n)^{1/3} \}\, n \log n}$. We then prove that our upper bound above is tight as long as $k \leqslant (n/\log n)^{1/4}$. This fact implies an exponential time-gap between the plurality-consensus process and the \emph{median} process studied by Doerr et al. in [ACM SPAA'11]. A natural question is whether looking at more (than three) random neighbors can significantly speed up the process. We provide a negative answer to this question: In particular, we show that samples of polylogarithmic size can speed up the process by a polylogarithmic factor only.Comment: Preprint of journal versio

Noisy Rumor Spreading and Plurality Consensus

Error-correcting codes are efficient methods for handling \emph{noisy} communication channels in the context of technological networks. However, such elaborate methods differ a lot from the unsophisticated way biological entities are supposed to communicate. Yet, it has been recently shown by Feinerman, Haeupler, and Korman {[}PODC 2014{]} that complex coordination tasks such as \emph{rumor spreading} and \emph{majority consensus} can plausibly be achieved in biological systems subject to noisy communication channels, where every message transferred through a channel remains intact with small probability $\frac{1}{2}+\epsilon$, without using coding techniques. This result is a considerable step towards a better understanding of the way biological entities may cooperate. It has been nevertheless be established only in the case of 2-valued \emph{opinions}: rumor spreading aims at broadcasting a single-bit opinion to all nodes, and majority consensus aims at leading all nodes to adopt the single-bit opinion that was initially present in the system with (relative) majority. In this paper, we extend this previous work to $k$-valued opinions, for any $k\geq2$. Our extension requires to address a series of important issues, some conceptual, others technical. We had to entirely revisit the notion of noise, for handling channels carrying $k$-\emph{valued} messages. In fact, we precisely characterize the type of noise patterns for which plurality consensus is solvable. Also, a key result employed in the bivalued case by Feinerman et al. is an estimate of the probability of observing the most frequent opinion from observing the mode of a small sample. We generalize this result to the multivalued case by providing a new analytical proof for the bivalued case that is amenable to be extended, by induction, and that is of independent interest.Comment: Minor revisio

Stabilizing Consensus with Many Opinions

We consider the following distributed consensus problem: Each node in a complete communication network of size $n$ initially holds an \emph{opinion}, which is chosen arbitrarily from a finite set $\Sigma$. The system must converge toward a consensus state in which all, or almost all nodes, hold the same opinion. Moreover, this opinion should be \emph{valid}, i.e., it should be one among those initially present in the system. This condition should be met even in the presence of an adaptive, malicious adversary who can modify the opinions of a bounded number of nodes in every round. We consider the \emph{3-majority dynamics}: At every round, every node pulls the opinion from three random neighbors and sets his new opinion to the majority one (ties are broken arbitrarily). Let $k$ be the number of valid opinions. We show that, if $k \leqslant n^{\alpha}$, where $\alpha$ is a suitable positive constant, the 3-majority dynamics converges in time polynomial in $k$ and $\log n$ with high probability even in the presence of an adversary who can affect up to $o(\sqrt{n})$ nodes at each round. Previously, the convergence of the 3-majority protocol was known for $|\Sigma| = 2$ only, with an argument that is robust to adversarial errors. On the other hand, no anonymous, uniform-gossip protocol that is robust to adversarial errors was known for $|\Sigma| > 2$

Majority Dynamics and Aggregation of Information in Social Networks

Consider n individuals who, by popular vote, choose among q >= 2 alternatives, one of which is "better" than the others. Assume that each individual votes independently at random, and that the probability of voting for the better alternative is larger than the probability of voting for any other. It follows from the law of large numbers that a plurality vote among the n individuals would result in the correct outcome, with probability approaching one exponentially quickly as n tends to infinity. Our interest in this paper is in a variant of the process above where, after forming their initial opinions, the voters update their decisions based on some interaction with their neighbors in a social network. Our main example is "majority dynamics", in which each voter adopts the most popular opinion among its friends. The interaction repeats for some number of rounds and is then followed by a population-wide plurality vote. The question we tackle is that of "efficient aggregation of information": in which cases is the better alternative chosen with probability approaching one as n tends to infinity? Conversely, for which sequences of growing graphs does aggregation fail, so that the wrong alternative gets chosen with probability bounded away from zero? We construct a family of examples in which interaction prevents efficient aggregation of information, and give a condition on the social network which ensures that aggregation occurs. For the case of majority dynamics we also investigate the question of unanimity in the limit. In particular, if the voters' social network is an expander graph, we show that if the initial population is sufficiently biased towards a particular alternative then that alternative will eventually become the unanimous preference of the entire population.Comment: 22 page

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 $K_n$

Opinion Dynamics and Bounded Confidence Models, Analysis and Simulation

When does opinion formation within an interacting group lead to consensus, polarization or fragmentation? The article investigates various models for the dynamics of continuous opinions by analytical methods as well as by computer simulations. Section 2 develops within a unified framework the classical model of consensus formation, the variant of this model due to Friedkin and Johnsen, a time-dependent version and a nonlinear version with bounded confidence of the agents. Section 3 presents for all these models major analytical results. Section 4 gives an extensive exploration of the nonlinear model with bounded confidence by a series of computer simulations. An appendix supplies needed mathematical definitions, tools, and theorems.opinion dynamics, consensus/dissent, bounded confidence, nonlinear dynamical systems.