Elsevier

Automatica

Volume 73, November 2016, Pages 110-116
Automatica

Brief paper
Behaviors of networks with antagonistic interactions and switching topologies

https://doi.org/10.1016/j.automatica.2016.06.022Get rights and content

Abstract

In this paper, we study the discrete-time consensus problem over networks with antagonistic and cooperative interactions. A cooperative interaction between two nodes takes place when one node receives the true state of the other while an antagonistic interaction happens when the former receives the opposite of the true state of the latter. We adopt a quite general model where the node communications can be either unidirectional or bidirectional, the network topology graph may vary over time, and the cooperative or antagonistic relations can be time-varying. It is proven that, the limits of all the node states exist, and the absolute values of the node states reach consensus if the switching interaction graph is uniformly jointly strongly connected for unidirectional topologies, or infinitely jointly connected for bidirectional topologies. These results are independent of the switching of the interaction relations. We construct a counterexample to indicate a rather surprising fact that quasi-strong connectivity of the interaction graph, i.e., the graph contains a directed spanning tree, is not sufficient to guarantee the consensus in absolute values even under fixed topologies. Based on these results, we also propose sufficient conditions for bipartite consensus to be achieved over the network with joint connectivity. Finally, simulation results using a discrete-time Kuramoto model are given to illustrate the convergence results showing that the proposed framework is applicable to a class of networks with general nonlinear dynamics.

Introduction

Distributed consensus algorithms were first introduced in the study of distributed optimization methods in Tsitsiklis, Bertsekas, and Athans (1986). Phase synchronization was observed in Vicsek, Czirok, Jacob, Cohen, and Schochet (1995) and its mathematical proof was given in Jadbabaie, Lin, and Morse (2003). The robustness of the consensus algorithm to link/node failures and time-delays was studied in Olfati-Saber, Fax, and Murray (2007). A central problem in consensus study is to investigate the influence of the interaction graph on the convergence or convergence speed of the multi-agent system dynamics. The interaction graph, which describes the information flow among the nodes, is often time-varying due to the complexity of the interaction patterns in practice. Both continuous-time and discrete-time models were studied for consensus algorithms with switching interaction graphs and joint connectivity conditions were established for linear models (Blondel et al., 2005, Cao et al., 2008b, Hendrickx and Tsitsiklis, 2013, Ren and Beard, 2005). Nonlinear multi-agent dynamics have also drawn much attention (Meng et al., 2013, Shi and Hong, 2009) since in many practical problems the node dynamics are naturally nonlinear, e.g., the Kuramoto model (Strogatz, 2000).

Although great progress has been made, most of the existing results are based on the assumption that the agents in the network are cooperative. Recently, motivated by opinion dynamics over social networks (Cartwright and Harary, 1956, Easley and Kleinberg, 2010, Hegselmann and Krause, 2002), consensus algorithms over cooperative–antagonistic networks drew much attention (Altafini, 2012, Altafini, 2013, Shi et al., 2013). Altafini (2013) assumed that a node receives the opposite of the true state of its neighboring node if they are antagonistic. Therefore, the modeling of such an antagonistic input for agent i is of the form (xi+xj) (in contrast to the form (xixj) for cooperative input), where j denotes the antagonistic neighbor of agent i. On the other hand, the authors of Shi et al. (2013) assumed that a node receives the opposite of the relative state from its neighboring node if they are antagonistic. Then, the antagonistic input for agent i is modeled by the form (xixj) in this case. The extension to the case of homogeneous single-input high-order dynamical systems was discussed in Valcher and Misra (2014). The graph was assumed to be fixed and a spectral analysis approach was used. Instead, we will focus on switching topologies with joint connectivity and take advantage of a detailed state-space analysis approach in this paper. A lifting approach was proposed in Hendrickx (2014) to study opinion dynamics with antagonisms over switching interaction graphs. Some general conditions were established by applying the rich results from the consensus literature. The dissensus problem was studied in Bauso, Giarre, and Pesenti (2012), where the focus was to understand when consensus is or is not achieved if death and duplication phenomena occur. Note that in dissensus the control terms for death and duplication phenomena are added to the classical consensus algorithm. However, consensus or bipartite consensus in our study does not denote a control goal, but a final behavior of multi-agent systems. In particular, consensus denotes the final states of all the agents converging to the same value while bipartite consensus denotes the final states of all the agents converging to two opposite values.

Note that most of the existing works on antagonistic interactions are based on the assumption that the interaction graph is fixed. In many practical cases, however, the interactions between agents may vary over time or be dependent on the states. In this paper, we focus on the behavior of multiple agents with antagonistic interactions, discrete-time dynamics, and switching interaction graphs. Both unidirectional and bidirectional topologies are considered. We show that the limits of all node states exist and reach a consensus in absolute values if the switching interaction graph is uniformly jointly strongly connected for unidirectional topologies, or infinitely jointly connected for bidirectional topologies. Here, reaching consensus in absolute values is not a design objective, but rather an emergent behavior that we can observe for cooperative–antagonistic networks. By noting that an antagonistic interaction also represents an arc in the graph, we know that the connectivity of the network may not be guaranteed without antagonistic interactions. Therefore, we actually show that the antagonistic interaction has a similar role as the cooperative interaction in contributing to the consensus of the absolute values of the node states. In addition, a counterexample is constructed that indicates that quasi-strong connectivity of the interaction graph, i.e., the graph has a directed spanning tree, is not sufficient to guarantee consensus of the node states in absolute values even under a fixed topology. Based on these results, we propose sufficient conditions for bipartite consensus to be achieved over a network with joint connectivity. It turns out that the structural balance condition is essentially important and this part of the result can be viewed as an extension of the work (Altafini, 2013) to the case of general time-varying graphs with joint connectivity. A detailed asymptotic analysis is performed with a contradiction argument to show the main results.

Section snippets

Problem formulation and main results

Consider a multi-agent network with agent set V={1,,n}. In the rest of the paper we use agent and node interchangeably. The state-space for the agents is R, and we let xiR denote the state of node i. Set x=(x1,x2,,xn)T.

Proofs

In this section, we present the proofs of the statements. First a key technical lemma is established, and then the proofs of Theorem 2.1, Theorem 2.2, Theorem 2.3 are presented. Since the proofs do not rely on whether or not aij depends on x, without loss of generality, we use aij(k) to denote aij(x,k).

Lemma 1

Suppose that   Assumption  2.1   holds. For system   (1), it holds that x(k+1)x(k), for all k=0,1,.

Proof

It follows from Assumption 2.1 that |xi(k+1)|jNi(k)|aij(k)||xj(k)|(jNi(k)|aij(k)|)

Numerical example

Consider the following discrete-time Kuramoto oscillator system with antagonistic and cooperative links: θi(k+1)=θi(k)μjNi(k){i}sin(θi(k)Rij(k)θj(k)), where θi(k) denotes the state of node i at time k, μ>0 is the stepsize, and Rij(k){1,1} represents the cooperative or antagonistic relationship between node i and node j. Note that with Rij(k)1, system (8) corresponds to the classical Kuramoto oscillator model (Strogatz, 2000). Let δ(0,π2) be a given constant and suppose θi(0)(π2+δ,π2δ

Conclusions

In this paper, we studied the consensus problem of multi-agent systems over cooperative–antagonistic networks in a discrete-time setting. Both unidirectional and bidirectional topologies were considered. It was proven that the limits of all agent states exist and reach a consensus in absolute values if the topology is uniformly jointly strongly connected or infinitely jointly connected. We also gave an example to show that uniform quasi-strong connectedness is not sufficient to guarantee

Ziyang Meng received his Bachelor degree with honors from Huazhong University of Science & Technology, Wuhan, China, in 2006, and Ph.D. degree from Tsinghua University, Beijing, China, in 2010. He was an exchange Ph.D. student at Utah State University, Logan, USA from Sept. 2008 to Sept. 2009. From 2010 to 2015, he held postdoc, researcher, and Humboldt research fellow positions at, respectively, Shanghai Jiao Tong University, Shanghai, China, KTH Royal Institute of Technology, Stockholm,

References (22)

  • D. Cartwright et al.

    Structural balance: a generalization of Heiders theory

    Psychological Review

    (1956)
  • Cited by (164)

    • Edge controllability of signed networks

      2023, Automatica
      Citation Excerpt :

      It is noteworthy that the aforementioned studies (Guan et al., 2017; Ji et al., 2020, 2009; Rahmani et al., 2009; Tanner, 2004; Yazicioǧlu et al., 2016; Zhang et al., 2014) focus only on controllability of MAN with cooperative interactions. However, apart from cooperative interactions, antagonistic interactions between agents are ubiquitous in practice (Altafini, 2013; Meng, Shi, Johansson, Cao, & Hong, 2016; She & Kan, 2019). And typical examples include the enemy relationships in social networks (Altafini, 2012), the predator–prey interactions in ecological systems (Coyte, Schluter, & Foster, 2015), and the inhibition of enzymes in biological cells (Almaas, Kovacs, Vicsek, Oltvai, & Barabási, 2004).

    • Leader selection in networks under switching topologies with antagonistic interactions

      2022, Automatica
      Citation Excerpt :

      Antagonistic interactions are usually modeled as negative weights on a communication graph, and indicate collaboration disruption between agents during the evolution process (Shi, Altafini, & J.S., 2019). Antagonistic interactions are present in many real-world systems, e.g., disagreements in social networks (Meng et al., 2016), and repressive connections in biological regulatory networks (Clark, Hou, Bushnell, & Poovendran, 2017). In MASs, antagonistic interactions may arise from adversarial attacks on a network or corrupted signals between agents (Chen et al., 2016).

    • Spreading and Structural Balance on Signed Networks

      2024, SIAM Journal on Applied Dynamical Systems
    View all citing articles on Scopus

    Ziyang Meng received his Bachelor degree with honors from Huazhong University of Science & Technology, Wuhan, China, in 2006, and Ph.D. degree from Tsinghua University, Beijing, China, in 2010. He was an exchange Ph.D. student at Utah State University, Logan, USA from Sept. 2008 to Sept. 2009. From 2010 to 2015, he held postdoc, researcher, and Humboldt research fellow positions at, respectively, Shanghai Jiao Tong University, Shanghai, China, KTH Royal Institute of Technology, Stockholm, Sweden, and Technical University of Munich, Munich, Germany. He joined Department of Precision Instrument, Tsinghua University, China as an associate professor since Sept. 2015. His research interests include multiagent systems, small satellite systems, distributed control and optimization, nonlinear systems and information fusion techniques. He was selected to the national “1000-Youth Talent Program” of China in 2015.

    Guodong Shi received his B.Sc. degree in Mathematics and Applied Mathematics from School of Mathematics, Shandong University, Jinan, China, in July 2005, and his Ph.D. in Systems Theory from the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China, in July 2010, respectively. From Aug. 2010 to Apr. 2014 he was a postdoctoral researcher at the ACCESS Linnaeus Centre, School of Electrical Engineering, KTH Royal Institute of Technology, Stockholm, Sweden. He held a visiting position from Oct. 2013 to Dec. 2013 at the School of Information and Engineering Technology, University of New South Wales, Canberra, Australia.

    Since May 2014 he has been with the Research School of Engineering, College of Engineering and Computer Science, The Australian National University, Canberra, Australia, as a Lecturer and Future Engineering Research Leadership Fellow. Dr. Shi was selected in the Triennial IFAC Young Author Prize Finalist in 2011, and was a co-recipient of Best Paper Award in Control Theory from the World Congress on Intelligent Control and Automation in 2014 and the Guan Zhao-Zhi Best Paper Award from the Chinese Control Conference in 2015. His current research interests include distributed control systems, quantum networking and decisions, and social opinion dynamics.

    Karl H. Johansson is Director of the ACCESS Linnaeus Centre and Professor at the School of Electrical Engineering, KTH Royal Institute of Technology, Sweden. He heads the Stockholm Strategic Research Area ICT The Next Generation. He received M.Sc. and Ph.D. degrees in Electrical Engineering from Lund University. He has held visiting positions at UC Berkeley, California Institute of Technology, Nanyang Technological University, and Institute of Advanced Studies Hong Kong University of Science and Technology. His research interests are in networked control systems, cyber–physical systems, and applications in transportation, energy, and automation systems. He is a member of the IEEE Control Systems Society Board of Governors and the European Control Association Council. He is past Chair of the IFAC Technical Committee on Networked Systems. He has been on the Editorial Boards of Automatica, IEEE Transactions on Automatic Control, and IET Control Theory and Applications. He is currently a Senior Editor of IEEE Transactions on Control of Network Systems and Associate Editor of European Journal of Control. In 2009 he was awarded Wallenberg Scholar, as one of the first ten scholars from all sciences, by the Knut and Alice Wallenberg Foundation. He was awarded Future Research Leader from the Swedish Foundation for Strategic Research in 2005. He received the triennial Young Author Prize from IFAC in 1996 and the Peccei Award from the International Institute of System Analysis, Austria, in 1993. He received Young Researcher Awards from Scania in 1996 and from Ericsson in 1998 and 1999. He is a Fellow of the IEEE.

    Ming Cao is currently an associate professor of network analysis and control with the Engineering and Technology Institute (ENTEG) at the University of Groningen, the Netherlands, where he started as a tenure-track assistant professor in 2008. He received the Bachelor degree in 1999 and the Master degree in 2002 from Tsinghua University, Beijing, China, and the Ph.D. degree in 2007 from Yale University, New Haven, CT, USA, all in electrical engineering. From September 2007 to August 2008, he was a postdoctoral research associate with the Department of Mechanical and Aerospace Engineering at Princeton University, Princeton, NJ, USA. He worked as a research intern during the summer of 2006 with the Mathematical Sciences Department at the IBM T. J. Watson Research Center, NY, USA. He won the European Control Award sponsored by the European Control Association (EUCA) in 2016. He is an associate editor for IEEE Transactions on Circuits and Systems and Systems and Control Letters, and for the Conference Editorial Board of the IEEE Control Systems Society. He is also a member of the IFAC Technical Committee on Networked Systems. His main research interest is in autonomous agents and multi-agent systems, mobile sensor networks and complex networks.

    Yiguang Hong received his B.S. and M.S. degrees from Peking University, China, and the Ph.D. degree from the Chinese Academy of Sciences (CAS), China. He is currently a Professor in Academy of Mathematics and Systems Science, CAS, and serves as the Director of Key Lab of Systems and Control, CAS and the Director of the Information Technology Division, National Center for Mathematics and Interdisciplinary Sciences, CAS. His current research interests include nonlinear control, multi-agent systems, distributed optimization and social networks.

    Prof. Hong serves as Editor-in-Chief of Control Theory and Technology and Deputy Editor-in-Chief of Acta Automatica Sinca. He also serves or served as Associate Editors for many journals including the IEEE Transactions on Automatic Control, IEEE Transactions on Control of Network Systems, IEEE Control Systems Magazine, and Nonlinear Analysis: Hybrid Systems. He is a recipient of the Guang Zhaozhi Award at the Chinese Control Conference, Young Author Prize of the IFAC World Congress, Young Scientist Award of CAS, and the Youth Award for Science and Technology of China, and the National Natural Science Prize of China.

    The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Claudio De Persis under the direction of Editor Christos G. Cassandras.

    View full text