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

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→∞. Our interest in this article 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→∞? 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

Pooling or sampling: Collective dynamics for electrical flow estimation

The computation of electrical flows is a crucial primitive for many recently proposed optimization algorithms on weighted networks. While typically implemented as a centralized subroutine, the ability to perform this task in a fully decentralized way is implicit in a number of biological systems. Thus, a natural question is whether this task can provably be accomplished in an efficient way by a network of agents executing a simple protocol. We provide a positive answer, proposing two distributed approaches to electrical flow computation on a weighted network: a deterministic process mimicking Jacobi's iterative method for solving linear systems, and a randomized token diffusion process, based on revisiting a classical random walk process on a graph with an absorbing node. We show that both processes converge to a solution of Kirchhoff's node potential equations, derive bounds on their convergence rates in terms of the weights of the network, and analyze their time and message complexity

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

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

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 $|\Sigma| > 2$

Minority Becomes Majority in Social Networks

It is often observed that agents tend to imitate the behavior of their neighbors in a social network. This imitating behavior might lead to the strategic decision of adopting a public behavior that differs from what the agent believes is the right one and this can subvert the behavior of the population as a whole. In this paper, we consider the case in which agents express preferences over two alternatives and model social pressure with the majority dynamics: at each step an agent is selected and its preference is replaced by the majority of the preferences of her neighbors. In case of a tie, the agent does not change her current preference. A profile of the agents' preferences is stable if the preference of each agent coincides with the preference of at least half of the neighbors (thus, the system is in equilibrium). We ask whether there are network topologies that are robust to social pressure. That is, we ask if there are graphs in which the majority of preferences in an initial profile always coincides with the majority of the preference in all stable profiles reachable from that profile. We completely characterize the graphs with this robustness property by showing that this is possible only if the graph has no edge or is a clique or very close to a clique. In other words, except for this handful of graphs, every graph admits at least one initial profile of preferences in which the majority dynamics can subvert the initial majority. We also show that deciding whether a graph admits a minority that becomes majority is NP-hard when the minority size is at most 1/4-th of the social network size.Comment: To appear in WINE 201