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OPINION FLUCTUATIONS AND DISAGREEMENT IN SOCIAL NETWORKS
 SUBMITTED TO THE ANNALS OF APPLIED PROBABILITY
, 2010
"... We study a stochastic gossip model of continuous opinion dynamics in a society consisting of two types of agents: regular agents, who update their beliefs according to information that they receive from their social neighbors; and stubborn agents, who never update their opinions and might represent ..."
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Cited by 26 (5 self)
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We study a stochastic gossip model of continuous opinion dynamics in a society consisting of two types of agents: regular agents, who update their beliefs according to information that they receive from their social neighbors; and stubborn agents, who never update their opinions and might represent leaders, political parties or media sources attempting to influence the beliefs in the rest of the society. When the society contains stubborn agents with different opinions, opinion dynamics never lead to a consensus (among the regular agents). Instead, beliefs in the society almost surely fail to converge, and the belief of each regular agent converges in law to a nondegenerate random variable. The model thus generates longrun disagreement and continuous opinion fluctuations. The structure of the social network and the location of stubborn agents within it shape opinion dynamics. When the society is “highly fluid”, meaning that the mixing time of the random walk on the graph describing the social network is small relative to (the inverse of) the relative size of the linkages to stubborn agents, the ergodic beliefs of most of the agents concentrate around a certain common value. We also show that under additional conditions, the ergodic beliefs distribution becomes “approximately chaotic”, meaning that the variance of the aggregate belief of the society vanishes in the large population limit while individual opinions still fluctuate significantly.
NonBayesian Social Learning
, 2011
"... We develop a dynamic model of opinion formation in social networks when the information required for learning a payoffrelevant parameter may not be at the disposal of any single agent. Individuals engage in communication with their neighbors in order to learn from their experiences. However, instea ..."
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Cited by 25 (6 self)
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We develop a dynamic model of opinion formation in social networks when the information required for learning a payoffrelevant parameter may not be at the disposal of any single agent. Individuals engage in communication with their neighbors in order to learn from their experiences. However, instead of incorporating the views of their neighbors in a fully Bayesian manner, agents use a simple updating rule which linearly combines their personal experience and the views of their neighbors (even though the neighbors ’ views may be quite inaccurate). This nonBayesian learning rule is motivated by the formidable complexity required to fully implement Bayesian updating in networks. We show that, as long as individuals take their personal signals into account in a Bayesian way, repeated interactions lead them to successfully aggregate information and learn the true underlying state of the world. This result holds in spite of the apparent naïveté of agents’ updating rule, the agents ’ need for information from sources the existence of which they may not be aware of, the possibility that the most persuasive agents in the network are precisely those least informed and with worst prior views, and the assumption that no agent can
The Network Origins of Aggregate Fluctuations
 Econometrica
, 2012
"... This paper argues that in the presence of intersectoral inputoutput linkages, microeconomic idiosyncratic shocks may lead to aggregate fluctuations. In particular, it shows that, as the economy becomes more disaggregated, the rate at which aggregate volatility decays is determined by the structure ..."
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Cited by 18 (2 self)
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This paper argues that in the presence of intersectoral inputoutput linkages, microeconomic idiosyncratic shocks may lead to aggregate fluctuations. In particular, it shows that, as the economy becomes more disaggregated, the rate at which aggregate volatility decays is determined by the structure of the network capturing such linkages. Our main results provide a characterization of this relationship in terms of the importance of different sectors as suppliers to their immediate customers as well as their role as indirect suppliers to chains of downstream sectors. Such higherorder interconnections capture the possibility of “cascade effects ” whereby productivity shocks to a sector propagate not only to its immediate downstream customers, but also indirectly to the rest of the economy. Our results highlight that sizable aggregate volatility is obtained from sectoral idiosyncratic shocks only if there exists significant asymmetry in the roles that sectors play as suppliers to others, and that the “sparseness ” of the inputoutput matrix is unrelated to the nature of aggregate fluctuations.
Discrete Opinion Dynamics with Stubborn Agents
"... We study discrete opinion dynamics in a social network with ”stubborn agents” who influence others but do not change their opinions. We generalize the classical voter model by introducing nodes (stubborn agents) that have a fixed state. We show that the presence of stubborn agents with opposing opin ..."
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Cited by 17 (1 self)
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We study discrete opinion dynamics in a social network with ”stubborn agents” who influence others but do not change their opinions. We generalize the classical voter model by introducing nodes (stubborn agents) that have a fixed state. We show that the presence of stubborn agents with opposing opinions precludes convergence to consensus; instead, opinions converge in distribution with disagreement and fluctuations. In addition to the first moment of this distribution typically studied in the literature, we study the behavior of the second moment in terms of network properties and the opinions and locations of stubborn agents. We also study the problem of ”optimal placement of stubborn agents” where the location of a fixed number of stubborn agents is chosen to have the maximum impact on the longrun expected opinions of agents.
Opinion fluctuations and persistent disagreement in social networks
 in Decision and Control and European Control Conference (CDCECC), 2011 50th IEEE Conference on. IEEE, 2011
"... Abstract — Disagreement among individuals in a society, even on central questions that have been debated for centuries, is the rule; agreement is the rare exception. How can disagreement of this sort persist for so long? Most existing models of communication and learning, based on Bayesian or nonB ..."
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Cited by 13 (1 self)
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Abstract — Disagreement among individuals in a society, even on central questions that have been debated for centuries, is the rule; agreement is the rare exception. How can disagreement of this sort persist for so long? Most existing models of communication and learning, based on Bayesian or nonBayesian updating mechanisms, typically lead to consensus provided that communication takes place over a strongly connected network. These models are thus unable to explain persistent disagreements, and belief fluctuations. We propose a tractable model that generates longrun disagreements and persistent opinion fluctuations. Our model involves a stochastic gossip model of continuous opinion dynamics in a society consisting of two types of agents: regular agents, who update their beliefs according to information that they receive from their social neighbors; and stubborn agents, who never update their opinions and might represent leaders, political parties or media sources attempting to influence the beliefs in the rest of the society. When the society contains stubborn agents with different opinions, the belief dynamics never lead to a consensus (among the regular agents). Instead, beliefs in the society almost surely fail to converge, the belief profile keeps on oscillating in an ergodic fashion, and it converges in law to a nondegenerate random vector. The structure of the graph describing the social network and the location of stubborn agents within it shape the long run behavior of the opinion dynamics. We prove that, when the society is highly fluid, meaning that the mixing time of the random walk on the graph describing the social network is small relative to the inverse of the relative size of the linkages to stubborn agents, a condition of homogeneous influence emerges, whereby the ergodic beliefs of most of the regular agents have approximately equal marginal distributions. This clearly need not imply approximate consensus and in fact we show, under mild conditions, the ergodic belief distribution becomes approximately chaotic, meaning that the variance of the aggregate belief of the society vanishes in the large population limit while individual opinions still fluctuate significantly in an essentially uncorrelated way. I.
Algorithms for leader selection in stochastically forced consensus networks
 IEEE Trans. Automat. Control
"... Abstract—We are interested in assigning a prespecified number of nodes as leaders in order to minimize the meansquare deviation from consensus in stochastically forced networks. This problem arises in several applications including control of vehicular formations and localization in sensor networ ..."
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Cited by 12 (3 self)
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Abstract—We are interested in assigning a prespecified number of nodes as leaders in order to minimize the meansquare deviation from consensus in stochastically forced networks. This problem arises in several applications including control of vehicular formations and localization in sensor networks. For networks with leaders subject to noise, we show that the Boolean constraints (which indicate whether a node is a leader) are the only source of nonconvexity. By relaxing these constraints to their convex hull we obtain a lower bound on the global optimal value. We also use a simple but efficient greedy algorithm to identify leaders and to compute an upper bound. For networks with leaders that perfectly follow their desired trajectories, we identify an additional source of nonconvexity in the form of a rank constraint. Removal of the rank constraint and relaxation of the Boolean constraints yields a semidefinite program for which we develop a customized algorithm wellsuited for large networks. Several examples ranging from regular lattices to random graphs are provided to illustrate the effectiveness of the developed algorithms. Index Terms—Alternating direction method of multipliers (ADMMs), consensus networks, convex optimization, convex relaxations, greedy algorithm, leader selection, performance bounds, semidefinite programming (SDP), sensor selection, variance amplification. I.
Majority Dynamics and Aggregation of Information in Social Networks
, 2012
"... 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 ..."
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Cited by 11 (2 self)
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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 → ∞. Ouri nterest 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 populationwide 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.
A model of influence with a continuum of actions
, 2011
"... We generalize a twoaction (yesno) model of influence to a framework in which every player has a continuum of actions, among which he has to choose one. We assume the set of actions to be an interval. Each player has an inclination to choose one of the actions. Due to influence among players, the f ..."
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Cited by 7 (3 self)
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We generalize a twoaction (yesno) model of influence to a framework in which every player has a continuum of actions, among which he has to choose one. We assume the set of actions to be an interval. Each player has an inclination to choose one of the actions. Due to influence among players, the final decision of a player, i.e., his choice of one action, may be different from his original inclination. In particular, a coalition of players with the same inclination may influence another player with different inclination, and as a result of this influence, the decision of the player is closer to the inclination of the influencing coalition than his inclination was. We introduce a measure of such a positive influence of a coalition on a player. Several unanimous influence functions in this generalized framework are considered. Also the set of fixed points under a given influence function is analyzed. Furthermore, we study linear influence functions and discuss their convergence. For a linear unanimous function, we find necessary and sufficient conditions for the existence of the positive influence of a coalition on a player, and we calculate the value of the influence index. We also introduce a measure of a negative influence of a coalition on a player.
Tractable Bayesian Social Learning on Trees
, 2012
"... We study a model of Bayesian agents in social networks who learn from the actions of their neighbors. Most results concerning social learning in networks have been achieved either in ‘herd behavior ’ models, where each agent acts only once, or in models where agents are not Bayesian and use rules of ..."
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Cited by 6 (3 self)
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We study a model of Bayesian agents in social networks who learn from the actions of their neighbors. Most results concerning social learning in networks have been achieved either in ‘herd behavior ’ models, where each agent acts only once, or in models where agents are not Bayesian and use rules of thumb, or are boundedly rational. Models of Bayesian agents who act repeatedly have posed two related problems: (1) they have proved notoriously difficult to analyze; and (2) the calculations required of interacting Bayesian agents often seem intractable. We consider a set of Bayesian agents who are attempting to iteratively estimate an unknown ‘state of the world ’ s from their initial private signals, and the past actions of their neighbors in a social network. When private signals are independent conditioned on s, and when the social network graph is a tree, we provide an algorithm for the agents ’ calculations with running time that is exponentially lower than what is currently known. We use this algorithm to perform the first numerical simulations of interacting Bayesian agents on networks with hundreds of nodes, and observe rapid learning of s in some settings.