Consensus and polarization in Altafini's model with bidirectional time-varying network topologies

The mechanism of reaching consensus in multi-agent systems has been exhaustively studied in recent years, motivated by numerous applications in engineering and science. Most consensus algorithms examined in the literature are based on the assumption about mutual trust and cooperation between the agents, implemented in the form of attractive couplings between the agents that render the values of the agents' states closer. However, in opinion dynamics of social groups, competition or antagonism between some pairs of agents is ubiquitous, which is usually characterized by the repulsive coupling, and may lead to clustering and polarization of opinions. A simple yet insightful model of opinion dynamics with antagonistic interactions was proposed recently by C. Altafini, which examined conventional first-order consensus algorithms with static signed interaction graphs, where the positive weight of an arc implies cooperation between the two agents and the negative one corresponds to antagonism. This protocol establishes modulus consensus, where the opinions become the same in modulus but may differ in sign. In the present paper, we extend the modulus consensus model to the case where the network topology is time-varying and undirected. We give necessary and sufficient conditions under which the consensus protocol with the time-varying signed Laplacian establishes agreement of opinions in moduli, whose signs may be opposite, so that the agents' opinions either reach consensus or polarize.

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

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

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

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

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

[6]  Ziyang Meng,et al.  Modulus Consensus over Networks with Antagonistic Interactions and Switching Topologies , 2014, 1402.2766.

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

[8]  Jon M. Kleinberg,et al.  Networks, Crowds, and Markets: Reasoning about a Highly Connected World [Book Review] , 2013, IEEE Technol. Soc. Mag..

[9]  J.N. Tsitsiklis,et al.  Convergence in Multiagent Coordination, Consensus, and Flocking , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

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

[11]  Hal L. Smith.  Systems of ordinary differential equations which generate an order preserving flow. A survey of results , 1988 .

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

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

[14]  David Angeli,et al.  Stability of leaderless discrete-time multi-agent systems , 2006, Math. Control. Signals Syst..

[15]  Jiangping Hu,et al.  Bipartite Consensus Control of Multiagent Systems on Coopetition Networks , 2014 .

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

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

[18]  Rodolphe Sepulchre,et al.  Synchronization in networks of identical linear systems , 2009, Autom..

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

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

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

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

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

[24]  Rodolphe Sepulchre,et al.  Synchronization in networks of identical linear systems , 2008, 2008 47th IEEE Conference on Decision and Control.

[25]  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).

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

[27]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

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