[Paper Review] Modulus Consensus over Networks with Antagonistic Interactions and Switching Topologies

Ziyang Meng, Guodong Shi+3 more|arXiv (Cornell University)|Feb 12, 2014

Distributed Control Multi-Agent Systems18 references18 citations

TL;DR

This paper studies discrete-time consensus in multi-agent networks with cooperative and antagonistic interactions under switching topologies. It proves that absolute state values converge to consensus under uniform joint strong connectivity (unidirectional) or infinite joint connectivity (bidirectional), regardless of time-varying interaction signs, and establishes conditions for bipartite consensus. The results are validated via simulations using a Kuramoto model.

ABSTRACT

In this paper, we study the discrete-time consensus problem over networks with antagonistic and cooperative interactions. Following the work by Altafini [IEEE Trans. Automatic Control, 58 (2013), pp. 935--946], by an antagonistic interaction between a pair of nodes updating their scalar states we mean one node receives the opposite of the state of the other and naturally by an cooperative interaction we mean the former receives the true state of the latter. Here the pairwise communication can be either unidirectional or bidirectional and the overall network topology graph may change with time. The concept of modulus consensus is introduced to characterize the scenario that the moduli of the node states reach a consensus. It is proved that modulus consensus is achieved if the switching interaction graph is uniformly jointly strongly connected for unidirectional communications, or infinitely jointly connected for bidirectional communications. We construct a counterexample to underscore the 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 modulus consensus even under fixed topologies. 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 node dynamics.

Motivation & Objective

• To analyze the behavior of discrete-time multi-agent systems with both cooperative and antagonistic interactions.
• To investigate how switching topologies affect consensus in absolute values when interaction signs vary over time.
• To determine sufficient connectivity conditions for absolute value consensus and bipartite consensus in cooperative-antagonistic networks.
• To validate the theoretical results using simulations with a nonlinear Kuramoto model.

Proposed method

• The authors model antagonistic interactions as receiving the negative of a neighbor's state, while cooperative interactions involve receiving the true state.
• They analyze the system using a state-space approach with time-varying signed interaction matrices representing unidirectional or bidirectional communication.
• Key connectivity conditions are introduced: uniform joint strong connectivity for unidirectional graphs and infinite joint connectivity for bidirectional graphs.
• Theoretical analysis proves that agent state limits exist and their absolute values converge under these joint connectivity conditions.
• A lifting technique is applied to extend results to nonlinear dynamics, such as the Kuramoto model with antagonistic links.
• Simulations are conducted using periodic switching topologies and specific signed matrices to verify convergence of absolute values and bipartite consensus.

Experimental results

Research questions

• RQ1 Under what switching topology conditions does the absolute value of agent states converge in cooperative-antagonistic networks?
• RQ2 Is quasi-strong connectivity sufficient to guarantee consensus in absolute values, even under fixed topologies?
• RQ3 Can bipartite consensus be achieved under joint connectivity conditions in switching networks?
• RQ4 How do antagonistic interactions affect the convergence behavior of nonlinear multi-agent systems like the Kuramoto model?

Key findings

• The limits of all agent states exist and their absolute values converge to a common value under uniformly jointly strongly connected unidirectional topologies.
• For bidirectional topologies, absolute value consensus is achieved if the interaction graph is infinitely jointly connected.
• A counterexample shows that quasi-strong connectivity—containing a directed spanning tree—is not sufficient for absolute value consensus, even with fixed topologies.
• Bipartite consensus is achievable under joint connectivity conditions, and the simulations confirm this behavior in bidirectional switching topologies.
• The theoretical framework applies to nonlinear systems, as demonstrated by convergence in a discrete-time Kuramoto model with antagonistic links.
• The convergence of absolute values is an emergent behavior, not a design objective, and holds regardless of time-varying interaction signs.