Continuous-Time Opinion Dynamics With Stochastic Multiplicative Noises

Recently, the study of opinion dynamics on complex networks has attracted a great deal of attention. In this brief, we take an interest in how an opinion forms in a social network with stochastic noises. We investigate a stochastic opinion dynamics model, in which individuals obtain information from their neighbors in a social network. We model the social network as a strongly connected graph. Each individual has an initial opinion and iteratively exchanges her opinion based on a stubborn opinion, her neighbors’ opinions and small stochastic noises continually. We first construct a general stochastic opinion evolution problem with multiplicative noises as a stochastic differential equation and then investigate the stochastic stability of this equation. We consider whether an opinion may form as a result of stochastic disturbance and local interactions. We also establish two sufficient conditions which ensure that the average opinion and the error vector system for all individuals are stochastic stable, respectively. Numerical simulations are given to demonstrate the effectiveness of the theoretical results.

[1]  Qipeng Liu,et al.  Opinion dynamics with similarity-based random neighbors , 2013, Scientific Reports.

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

[3]  Antoine Girard,et al.  Opinion Dynamics With Decaying Confidence: Application to Community Detection in Graphs , 2009, IEEE Transactions on Automatic Control.

[4]  Guanrong Chen,et al.  Fully Distributed Event-Triggered Semiglobal Consensus of Multi-agent Systems With Input Saturation , 2017, IEEE Transactions on Industrial Electronics.

[5]  Xuerong Mao,et al.  Almost Sure Exponential Stability of Stochastic Differential Delay Equations , 2016, SIAM J. Control. Optim..

[6]  François Baccelli,et al.  Pairwise stochastic bounded confidence opinion dynamics: Heavy tails and stability , 2015, 2015 IEEE Conference on Computer Communications (INFOCOM).

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

[8]  Yongming Li,et al.  Adaptive Neural Network Output Feedback Control for Stochastic Nonlinear Systems With Unknown Dead-Zone and Unmodeled Dynamics , 2014, IEEE Transactions on Cybernetics.

[9]  Constantino M. Lagoa,et al.  Randomized opinion dynamics over networks: influence estimation from partial observations , 2018, 2018 IEEE Conference on Decision and Control (CDC).

[10]  Liang Chen,et al.  Synchronization: An Obstacle to Identification of Network Topology , 2009, IEEE Transactions on Circuits and Systems II: Express Briefs.

[11]  Frank Kozin,et al.  A survey of stability of stochastic systems , 1969, Autom..

[12]  E. Wong,et al.  Stochastic Processes in Engineering Systems , 1984 .

[13]  Xiangpeng Xie,et al.  Observer-Based Non-PDC Control for Networked T–S Fuzzy Systems With an Event-Triggered Communication , 2017, IEEE Transactions on Cybernetics.

[14]  Asuman E. Ozdaglar,et al.  Opinion Fluctuations and Disagreement in Social Networks , 2010, Math. Oper. Res..

[15]  Sulan Li,et al.  Bipartite Consensus in Networks of Agents With Antagonistic Interactions and Quantization , 2018, IEEE Transactions on Circuits and Systems II: Express Briefs.

[16]  Dario Bauso,et al.  Opinion dynamics and stubbornness through mean-field games , 2013, 52nd IEEE Conference on Decision and Control.

[17]  G. Weisbuch,et al.  Persuasion dynamics , 2004, cond-mat/0410200.

[18]  Dario Bauso,et al.  Opinion Dynamics and Stubbornness Via Multi-Population Mean-Field Games , 2016, J. Optim. Theory Appl..

[19]  James Lam,et al.  Positive Edge-Consensus for Nodal Networks via Output Feedback , 2019, IEEE Transactions on Automatic Control.

[20]  Brian D. O. Anderson,et al.  Continuous-time Opinion Dynamics on Multiple Interdependent Topics , 2018, Autom..

[21]  Housheng Su,et al.  Discrete-Time Positive Edge-Consensus for Undirected and Directed Nodal Networks , 2018, IEEE Transactions on Circuits and Systems II: Express Briefs.

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

[23]  Xiao Fan Wang,et al.  Nonnegative Edge Quasi-Consensus of Networked Dynamical Systems , 2017, IEEE Transactions on Circuits and Systems II: Express Briefs.

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

[25]  C. K. Michael Tse,et al.  Adaptive Feedback Synchronization of a General Complex Dynamical Network With Delayed Nodes , 2008, IEEE Transactions on Circuits and Systems II: Express Briefs.

[26]  R. Bucy,et al.  Stability and positive supermartingales , 1965 .

[27]  S. Aachen.  Stochastic Differential Equations An Introduction With Applications , 2016 .

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

[29]  Noah E. Friedkin,et al.  Social Influence Network Theory: A Sociological Examination of Small Group Dynamics , 2011 .

[30]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .