A General Model of Opinion Dynamics with Tunable Sensitivity

We present a model of continuous-time opinion dynamics for an arbitrary number of agents that communicate over a network and form real-valued opinions about an arbitrary number of options. The model generalizes linear and nonlinear models in the literature. Drawing from biology, physics, and social psychology, we introduce an attention parameter to modulate social influence and a saturation function to bound inter-agent and intra-agent opinion exchanges. This yields simply parameterized dynamics that exhibit the range of opinion formation behaviors predicted by model-independent bifurcation theory but not exhibited by linear models or existing nonlinear models. Behaviors include rapid and reliable formation of multistable consensus and dissensus states, even in homogeneous networks, as well as ultra-sensitivity to inputs, robustness to uncertainty, flexible transitions between consensus and dissensus, and opinion cascades. Augmenting the opinion dynamics with feedback dynamics for the attention parameter results in tunable thresholds that govern sensitivity and robustness. The model provides new means for systematic study of dynamics on natural and engineered networks, from information spread and political polarization to collective decision making and dynamic task allocation.

[2]  S. Strogatz.  From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators , 2000 .

[3]  N. Franks,et al.  A Mechanism for Value-Sensitive Decision-Making , 2013, PloS one.

[4]  Munakata,et al.  Neural-network model composed of multidimensional spin neurons. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[5]  Philip Holmes,et al.  Simple Neural Networks that Optimize Decisions , 2005, Int. J. Bifurc. Chaos.

[6]  Emilio Salinas,et al.  Gain Modulation A Major Computational Principle of the Central Nervous System , 2000, Neuron.

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

[8]  Naomi Ehrich Leonard,et al.  A Continuous Threshold Model of Cascade Dynamics , 2019, 2019 IEEE 58th Conference on Decision and Control (CDC).

[9]  Mauricio Barahona,et al.  Graph partitions and cluster synchronization in networks of oscillators , 2016, Chaos.

[10]  Paul B. Reverdy,et al.  Dynamical, value-based decision making among N options , 2020, ArXiv.

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

[12]  Andreagiovanni Reina,et al.  Model of the best-of-N nest-site selection process in honeybees. , 2016, Physical review. E.

[13]  Behrouz Touri,et al.  Multi-dimensional Hegselmann-Krause dynamics , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[14]  Claudio Altafini,et al.  Achieving a decision in antagonistic multi agent networks: frustration determines commitment strength* , 2018, 2018 IEEE Conference on Decision and Control (CDC).

[15]  Mattia Frasca,et al.  A criterion for stability of cluster synchronization in networks with external equitable partitions , 2019, Autom..

[16]  Francesco Bullo,et al.  Opinion Dynamics and the Evolution of Social Power in Influence Networks , 2015, SIAM Rev..

[17]  Iain D Couzin,et al.  Collective conflict resolution in groups on the move. , 2018, Physical review. E.

[18]  Rodolphe Sepulchre,et al.  Consensus on nonlinear spaces , 2010, Annu. Rev. Control..

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

[20]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[21]  Alain Sarlette,et al.  Consensus on nonlinear spaces and graph coloring , 2011, IEEE Conference on Decision and Control and European Control Conference.

[22]  J. Cowan,et al.  Excitatory and inhibitory interactions in localized populations of model neurons. , 1972, Biophysical journal.

[23]  Claudio Altafini,et al.  Spectral Conditions for Stability and Stabilization of Positive Equilibria for a Class of Nonlinear Cooperative Systems , 2018, IEEE Transactions on Automatic Control.

[24]  Chris Cosner,et al.  Book Review: Monotone dynamical systems: An introduction to the theory of competitive and cooperative systems , 1996 .

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

[26]  Vaibhav Srivastava,et al.  Multiagent Decision-Making Dynamics Inspired by Honeybees , 2017, IEEE Transactions on Control of Network Systems.

[27]  Yaneer Bar-Yam,et al.  Negative representation and instability in democratic elections , 2018, ArXiv.

[28]  Naomi Ehrich Leonard,et al.  Decision versus compromise for animal groups in motion , 2011, Proceedings of the National Academy of Sciences.

[29]  Steven H. Strogatz,et al.  Jump bifurcation and hysteresis in an infinite-dimensional dynamical system of coupled spins , 1990 .

[30]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Non-linear Systems , 1998, Autom..

[31]  Claudio Altafini,et al.  Dynamics over Signed Networks , 2017, SIAM Rev..

[32]  Naomi Ehrich Leonard,et al.  Patterns of Nonlinear Opinion Formation on Networks , 2020, 2021 American Control Conference (ACC).

[33]  Ye Yuan,et al.  Observability and coarse graining of consensus dynamics through the external equitable partition. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  Guillaume Deffuant,et al.  Mixing beliefs among interacting agents , 2000, Adv. Complex Syst..

[35]  Vaibhav Srivastava,et al.  A Realization Theory for Bio-inspired Collective Decision-Making , 2015, 1503.08526.

[36]  Roberto Tempo,et al.  Network science on belief system dynamics under logic constraints , 2016, Science.

[37]  Brian D. O. Anderson,et al.  Consensus and Disagreement of Heterogeneous Belief Systems in Influence Networks , 2018, IEEE Transactions on Automatic Control.

[38]  Jan Lorenz,et al.  Continuous opinion dynamics of multidimensional allocation problems under bounded confidence: More dimensions lead to better chances for consensus , 2007, 0708.2923.

[39]  Claudio Altafini,et al.  Multiequilibria Analysis for a Class of Collective Decision-Making Networked Systems , 2017, IEEE Transactions on Control of Network Systems.

[40]  Alexandre Pouget,et al.  Optimal policy for multi-alternative decisions , 2019 .

[41]  Roberto Tempo,et al.  A new model of opinion dynamics for social actors with multiple interdependent attitudes and prejudices , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[42]  J J Hopfield,et al.  Neurons with graded response have collective computational properties like those of two-state neurons. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

[43]  Ananthram Swami,et al.  Consensus, Polarization and Clustering of Opinions in Social Networks , 2013, IEEE Journal on Selected Areas in Communications.

[44]  Naomi Ehrich Leonard,et al.  Stability-Flexibility Dilemma in Cognitive Control: A Dynamical System Perspective , 2019, CogSci.

[45]  James L. McClelland,et al.  The time course of perceptual choice: the leaky, competing accumulator model. , 2001, Psychological review.

[46]  Brian D. O. Anderson,et al.  Evolution of Social Power in Social Networks With Dynamic Topology , 2017, IEEE Transactions on Automatic Control.

[47]  Wei Xing Zheng,et al.  Network-based practical set consensus of multi-agent systems subject to input saturation , 2018, Autom..

[48]  B. Piccoli,et al.  A nonlinear model of opinion formation on the sphere , 2015 .

[49]  Weiguo Xia,et al.  Analysis of a nonlinear opinion dynamics model with biased assimilation , 2019, Autom..

[50]  David Lee,et al.  Biased assimilation, homophily, and the dynamics of polarization , 2012, Proceedings of the National Academy of Sciences.

[51]  U. Krause,et al.  Opinion Dynamics Driven by Various Ways of Averaging , 2005 .

[52]  Kevin N. Gurney,et al.  The Basal Ganglia and Cortex Implement Optimal Decision Making Between Alternative Actions , 2007, Neural Computation.

[53]  Roberto Tempo,et al.  Novel Multidimensional Models of Opinion Dynamics in Social Networks , 2015, IEEE Transactions on Automatic Control.

[54]  Claudio Altafini,et al.  Consensus Problems on Networks With Antagonistic Interactions , 2013, IEEE Transactions on Automatic Control.

[55]  Jinde Cao,et al.  Event-triggered group consensus for multi-agent systems subject to input saturation , 2018, J. Frankl. Inst..

[56]  Francesco Bullo,et al.  Polarization and Fluctuations in Signed Social Networks , 2019, IEEE Transactions on Automatic Control.

[57]  A. Michel,et al.  Robustness and Perturbation Analysis of a Class of Nonlinear Systems with Applications to Neural Networks , 1993, 1993 American Control Conference.

[58]  Richard M. Murray,et al.  Consensus problems in networks of agents with switching topology and time-delays , 2004, IEEE Transactions on Automatic Control.

[59]  Marcus Pivato,et al.  Symmetry Groupoids and Patterns of Synchrony in Coupled Cell Networks , 2003, SIAM J. Appl. Dyn. Syst..

[60]  Vittorio Loreto,et al.  Opinion Dynamics with Disagreement and Modulated Information , 2012, ArXiv.

[61]  Tamer Basar,et al.  Exponential Convergence of the Discrete- and Continuous-Time Altafini Models , 2017, IEEE Transactions on Automatic Control.

[62]  Dewei Li,et al.  Bipartite Consensus on Matrix-Valued Weighted Networks , 2019, IEEE Transactions on Circuits and Systems II: Express Briefs.

[63]  Naomi Ehrich Leonard,et al.  Dynamics of Decision Making in Animal Group Motion , 2009, J. Nonlinear Sci..

[64]  J. Cowan,et al.  A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue , 1973, Kybernetik.

[65]  Naomi Ehrich Leonard,et al.  Stabilization of Planar Collective Motion With Limited Communication , 2008, IEEE Transactions on Automatic Control.

[66]  Naomi Ehrich Leonard,et al.  Stabilization of Planar Collective Motion: All-to-All Communication , 2007, IEEE Transactions on Automatic Control.

[67]  J J Hopfield,et al.  Neural networks and physical systems with emergent collective computational abilities. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

[68]  V. Latora,et al.  VECTOR OPINION DYNAMICS IN A BOUNDED CONFIDENCE CONSENSUS MODEL , 2005, physics/0504017.