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

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.

💡 Research Summary

This paper tackles the largely unexplored problem of non‑trivial consensus on directed signed matrix‑weighted networks, where agents interact through multi‑dimensional couplings that can be either cooperative (positive‑definite) or antagonistic (negative‑definite). The authors first extend a classical result on grounded Laplacians—originally proved for scalar‑weighted signed digraphs—to the matrix‑weighted setting. They introduce two novel graph‑theoretic concepts: (i) positive‑negative paths, which capture sequences of edges whose signs alternate, and (ii) in‑degree‑dominated vertices, a generalization of the balanced node notion. Under the condition that every vertex can be reached via a positive‑negative path from at least one in‑degree‑dominated vertex, they prove that every eigenvalue of the grounded Laplacian has a strictly positive real part. This spectral property guarantees that the dynamics of the autonomous network converge asymptotically to the null space of the signed Laplacian, thereby providing a solid foundation for stability analysis on rooted signed matrix‑weighted graphs.

Building on this spectral insight, the paper proposes a systematic design methodology to achieve a prescribed non‑zero consensus state θ (the “non‑trivial consensus”). The original fully‑autonomous network (FAN) described by (\dot{x}= -Lx) is augmented with external inputs to a selected subset of agents, called informed agents. Each informed agent i receives an external signal (x_0) through a coupling matrix (B_i) and a scalar gain (\delta_i). By carefully selecting the informed set (V_I), designing the external signal, and ensuring that the coupling gains satisfy explicit lower‑bound inequalities (derived from the smallest real part of the grounded Laplacian eigenvalues), the authors guarantee global asymptotic convergence of all agents to the desired θ, regardless of whether the underlying graph is structurally balanced or unbalanced. The paper also defines a “non‑trivial consensus space” that provides an equivalent algebraic characterization of the convergence condition.

For undirected signed matrix‑weighted networks, the authors show that the required lower bounds on the coupling gains are even less restrictive, reflecting the symmetry of the Laplacian. This contrasts with many existing works that demand a positive spanning tree or a positive‑negative spanning tree, which are stronger connectivity requirements.

The study further extends to switching topologies. A necessary condition for convergence is established: the null spaces of all Laplacians that appear infinitely often during switching must intersect non‑trivially. Using logarithmic norm analysis, the authors derive sufficient conditions and propose a dynamic gain‑adjustment protocol that adapts the coupling coefficients whenever the topology changes. This approach avoids the need for pre‑selected time subsequences used in prior literature and ensures that the network remains on a trajectory toward the preset non‑trivial consensus despite arbitrary switching among a family of directed graphs.

Numerical simulations illustrate the theory. A six‑agent network with three‑dimensional states and mixed positive/negative matrix weights is driven to a target consensus (\theta =