Differential Inequalities in Multi-Agent Coordination and Opinion Dynamics Modeling

Abstract Many distributed algorithms for multi-agent coordination employ the simple averaging dynamics, referred to as the Laplacian flow . Besides the standard consensus protocols, examples include, but are not limited to, algorithms for aggregation and containment control, target surrounding, distributed optimization and models of opinion formation in social groups. In spite of their similarities, each of these algorithms has been studied using separate mathematical techniques. In this paper, we show that stability and convergence of many coordination algorithms involving the Laplacian flow dynamics follow from the general consensus dichotomy property of a special differential inequality. The consensus dichotomy implies that any solution to the differential inequality is either unbounded or converges to a consensus equilibrium. In this paper, we establish the dichotomy criteria for differential inequalities and illustrate their applications to multi-agent coordination and opinion dynamics modeling.

[1]  Youcheng Lou,et al.  Distributed Surrounding Design of Target Region With Complex Adjacency Matrices , 2015, IEEE Transactions on Automatic Control.

[2]  A. Jadbabaie,et al.  Effects of Delay in Multi-Agent Consensus and Oscillator Synchronization , 2010, IEEE Transactions on Automatic Control.

[3]  Frank Allgöwer,et al.  Consensus in Multi-Agent Systems With Coupling Delays and Switching Topology , 2011, IEEE Transactions on Automatic Control.

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

[5]  Michaell Taylor,et al.  Towards a Mathematical Theory of Influence and Attitude Change , 1968 .

[6]  Jie Lin,et al.  Coordination of groups of mobile autonomous agents using nearest neighbor rules , 2003, IEEE Trans. Autom. Control..

[7]  Miguel Parada Contzen.  Consensus in multiagent systems , 2017 .

[8]  Alexey S. Matveev,et al.  Opinion Dynamics in Social Networks With Hostile Camps: Consensus vs. Polarization , 2015, IEEE Transactions on Automatic Control.

[9]  W. Zheng,et al.  Emergent collective behaviors on coopetition networks , 2014 .

[10]  Karl Henrik Johansson,et al.  Robust Consensus for Continuous-Time Multiagent Dynamics , 2013, SIAM J. Control. Optim..

[11]  Anton V. Proskurnikov,et al.  Average consensus in networks with nonlinearly delayed couplings and switching topology , 2013, Autom..

[12]  Claudio Altafini,et al.  Dynamics of Opinion Forming in Structurally Balanced Social Networks , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[13]  John N. Tsitsiklis,et al.  Convergence of Type-Symmetric and Cut-Balanced Consensus Seeking Systems , 2011, IEEE Transactions on Automatic Control.

[14]  Magnus Egerstedt,et al.  Graph Theoretic Methods in Multiagent Networks , 2010, Princeton Series in Applied Mathematics.

[15]  Karl Henrik Johansson,et al.  Reaching an Optimal Consensus: Dynamical Systems That Compute Intersections of Convex Sets , 2011, IEEE Transactions on Automatic Control.

[16]  L. Moreau,et al.  Stability of continuous-time distributed consensus algorithms , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[17]  Brian D. O. Anderson,et al.  Reaching a Consensus in a Dynamically Changing Environment: Convergence Rates, Measurement Delays, and Asynchronous Events , 2008, SIAM J. Control. Optim..

[18]  Randal W. Beard,et al.  Distributed Consensus in Multi-vehicle Cooperative Control - Theory and Applications , 2007, Communications and Control Engineering.

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

[20]  Alexey S. Matveev,et al.  Consensus and polarization in Altafini's model with bidirectional time-varying network topologies , 2014, 53rd IEEE Conference on Decision and Control.

[21]  Anton V. Proskurnikov,et al.  Consensus in switching networks with sectorial nonlinear couplings: Absolute stability approach , 2013, Autom..

[22]  Ziyang Meng,et al.  Behaviors of networks with antagonistic interactions and switching topologies , 2016, Autom..

[23]  Andrey V. Savkin,et al.  Decentralized Coverage Control Problems for Mobile Robotic Sensor and Actuator Networks: Savkin/Decentralized , 2015 .

[24]  Frank Allgöwer,et al.  An internal model principle is necessary and sufficient for linear output synchronization , 2011, Autom..

[25]  Abhijit Das,et al.  Cooperative Control of Multi-Agent Systems , 2014 .

[26]  Julien M. Hendrickx,et al.  A lifting approach to models of opinion dynamics with antagonisms , 2014, 53rd IEEE Conference on Decision and Control.

[27]  Mehmet E. Yildiz,et al.  Binary Opinion Dynamics with Stubborn Agents , 2013, TEAC.

[28]  Ming Cao,et al.  Modulus consensus in discrete-time signed networks and properties of special recurrent inequalities , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[29]  V. Yakubovich.  Dichotomy and absolute stability of nonlinear systems with periodically nonstationary linear part , 1988 .

[30]  G. Pecelli,et al.  Dichotomies for linear functional-differential equations , 1971 .

[31]  Mingjun Du,et al.  Interval Bipartite Consensus of Networked Agents Associated With Signed Digraphs , 2016, IEEE Transactions on Automatic Control.

[32]  Maria Elena Valcher,et al.  On the consensus and bipartite consensus in high-order multi-agent dynamical systems with antagonistic interactions , 2014, Syst. Control. Lett..

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

[34]  Manfredi Maggiore,et al.  State Agreement for Continuous-Time Coupled Nonlinear Systems , 2007, SIAM J. Control. Optim..

[35]  Li Qiu,et al.  Complex Laplacians and Applications in Multi-Agent Systems , 2014 .

[36]  Brian D. O. Anderson,et al.  Reaching a Consensus in a Dynamically Changing Environment: Convergence Rates, Measurement Delays, and Asynchronous Events , 2008, SIAM J. Control. Optim..

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

[38]  Ming Cao,et al.  CONSENSUS IN MULTI-AGENT SYSTEMS , 2016 .

[39]  Karl Henrik Johansson,et al.  The Role of Persistent Graphs in the Agreement Seeking of Social Networks , 2011, IEEE Journal on Selected Areas in Communications.

[40]  Ming Cao,et al.  Opinion dynamics using Altafini's model with a time-varying directed graph , 2014, 2014 IEEE International Symposium on Intelligent Control (ISIC).

[41]  Karl Henrik Johansson,et al.  Structural Balance and Opinion Separation in Trust–Mistrust Social Networks , 2016, IEEE Transactions on Control of Network Systems.

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

[43]  John N. Tsitsiklis,et al.  Continuous-Time Average-Preserving Opinion Dynamics with Opinion-Dependent Communications , 2009, SIAM J. Control. Optim..

[44]  Yongcan Cao,et al.  Distributed Coordination of Multi-agent Networks , 2011 .

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

[46]  Ming Cao,et al.  Clustering in diffusively coupled networks , 2011, Autom..

[47]  Samuel Martin,et al.  Continuous-Time Consensus Under Non-Instantaneous Reciprocity , 2014, IEEE Transactions on Automatic Control.

[48]  Frank L. Lewis,et al.  Cooperative Control of Multi-Agent Systems: Optimal and Adaptive Design Approaches , 2013 .

[49]  M. Macy,et al.  Small Worlds and Cultural Polarization , 2011 .

[50]  Norman,et al.  Structural Models: An Introduction to the Theory of Directed Graphs. , 1966 .

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

[52]  Giancarlo Ferrari-Trecate,et al.  Analysis of coordination in multi-agent systems through partial difference equations , 2006, IEEE Transactions on Automatic Control.

[53]  P. Chebotarev,et al.  On of the Spectra of Nonsymmetric Laplacian Matrices , 2004, math/0508176.

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

[55]  B. A. Farbey,et al.  Structural Models: An Introduction to the Theory of Directed Graphs , 1966 .

[56]  Luc Moreau,et al.  Stability of multiagent systems with time-dependent communication links , 2005, IEEE Transactions on Automatic Control.

[57]  Guodong Shi,et al.  Global target aggregation and state agreement of nonlinear multi-agent systems with switching topologies , 2009, Autom..

[58]  Roberto Tempo,et al.  A Tutorial on Modeling and Analysis of Dynamic Social Networks. Part II , 2018, Annu. Rev. Control..

[59]  Alexey S. Matveev,et al.  Stability of continuous-time consensus algorithms for switching networks with bidirectional interaction , 2013, 2013 European Control Conference (ECC).

[60]  Wenwu Yu,et al.  An Overview of Recent Progress in the Study of Distributed Multi-Agent Coordination , 2012, IEEE Transactions on Industrial Informatics.

[61]  Tamer Basar,et al.  Stability of the continuous-time Altafini model , 2016, 2016 American Control Conference (ACC).

[62]  Tamer Basar,et al.  Stability of discrete-time Altafini's model: A graphical approach , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[63]  Ming Cao,et al.  Dichotomic differential inequalities and multi-agent coordination , 2016, 2016 European Control Conference (ECC).

[64]  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.