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.

翻译：本文研究了有向符号矩阵加权网络上的非平凡一致性问题——这是一种新颖的收敛状态，尽管先前已有关于二分一致性和平凡一致性的研究，但该状态在很大程度上尚未被探索。值得注意的是，我们首先证明了对于有向符号矩阵加权网络，在特定条件下，接地拉普拉斯矩阵的每个特征值均具有正实部。这一关键发现确保了系统状态全局渐近收敛到符号矩阵加权拉普拉斯矩阵的零空间，为分析具有根节点的符号矩阵加权网络上的动力学行为提供了基础工具。为实现非平凡一致性，我们提出了一种系统方法，包括策略性地选择信息代理、精心设计外部信号以及精确确定耦合项。至关重要的是，我们推导出了耦合系数的下界。我们的一致性算法在较弱的连通性条件下运行，且不限制网络是结构平衡还是非平衡的。此外，非平凡一致性状态可根据需要任意预设。我们还对无向网络进行了上述分析，其耦合系数的条件相较于有向情形更为宽松。本文进一步研究了切换拓扑下的非平凡一致性，并提出了切换网络收敛的必要条件。本文的研究表明，具有合作与对抗性多维交互的群体能够达成一致性，而这一能力先前被认为仅存在于完全合作的群体中。