Fuzzy Opinion Networks: A Mathematical Framework for the Propagation of Opinions and Their Uncertainties Across Social Networks

We propose a new mathematical framework for the evolution and propagation of opinions, called fuzzy opinion network, which is the connection of a number of Gaussian nodes, possibly through some weighted average, time delay, or logic operators, where a Gaussian node is a Gaussian fuzzy set with the center and the standard deviation being the node inputs and the fuzzy set itself being the node output. In this framework, an opinion is modeled as a Gaussian fuzzy set with the center representing the opinion itself and the standard deviation characterizing the uncertainty about the opinion. We study the basic connections of fuzzy opinion networks, including basic center, basic standard deviation (sdv), basic center-sdv, chain-in-center, and chain-in-sdv connections, and we analyze a number of dynamic connections to show how opinions and their uncertainties propagate and evolve across different network structures and scenarios. We explain what insights we might gain from these mathematical results about the formation and evolution of human opinions.

[1]  J. Keynes,et al.  The General Theory of Employment, Interest and Money. , 1936 .

[2]  H. Barger.  The General Theory of Employment, Interest and Money , 1936, Nature.

[3]  J. Keynes,et al.  The General Theory of Employment, Interest and Money. , 1936 .

[4]  Lotfi A. Zadeh,et al.  Outline of a New Approach to the Analysis of Complex Systems and Decision Processes , 1973, IEEE Trans. Syst. Man Cybern..

[5]  M. Degroot.  Reaching a Consensus , 1974 .

[6]  L. A. ZADEH,et al.  The concept of a linguistic variable and its application to approximate reasoning - I , 1975, Inf. Sci..

[7]  Lotfi A. Zadeh,et al.  The Concepts of a Linguistic Variable and its Application to Approximate Reasoning , 1975 .

[8]  Lotfi A. Zadeh,et al.  The concept of a linguistic variable and its application to approximate reasoning-III , 1975, Inf. Sci..

[9]  S. Fiske,et al.  Social Psychology , 2019, Encyclopedia of Personality and Individual Differences.

[10]  B. Bollobás.  The evolution of random graphs , 1984 .

[11]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[12]  John G. Proakis,et al.  Probability, random variables and stochastic processes , 1985, IEEE Trans. Acoust. Speech Signal Process..

[13]  D. Isenberg.  Group polarization: A critical review and meta-analysis. , 1986 .

[14]  Philip Mirowski,et al.  More Heat Than Light: Economics as Social Physics , 1989 .

[15]  Philip Mirowski,et al.  More Heat Than Light: Economics as Social Physics , 1989 .

[16]  George J. Klir,et al.  A principle of uncertainty and information invariance , 1990 .

[17]  Eugene Charniak,et al.  Bayesian Networks without Tears , 1991, AI Mag..

[18]  Philip Mirowski,et al.  More Heat Than Light. Economics as Social Physics: Physics as Nature's Economics. , 1991 .

[19]  Peter Norvig,et al.  Artificial Intelligence: A Modern Approach , 1995 .

[20]  Sanjeev Goyal,et al.  Learning from Neighbors , 1995 .

[21]  Li-Xin Wang,et al.  A Course In Fuzzy Systems and Control , 1996 .

[22]  S. Goyal,et al.  Learning from neighbours , 1998 .

[23]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[24]  D. Sornette,et al.  Stretched exponential distributions in nature and economy: “fat tails” with characteristic scales , 1998, cond-mat/9801293.

[25]  C. Sunstein.  The Law of Group Polarization , 1999, How Change Happens.

[26]  Zoubin Ghahramani,et al.  A Unifying Review of Linear Gaussian Models , 1999, Neural Computation.

[27]  L. Zadeh.  Fuzzy sets as a basis for a theory of possibility , 1999 .

[28]  M. Ferguson,et al.  Beyond behaviorism: on the automaticity of higher mental processes. , 2000, Psychological bulletin.

[29]  Finn V. Jensen,et al.  Bayesian Networks and Decision Graphs , 2001, Statistics for Engineering and Information Science.

[30]  S. Strogatz.  Exploring complex networks , 2001, Nature.

[31]  Valdis E. Krebs,et al.  Mapping Networks of Terrorist Cells , 2001 .

[32]  Rainer Hegselmann,et al.  Opinion dynamics and bounded confidence: models, analysis and simulation , 2002, J. Artif. Soc. Soc. Simul..

[33]  D. Kahneman.  Maps of Bounded Rationality , 2002 .

[34]  D. Kahneman.  Maps of Bounded Rationality: Psychology for Behavioral Economics , 2003 .

[35]  P. Bearman,et al.  Cloning Headless Frogs and Other Important Matters: Conversation Topics and Network Structure , 2004 .

[36]  James A. Robinson,et al.  Institutions as a Fundamental Cause of Long-Run Growth , 2005 .

[37]  James Surowiecki.  The wisdom of crowds: Why the many are smarter than the few and how collective wisdom shapes business, economies, societies, and nations Doubleday Books. , 2004 .

[38]  Daron Acemoglu,et al.  Economic Origins of Dictatorship and Democracy , 2005 .

[39]  C. Hommes.  Heterogeneous Agent Models in Economics and Finance , 2005 .

[40]  Jan Lorenz,et al.  A stabilization theorem for dynamics of continuous opinions , 2005, 0708.2981.

[41]  Lotfi A. Zadeh,et al.  From Computing with Numbers to Computing with Words - from Manipulation of Measurements to Manipulation of Perceptions , 2005, Logic, Thought and Action.

[42]  Antoni Calvó-Armengol,et al.  Centre De Referència En Economia Analítica Barcelona Economics Working Paper Series Working Paper Nº 178 Who's Who in Networks. Wanted: the Key Player Who's Who in Networks. Wanted: the Key Player Barcelona Economics Wp Nº 178 , 2022 .

[43]  Mark E. J. Newman,et al.  Structure and Dynamics of Networks , 2009 .

[44]  Jerry Mendel,et al.  Type-2 Fuzzy Sets and Systems: An Overview [corrected reprint] , 2007, IEEE Computational Intelligence Magazine.

[45]  Eyke Hüllermeier,et al.  Comparing probability measures using possibility theory: A notion of relative peakedness , 2007, Int. J. Approx. Reason..

[46]  Jan Lorenz,et al.  Continuous Opinion Dynamics under Bounded Confidence: A Survey , 2007, 0707.1762.

[47]  B. Golub,et al.  Naive Learning in Social Networks: Convergence, Influence and Wisdom of Crowds , 2007 .

[48]  Stanley Wasserman,et al.  Social Network Analysis: Methods and Applications , 1994, Structural analysis in the social sciences.

[49]  Reza Olfati-Saber,et al.  Consensus and Cooperation in Networked Multi-Agent Systems , 2007, Proceedings of the IEEE.

[50]  Eliot R. Smith,et al.  Situating Social Influence Processes: Dynamic, Multidirectional Flows of Influence Within Social Networks , 2007, Personality and social psychology review : an official journal of the Society for Personality and Social Psychology, Inc.

[51]  Jerry M. Mendel,et al.  Type-2 fuzzy sets and systems: an overview , 2007, IEEE Computational Intelligence Magazine.

[52]  Thilo Gross,et al.  Adaptive coevolutionary networks: a review , 2007, Journal of The Royal Society Interface.

[53]  A. Barabasi,et al.  Quantifying social group evolution , 2007, Nature.

[54]  Andre C. R. Martins,et al.  Bayesian updating rules in continuous opinion dynamics models , 2008, 0807.4972.

[55]  Pawel Sobkowicz,et al.  Modelling Opinion Formation with Physics Tools: Call for Closer Link with Reality , 2009, J. Artif. Soc. Soc. Simul..

[56]  Asuman Ozdaglar,et al.  Spread of (Mis)Information in Social Networks , 2009 .

[57]  John N. Tsitsiklis,et al.  On Krause's Multi-Agent Consensus Model With State-Dependent Connectivity , 2008, IEEE Transactions on Automatic Control.

[58]  A. Vespignani.  Predicting the Behavior of Techno-Social Systems , 2009, Science.

[59]  Albert-László Barabási,et al.  Scale-Free Networks: A Decade and Beyond , 2009, Science.

[60]  C. T. Butts,et al.  Revisiting the Foundations of Network Analysis , 2009, Science.

[61]  Arthur L. Stinchcombe,et al.  Going to Extremes: How Like Minds Unite and Divide , 2010 .

[62]  Jan Lorenz,et al.  Heterogeneous bounds of confidence: Meet, discuss and find consensus! , 2008, Complex..

[63]  Asuman E. Ozdaglar,et al.  Spread of (Mis)Information in Social Networks , 2009, Games Econ. Behav..

[64]  Jerry M. Mendel,et al.  Perceptual Computing: Aiding People in Making Subjective Judgments , 2010 .

[65]  Asuman E. Ozdaglar,et al.  Constrained Consensus and Optimization in Multi-Agent Networks , 2008, IEEE Transactions on Automatic Control.

[66]  Tom A. B. Snijders,et al.  Social Network Analysis , 2011, International Encyclopedia of Statistical Science.

[67]  Asuman E. Ozdaglar,et al.  Opinion Dynamics and Learning in Social Networks , 2010, Dyn. Games Appl..

[68]  A. Ozdaglar,et al.  Discrete Opinion Dynamics with Stubborn Agents , 2011 .

[69]  Francesco Bullo,et al.  On opinion dynamics in heterogeneous networks , 2010, Proceedings of the 2011 American Control Conference.

[70]  M. Kocher,et al.  Are Groups More Rational than Individuals? A Review of Interactive Decision Making in Groups , 2012, Wiley interdisciplinary reviews. Cognitive science.

[71]  Erik J. Olsson,et al.  Trust and the value of overconfidence: a Bayesian perspective on social network communication , 2013, Synthese.

[72]  Zoé Christoff,et al.  A Two-Tiered Formalization of Social Influence , 2013, LORI.

[73]  Anna Scaglione,et al.  Models for the Diffusion of Beliefs in Social Networks: An Overview , 2013, IEEE Signal Processing Magazine.

[74]  R. Srikant,et al.  Opinion dynamics in social networks with stubborn agents: Equilibrium and convergence rate , 2014, Autom..

[75]  Jerry M. Mendel,et al.  Fuzzy networks: What happens when fuzzy people are connected through social networks , 2014, 2014 IEEE Symposium on Foundations of Computational Intelligence (FOCI).

[76]  Fenrong Liu,et al.  Logical dynamics of belief change in the community , 2014, Synthese.

[77]  N. McGlynn.  Thinking fast and slow. , 2014, Australian veterinary journal.

[78]  Miguel A. Vadillo,et al.  Illusion of Control , 2013, Experimental psychology.

[79]  Noah E. Friedkin,et al.  The Problem of Social Control and Coordination of Complex Systems in Sociology: A Look at the Community Cleavage Problem , 2015, IEEE Control Systems.

[80]  Rainer Hegselmann,et al.  Opinion dynamics under the influence of radical groups, charismatic leaders, and other constant signals: A simple unifying model , 2015, Networks Heterog. Media.

[81]  Dror G. Feitelson,et al.  From Repeatability to Reproducibility and Corroboration , 2015, OPSR.

[82]  Li-Xin Wang,et al.  Dynamical Models of Stock Prices Based on Technical Trading Rules---Part II: Analysis of the Model , 2014, IEEE Transactions on Fuzzy Systems.

[83]  Li-Xin Wang,et al.  Dynamical Models of Stock Prices Based on Technical Trading Rules—Part III: Application to Hong Kong Stocks , 2014, IEEE Transactions on Fuzzy Systems.

[84]  John Bohannon,et al.  REPRODUCIBILITY. Many psychology papers fail replication test. , 2015, Science.

[85]  Camilo Hurtado-Parrado,et al.  Single-Case Research Methods: History and Suitability for a Psychological Science in Need of Alternatives , 2015, Integrative psychological & behavioral science.

[86]  Li-Xin Wang,et al.  Dynamical Models of Stock Prices Based on Technical Trading Rules Part I: The Models , 2014, IEEE Transactions on Fuzzy Systems.