Reaching a Consensus
01 Mar 1974-Journal of the American Statistical Association (Taylor & Francis Group)-Vol. 69, Iss: 345, pp 118-121
TL;DR: In this article, the authors consider a group of individuals who must act together as a team or committee, and assume that each individual in the group has his own subjective probability distribution for the unknown value of some parameter.
Abstract: Consider a group of individuals who must act together as a team or committee, and suppose that each individual in the group has his own subjective probability distribution for the unknown value of some parameter. A model is presented which describes how the group might reach agreement on a common subjective probability distribution for the parameter by pooling their individual opinions. The process leading to the consensus is explicitly described and the common distribution that is reached is explicitly determined. The model can also be applied to problems of reaching a consensus when the opinion of each member of the group is represented simply as a point estimate of the parameter rather than as a probability distribution.
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TL;DR: Developments in this field are reviewed, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
Abstract: Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
17,647 citations
05 Mar 2007
TL;DR: A theoretical framework for analysis of consensus algorithms for multi-agent networked systems with an emphasis on the role of directed information flow, robustness to changes in network topology due to link/node failures, time-delays, and performance guarantees is provided.
Abstract: This paper provides a theoretical framework for analysis of consensus algorithms for multi-agent networked systems with an emphasis on the role of directed information flow, robustness to changes in network topology due to link/node failures, time-delays, and performance guarantees. An overview of basic concepts of information consensus in networks and methods of convergence and performance analysis for the algorithms are provided. Our analysis framework is based on tools from matrix theory, algebraic graph theory, and control theory. We discuss the connections between consensus problems in networked dynamic systems and diverse applications including synchronization of coupled oscillators, flocking, formation control, fast consensus in small-world networks, Markov processes and gossip-based algorithms, load balancing in networks, rendezvous in space, distributed sensor fusion in sensor networks, and belief propagation. We establish direct connections between spectral and structural properties of complex networks and the speed of information diffusion of consensus algorithms. A brief introduction is provided on networked systems with nonlocal information flow that are considerably faster than distributed systems with lattice-type nearest neighbor interactions. Simulation results are presented that demonstrate the role of small-world effects on the speed of consensus algorithms and cooperative control of multivehicle formations
9,715 citations
Cites background from "Reaching a Consensus"
...Formal study of consensus problems in groups of experts originated in management science and statistics in 1960’s (See DeGroot [19] and references therein)....
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TL;DR: It is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence, even for the simple models discussed in the present paper.
Abstract: We study a simple but compelling model of network of agents interacting via time-dependent communication links. The model finds application in a variety of fields including synchronization, swarming and distributed decision making. In the model, each agent updates his current state based upon the current information received from neighboring agents. Necessary and/or sufficient conditions for the convergence of the individual agents' states to a common value are presented, thereby extending recent results reported in the literature. The stability analysis is based upon a blend of graph-theoretic and system-theoretic tools with the notion of convexity playing a central role. The analysis is integrated within a formal framework of set-valued Lyapunov theory, which may be of independent interest. Among others, it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence, even for the simple models discussed in the present paper.
2,828 citations
Additional excerpts
...6] and [33]....
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Posted Content•
TL;DR: This article investigates various models for the dynamics of continuous opinions by analytical methods as well as by computer simulations for the classical model of consensus formation, a time-dependent version and a nonlinear version with bounded confidence of the agents.
Abstract: 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. Sec- tion 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 ap- pendix supplies needed mathematical definitions, tools, and theorems.
2,425 citations
Cites background or methods from "Reaching a Consensus"
...Another source of opinion dynamics is the work by M.H. De Groot in 1974 (De Groot 1974) and by K. Lehrer in 1975 (Lehrer 1975) and subsequent work by C. G. Wagner, S. Chatterjee, E. Seneta, J. Cohen among others....
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...This model has been proposed and employed to assess opinion pooling by a dialogue among experts (De Groot 1974), (Lehrer 1975)....
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...For the first part of Theorem 1 see (De Groot 1974), for the second part see (Berger 1981)....
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...In this case A is the identity matrix and, of course, there is no consensus, except the special case of a consensus already in the initial opinion profile....
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...There is an interesting connection to the Delphi technique for pooling opinions of experts, (cf. (De Groot 1974))....
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TL;DR: In this article, the authors provide a review and annotated bibliography of that literature, including contributions from the forecasting, psychology, statistics, and management science literatures, providing a guide to the literature for students and researchers and to help researchers locate contributions in specific areas, both theoretical and applied.
2,269 citations
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Book•
01 Jan 1966
TL;DR: In this paper, the Basic Limit Theorem of Markov Chains and its applications are discussed and examples of continuous time Markov chains are presented. But they do not cover the application of continuous-time Markov chain in matrix analysis.
Abstract: Preface. Elements of Stochastic Processes. Markov Chains. The Basic Limit Theorem of Markov Chains and Applications. Classical Examples of Continuous Time Markov Chains. Renewal Processes. Martingales. Brownian Motion. Branching Processes. Stationary Processes. Review of Matrix Analysis. Index.
3,881 citations