Non-trivial consensus on directed signed matrix-weighted networks with compound measurement noises and time-varying topologies

This paper studies non-trivial consensus–a relatively novel and unexplored convergence behavior–on directed signed matrix-weighted networks subject to both additive and multiplicative measurement noises under time-varying topologies. Building upon grounded matrix-weighted Laplacian properties, a stochastic dynamic model is established that simultaneously captures inter-dimensional cooperative and antagonistic interactions, compound measurement noises and time-varying network structures. Based on stochastic differential equations theory, protocols that guarantee mean square and almost sure non-trivial consensus are proposed. Specifically, for any predetermined non-trivial consensus state, all agents are proven to converge toward this non-zero value in the mean-square and almost-sure senses. The design of control gain function in our protocols highlights a balanced consideration of the cumulative effect over time, the asymptotic decay property and the finite energy corresponding to measurement noises. Notably, the conditions on time-varying topologies in our protocols only require boundedness of elements in edge weight matrices, which facilitate the practicality of concept “time-varying topology” in matrix-weighted network consensus algorithms. Furthermore, the proposed protocols operate under milder connectivity conditions and no requirements on structural (un)balance properties. The work in this paper demonstrates that groups with both cooperative and antagonistic inter-dimensional interactions can achieve consensus even in the presence of compound measurement noises and time-varying topologies, challenging the conventional belief that consensus is attainable only in fully cooperative settings.

💡 Research Summary

This paper tackles a largely unexplored problem in multi‑agent systems: achieving non‑trivial consensus (convergence to a pre‑specified non‑zero vector) on directed signed matrix‑weighted networks that are simultaneously subject to additive and multiplicative measurement noises and time‑varying topologies. The authors first formalize the network as a triple (G(t)=(\mathcal V,\mathcal E(t),\mathcal A(t))) where each edge ((j,i)) carries a symmetric matrix weight (A_{ij}(t)\in\mathbb R^{d\times d}). The sign of the edge is given by (\operatorname{sgn}(A_{ij}(t))): positive semi‑definite edges are cooperative, negative semi‑definite edges are antagonistic, and zero matrices represent the absence of a link.

The agents’ continuous‑time dynamics extend the classic consensus law by adding two independent white‑noise processes: \