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Non-Trivial Consensus on Directed Matrix-Weighted Networks with Cooperative and Antagonistic Interactions

Tianmu Niu, Bing Mao, Xiaoqun Wu, Tingwen Huang

TL;DR

This paper tackles non-trivial consensus on directed signed matrix-weighted networks whose interventions combine cooperative and antagonistic interactions. It develops a grounded-Laplacian framework and an external-input design that yields a prescribed non-zero consensus state, with conditions guaranteeing convergence to the Laplacian null space. The main contributions include proving positive real parts of grounded-Laplacian eigenvalues under milder connectivity, deriving lower bounds on coupling coefficients, and providing fixed- and switching-topology algorithms that do not require structural balance. The results extend to undirected networks with relaxed conditions and offer a switching-topology strategy with a necessary condition for convergence, broadening the applicability to multi-dimensional agent systems in formation control and social dynamics.The work enables groups of agents with both cooperative and antagonistic interactions to reach a unified, non-zero consensus by carefully selecting informed agents and external signals, and by designing coupling gains and matrix weights. It introduces the concept of non-trivial consensus space through a compact signaling framework and demonstrates how expanded-system analysis and logarithmic-norm methods yield practical convergence guarantees. Overall, the paper provides a rigorous, actionable pathway to steer multi-dimensional agent networks toward a preset consensus despite sign-heterogeneous interconnections.

Abstract

This paper investigates the non-trivial consensus problem on directed signed matrix-weighted networks\textemdash a novel convergence state that has remained largely unexplored despite prior studies on bipartite consensus and trivial consensus. Notably, we first prove that for directed signed matrix-weighted networks, every eigenvalue of the grounded Laplacians has positive real part under certain conditions. This key finding ensures the global asymptotic convergence of systems states to the null spaces of signed matrix-weighted Laplacians, providing a foundational tool for analyzing dynamics on rooted signed matrix-weighted networks. To achieve non-trivial consensus, we propose a systematic approach involving the strategic selection of informed agents, careful design of external signals, and precise determination of coupling terms. Crucially, we derive the lower bounds of the coupling coefficients. Our consensus algorithm operates under milder connectivity conditions, and does not impose restrictions on whether the network is structurally balanced or unbalanced. Moreover, the non-trivial consensus state can be preset arbitrarily as needed. We also carry out the above analysis for undirected networks, with more relaxed conditions on the coupling coefficients comparing to the directed case. This paper further studies non-trivial consensus with switching topologies, and propose the necessary condition for the convergence of switching networks. The work in this paper demonstrates that groups with both cooperative and antagonistic multi-dimensional interactions can achieve consensus, which was previously deemed exclusive to fully cooperative groups.

Non-Trivial Consensus on Directed Matrix-Weighted Networks with Cooperative and Antagonistic Interactions

TL;DR

This paper tackles non-trivial consensus on directed signed matrix-weighted networks whose interventions combine cooperative and antagonistic interactions. It develops a grounded-Laplacian framework and an external-input design that yields a prescribed non-zero consensus state, with conditions guaranteeing convergence to the Laplacian null space. The main contributions include proving positive real parts of grounded-Laplacian eigenvalues under milder connectivity, deriving lower bounds on coupling coefficients, and providing fixed- and switching-topology algorithms that do not require structural balance. The results extend to undirected networks with relaxed conditions and offer a switching-topology strategy with a necessary condition for convergence, broadening the applicability to multi-dimensional agent systems in formation control and social dynamics.The work enables groups of agents with both cooperative and antagonistic interactions to reach a unified, non-zero consensus by carefully selecting informed agents and external signals, and by designing coupling gains and matrix weights. It introduces the concept of non-trivial consensus space through a compact signaling framework and demonstrates how expanded-system analysis and logarithmic-norm methods yield practical convergence guarantees. Overall, the paper provides a rigorous, actionable pathway to steer multi-dimensional agent networks toward a preset consensus despite sign-heterogeneous interconnections.

Abstract

This paper investigates the non-trivial consensus problem on directed signed matrix-weighted networks\textemdash a novel convergence state that has remained largely unexplored despite prior studies on bipartite consensus and trivial consensus. Notably, we first prove that for directed signed matrix-weighted networks, every eigenvalue of the grounded Laplacians has positive real part under certain conditions. This key finding ensures the global asymptotic convergence of systems states to the null spaces of signed matrix-weighted Laplacians, providing a foundational tool for analyzing dynamics on rooted signed matrix-weighted networks. To achieve non-trivial consensus, we propose a systematic approach involving the strategic selection of informed agents, careful design of external signals, and precise determination of coupling terms. Crucially, we derive the lower bounds of the coupling coefficients. Our consensus algorithm operates under milder connectivity conditions, and does not impose restrictions on whether the network is structurally balanced or unbalanced. Moreover, the non-trivial consensus state can be preset arbitrarily as needed. We also carry out the above analysis for undirected networks, with more relaxed conditions on the coupling coefficients comparing to the directed case. This paper further studies non-trivial consensus with switching topologies, and propose the necessary condition for the convergence of switching networks. The work in this paper demonstrates that groups with both cooperative and antagonistic multi-dimensional interactions can achieve consensus, which was previously deemed exclusive to fully cooperative groups.
Paper Structure (19 sections, 10 theorems, 82 equations, 6 figures)

This paper contains 19 sections, 10 theorems, 82 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Graph Theory
  5. Model Description
  6. Problem Statement
  7. Main Results
  8. Non-Trivial Consensus with Fixed Topology
  9. Non-Trivial Consensus with Switching Topology
  10. Simulation
  11. Non-Trivial Consensus with Fixed Topology
  12. Non-Trivial Consensus with Switching Topology
  13. Conclusion
  14. Proof of Lemma \ref{['PhiB Lemma']}
  15. Proof of Theorem \ref{['directed proposition']}
  16. ...and 4 more sections

Key Result

Lemma 1

matrix analysis Let $A,B\in\mathbb{R}^{d\times d}$ be symmetric, if $A$ is positive definite, then there is a non-singular $S\in\mathbb{C}^{d\times d}$ such that $A=SIS^{*}$ and $B=S\Lambda S^{*}$, in which $\Lambda$ is real diagonal, and the main diagonal entries of $\Lambda$ are the eigenvalues of

Figures (6)

  • Figure 1: An illustration example of the original FAN topology $\mathcal{G}$, the augmented graph $\widehat{\mathcal{G}}$ and its expanded topology $\widehat{\overline{\mathcal{G}}}$. The blue and red solid lines represent positive and negative definite edges, respectively.
  • Figure 2: Directed topology $\mathcal{G}$ for original FAN \ref{['original system']} and $\widehat{\mathcal{G}}$ for the corresponding SAN \ref{['augmented system2']} under Theorem \ref{['NTC thm']}. The blue and red solid (dashed) lines represent positive and negative (semi-) definite edges, respectively.
  • Figure 3: States evolution of directed matrix-weighted network with topology $\widehat{\mathcal{G}}$ in Fig. \ref{['directed topology']}.
  • Figure 4: States evolution of directed matrix-weighted network with topology $\widehat{\mathcal{G}}$ in Fig. \ref{['directed topology']}, after changing matrix weight $A_{65}$ into a positive semi-definite matrix.
  • Figure 5: Topology $\mathcal{G}_{2},\ \widehat{\mathcal{G}}_{2}$ and $\mathcal{G}_{3},\ \widehat{\mathcal{G}}_{3}$ for switching network \ref{['switching SAN']}\ref{['switching FAN']}\ref{['compact switching FAN']}.
  • ...and 1 more figures

Theorems & Definitions (21)

  • Definition 1: Non-Trivial Consensus
  • Definition 2: Non-Trivial Consensus Space
  • Remark 1
  • Lemma 1
  • Definition 3: positive-negative path
  • Definition 4: in-degree-dominated
  • Remark 2
  • Theorem 1
  • Remark 3
  • Lemma 2
  • ...and 11 more