Opinion Fluctuations and Disagreement in Social Networks
TL;DR: In large-scale societies, which are highly fluid, the product of the mixing time of the Markov chain on the graph describing the social network and the relative size of the linkages to stubborn agents vanishes as the population size grows large, a condition of homogeneous influence emerges.
Abstract: We study a tractable opinion dynamics model that generates long-run disagreements and persistent opinion fluctuations. Our model involves an inhomogeneous stochastic gossip process of continuous opinion dynamics in a society consisting of two types of agents: 1 regular agents who update their beliefs according to information that they receive from their social neighbors and 2 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 fail to converge almost surely, the belief profile keeps on fluctuating 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 the stubborn agents within it shape the opinion dynamics. The expected belief vector is proved to evolve according to an ordinary differential equation coinciding with the Kolmogorov backward equation of a continuous-time Markov chain on the graph with absorbing states corresponding to the stubborn agents, and hence to converge to a harmonic vector, with every regular agent's value being the weighted average of its neighbors' values, and boundary conditions corresponding to the stubborn agents' beliefs. Expected cross products of the agents' beliefs allow for a similar characterization in terms of coupled Markov chains on the graph describing the social network.
We prove that, in large-scale societies, which are highly fluid, meaning that the product of the mixing time of the Markov chain on the graph describing the social network and the relative size of the linkages to stubborn agents vanishes as the population size grows large, a condition of homogeneous influence emerges, whereby the stationary beliefs' marginal distributions of most of the regular agents have approximately equal first and second moments.
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TL;DR: It is shown that all opinions in a large society converge to the truth if and only if the influence of the most influential agent vanishes as the society grows.
Abstract: We study learning in a setting where agents receive independent noisy signals about the true value of a variable and then communi cate in a network. They naively update beliefs by repeatedly taking weighted averages of neighbors' opinions. We show that all opinions in a large society converge to the truth if and only if the influence of the most influential agent vanishes as the society grows. We also identify obstructions to this, including prominent groups, and pro vide structural conditions on the network ensuring efficient learn ing. Whether agents converge to the truth is unrelated to how quickly consensus is approached. (JEL D83, D85, Z13)
1,044 citations
TL;DR: The aim of this tutorial is to highlight a novel chapter of control theory, dealing with applications to social systems, to the attention of the broad research community.
382 citations
01 Dec 2013
TL;DR: It is shown that the presence of stubborn agents with opposing opinions precludes convergence to consensus; instead, opinions converge in distribution with disagreement and fluctuations.
Abstract: We study binary opinion dynamics in a social network with stubborn agents who influence others but do not change their opinions. We focus on a generalization of 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 long-run expected opinions of agents.
310 citations
Cites background from "Opinion Fluctuations and Disagreeme..."
...Finally, another literature also discusses opinion dynamics in a Bayesian setting, where individuals either observe the actions of others and/or communicate with them and update their beliefs about an underlying state variable (see, e.g., [Acemoglu et al. 2010b; Banerjee 1992; Jackson 2008])....
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...Finally, Acemoglu et al. study opinion .uctuations and disagreement over a general social network in a model with continuous opinions ([Acemoglu et al. 2010a])....
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TL;DR: The second edition of Stroock's text as discussed by the authors is suitable for first-year graduate students with a good grasp of introductory, undergraduate probability theory and a sound grounding in analysis and includes more than 750 exercises.
Abstract: This second edition of Daniel W. Stroock's text is suitable for first-year graduate students with a good grasp of introductory, undergraduate probability theory and a sound grounding in analysis. It is intended to provide readers with an introduction to probability theory and the analytic ideas and tools on which the modern theory relies. It includes more than 750 exercises. Much of the content has undergone significant revision. In particular, the treatment of Levy processes has been rewritten, and a detailed account of Gaussian measures on a Banach space is given. The first part of the book deals with independent random variables, Central Limit phenomena, and the construction of Levy processes, including Brownian motion. Conditioning is developed and applied to discrete parameter martingales in Chapter 5, Chapter 6 contains the ergodic theorem and Burkholder's inequality, and continuous parameter martingales are discussed in Chapter 7. Chapter 8 is devoted to Gaussian measures on a Banach space, where they are treated from the abstract Wiener space perspective. The abstract theory of weak convergence is developed in Chapter 9, which ends with a proof of Donsker's Invariance Principle. The concluding two chapters contain applications of Brownian motion to the analysis of partial differential equations and potential theory.
269 citations
10 Dec 2012
TL;DR: In this article, the authors generalize DeGroot's model to account for a phenomenon well-known in social psychology as biased assimilation: when presented with mixed or inconclusive evidence on a complex issue, individuals draw undue support for their initial position thereby arriving at a more extreme opinion.
Abstract: Are we as a society getting more polarized, and if so, why? We try to answer this question through a model of opinion formation. Empirical studies have shown that homophily results in polarization. However, we show that DeGroot's well-known model of opinion formation based on repeated averaging can never be polarizing, even if individuals are arbitrarily homophilous. We generalize DeGroot's model to account for a phenomenon well-known in social psychology as biased assimilation: when presented with mixed or inconclusive evidence on a complex issue, individuals draw undue support for their initial position thereby arriving at a more extreme opinion. We show that in a simple model of homophilous networks, our biased opinion formation process results in either polarization, persistent disagreement or consensus depending on how biased individuals are. In other words, homophily alone, without biased assimilation, is not sufficient to polarize society. Quite interestingly, biased assimilation also provides insight into the following related question: do internet based recommender algorithms that show us personalized content contribute to polarization? We make a connection between biased assimilation and the polarizing effects of some random-walk based recommender algorithms that are similar in spirit to some commonly used recommender algorithms.
267 citations
References
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TL;DR: Simple models of networks that can be tuned through this middle ground: regular networks ‘rewired’ to introduce increasing amounts of disorder are explored, finding that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs.
Abstract: Networks of coupled dynamical systems have been used to model biological oscillators, Josephson junction arrays, excitable media, neural networks, spatial games, genetic control networks and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks 'rewired' to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them 'small-world' networks, by analogy with the small-world phenomenon (popularly known as six degrees of separation. The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.
39,297 citations
"Opinion Fluctuations and Disagreeme..." refers background in this paper
...(Watts & Strogatz’s small world) Watts and Strogatz [50], and then Newman and Watts [40] proposed simple models of random graphs to explain the empirical evidence that most social networks contain a large number of triangles and have a small diameter (the latter has become known as the small-world phenomenon)....
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...Example 6.7 (Watts & Strogatz’s small world) Watts and Strogatz [47], and then Newman and Watts [37] proposed simple models of random graphs to explain the empirical evidence that most social networks contain a large number of triangles and have a small diameter (the latter has become known as the small-world phenomenon)....
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TL;DR: A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.
Abstract: Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature was found to be a consequence of two generic mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.
33,771 citations
Additional excerpts
...(Preferential attachment) The preferential attachment model was introduced by Barabasi and Albert [8] to model real-world networks which typically exhibit a power law degree distribution....
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...Example 6.6 (Preferential attachment) The preferential attachment model was introduced by Barabasi and Albert [8] to model real-world networks which typically exhibit a power law degree dis- tribution....
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TL;DR: A distinctive feature of this work is to address consensus problems for networks with directed information flow by establishing a direct connection between the algebraic connectivity of the network and the performance of a linear consensus protocol.
Abstract: In this paper, we discuss consensus problems for networks of dynamic agents with fixed and switching topologies. We analyze three cases: 1) directed networks with fixed topology; 2) directed networks with switching topology; and 3) undirected networks with communication time-delays and fixed topology. We introduce two consensus protocols for networks with and without time-delays and provide a convergence analysis in all three cases. We establish a direct connection between the algebraic connectivity (or Fiedler eigenvalue) of the network and the performance (or negotiation speed) of a linear consensus protocol. This required the generalization of the notion of algebraic connectivity of undirected graphs to digraphs. It turns out that balanced digraphs play a key role in addressing average-consensus problems. We introduce disagreement functions for convergence analysis of consensus protocols. A disagreement function is a Lyapunov function for the disagreement network dynamics. We proposed a simple disagreement function that is a common Lyapunov function for the disagreement dynamics of a directed network with switching topology. A distinctive feature of this work is to address consensus problems for networks with directed information flow. We provide analytical tools that rely on algebraic graph theory, matrix theory, and control theory. Simulations are provided that demonstrate the effectiveness of our theoretical results.
11,658 citations
"Opinion Fluctuations and Disagreeme..." refers background in this paper
...Our work is also related to work on consensus and gossip algorithms, which is motivated by different problems, but typically leads to a similar mathematical formulation (Tsitsiklis [48], Tsitsiklis, Bertsekas and Athans [49], Jadbabaie, Lin and Morse [28], Olfati-Saber and Murray [42], Olshevsky and Tsitsiklis [43], Fagnani and Zampieri [24], Nedić and Ozdaglar [39])....
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...Our work is also related to work on consensus and gossip algorithms, which is motivated by different problems, but typically leads to a similar mathematical formulation (Tsitsiklis [45], Tsitsiklis, Bertsekas and Athans [46], Jadbabaie, Lin and Morse [26], Olfati-Saber and Murray [39], Olshevsky and Tsitsiklis [40], Fagnani and Zampieri [22], Nedić and Ozdaglar [36])....
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TL;DR: A theoretical explanation for the observed behavior of the Vicsek model, which proves to be a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function.
Abstract: In a recent Physical Review Letters article, Vicsek et al. propose a simple but compelling discrete-time model of n autonomous agents (i.e., points or particles) all moving in the plane with the same speed but with different headings. Each agent's heading is updated using a local rule based on the average of its own heading plus the headings of its "neighbors." In their paper, Vicsek et al. provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models. The Vicsek model proves to be a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function.
8,233 citations
"Opinion Fluctuations and Disagreeme..." refers background in this paper
...Our work is also related to work on consensus and gossip algorithms, which is motivated by different problems, but typically leads to a similar mathematical formulation (Tsitsiklis [48], Tsitsiklis, Bertsekas and Athans [49], Jadbabaie, Lin and Morse [28], Olfati-Saber and Murray [42], Olshevsky and Tsitsiklis [43], Fagnani and Zampieri [24], Nedić and Ozdaglar [39])....
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...Our work is also related to work on consensus and gossip algorithms, which is motivated by different problems, but typically leads to a similar mathematical formulation (Tsitsiklis [45], Tsitsiklis, Bertsekas and Athans [46], Jadbabaie, Lin and Morse [26], Olfati-Saber and Murray [39], Olshevsky and Tsitsiklis [40], Fagnani and Zampieri [22], Nedić and Ozdaglar [36])....
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Book•
02 Jan 2013
TL;DR: In this paper, the authors provide a detailed description of the basic properties of optimal transport, including cyclical monotonicity and Kantorovich duality, and three examples of coupling techniques.
Abstract: Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.
5,524 citations
"Opinion Fluctuations and Disagreeme..." refers background in this paper
...The latter is a metric between probability measures on RV which metrizes weak convergence, and has been made popular by optimal transportation theory: we refer to [45] for definition, and an extensive survey of its properties....
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