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

Tianmu Niu; Xiaoqun Wu

eess.SYMar 14, 2026

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

Authors: Tianmu Niu, Xiaoqun Wu

Organizations: School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China · College of Computer Science and Software Engineering, Shenzhen University, Shenzhen, 518060, China

Abstract

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.

Figures & tables

Figure 1: Directed topology $\mathcal{G}_{1}$ for original FAN ( 2 ) and $\widehat{\mathcal{G}}_{1}$ for the corresponding SAN ( 5 ) under Lemma 1 . The blue and red solid (dashed) lines represent positive and negative (semi-) definite edges, respectively.

Figure 2: Mean square non-trivial consensus errors of directed signed matrix-weighted network with fixed topology $\mathcal{G}_{1},\ \widehat{\mathcal{G}}_{1}$ in Figure 1 .

Panel of Figure 3 — Figure 3: States evolution of directed matrix-weighted network with fixed topology $\widehat{\mathcal{G}}_{1}$ in Figure 1 .

Panel of Figure 4 — Figure 4: Topology $\mathcal{G}_{2},\ \widehat{\mathcal{G}}_{2}$ and $\mathcal{G}_{3},\ \widehat{\mathcal{G}}_{3}$ for switching network ( 2 ) ( 5 ) under Lemma 1 . The blue and red solid (dashed) lines represent positive and negative (semi-) definite edges, respectively.

Figure 5: Mean square non-trivial consensus error of directed matrix-weighted network with time-varying topology $\widehat{\mathcal{G}}(t)$ switching among $\widehat{\mathcal{G}}_{1}$ in Figure 1 and $\widehat{\mathcal{G}}_{2}$ , $\widehat{\mathcal{G}}_{3}$ in Figure 4 .

Panel of Figure 6 — Figure 6: States evolution of directed matrix-weighted network with time-varying topology $\widehat{\mathcal{G}}(t)$ switching among $\widehat{\mathcal{G}}_{1}$ in Figure 1 and $\widehat{\mathcal{G}}_{2}$ , $\widehat{\mathcal{G}}_{3}$ Figure 4 .

Panel of Figure 7 — Figure 7: States evolution of directed matrix-weighted network with time-varying topology $\widehat{\mathcal{G}}(t)$ switching among $\widehat{\mathcal{G}}_{1}$ in Figure 1 and $\widehat{\mathcal{G}}_{2}$ , $\widehat{\mathcal{G}}_{3}$ Figure 4 . The control gain $c(t)$ is chosen as $(1+t)^{-2}$ .

Multi-Agent Consensus

Multi-agent consensus studies how interacting agents reach shared decisions or states across networks, and how to make agreement reliable.

See all 125 papers

Feb 12, 2026

Grounded Laplacians of Directed Signed Matrix-Weighted Networks: Spectral Properties and Applications to Non-Trivial Consensus

Grounded Laplacians provide the spectral link between external information and network convergence. This paper establishes positive-stability results for grounded Laplacians in directed signed matrix-weighted networks, where directionality, antagonism, and singular edge weight matrices coexist. First, under in-degree dominance and positive-negative reachability, we derive explicit local thresholds for the grounding gains. Second, a scaled, kernel-based certificate replaces the unscaled degree condition with a signed matrix-weighted Dirichlet decomposition and a joint-kernel test for the scaled symmetric part. The computable margin

γ_p

lower-bounds the minimum real part of the spectrum and certifies exponential contraction in the

P

-norm. Under absolute generalized balance, the kernel-intersection test is given; the balanced and definite-edge unbalanced undirected cases follow. As an application, non-trivial consensus (NTC) on signed matrix-weighted networks is studied. Informed agents, external signals and coupling terms are designed to steer all agents to any prescribed nonzero state without requiring structural balance. Switching topology case retains non-trivial consensus result under certain conditions. Realizing NTC on signed matrix-weighted networks demonstrates that groups with both cooperative and antagonistic multi-dimensional interactions can achieve consensus, which was previously deemed exclusive to fully cooperative groups.

Tianmu Niu, Bing Mao, Hao Liao +2

Multi-Agent Coordination Opinion Dynamics

Ask PDF

Jun 24, 2026

Robustness and Leadership in Markov-switching Consensus Networks

We investigate how time-varying interactions, modeled via a Markov switching graph (MSG), impact the robustness of noisy multi-agent dynamics in both continuous- and discrete-time settings. Our focus is on the steady-state performance of consensus and leader-follower tracking dynamics subject to stochastic noise. Using the framework of Markov jump linear systems (MJLS), we derive expressions for the steady-state covariance of each agent's deviation from consensus and tracking error, respectively, and use them to quantify individual and group performance as a function of the interaction graphs and the switching dynamics. We extend established notions of robustness, certainty indices, and joint centrality from static graphs to the MSG setting. To gain analytical insight, we specialize our results to systems switching between two topologies and characterize how switching influences performance. Numerical simulations further illustrate how switching topologies affects system robustness in both coordination tasks.

Sarah H. Cen, Vaibhav Srivastava, Naomi Ehrich Leonard

Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA · Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USA · Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA

Multi-Agent Coordination Regime-Switching Dynamical Systems Robust Control Multi-Agent Systems Multi-Agent Consensus

Ask PDF

Jul 21, 2026

How network perturbations distort agreement trajectories in LTI multi-agent systems

Distributed coordination of multi-agent systems frequently relies on cooperative protocols designed to achieve agreement on a prescribed, non-trivial trajectory. While the robustness of such protocols to various uncertainties is well documented, existing literature universally assumes that the target agreement trajectory itself remains invariant. This assumption may hold in ideal cases, but we prove that network perturbations can vastly modify the asymptotic agreement trajectory. We first investigate the exact trajectories of Linear Time-Invariant (LTI) agents subjected to dynamic coupling uncertainties by establishing a new Laplace-domain criterion that characterizes the specific closed-loop poles governing the perturbed agreement manifold. To formalize our analysis, we introduce the notion of structure-preserving dynamics, perturbations that maintain the null space of the communication graph's Laplacian, and contrast them with transmission only dynamics, affecting only the adjacency matrix. We prove a critical fragility within standard cooperative output regulation schemes: while static consensus is uniquely robust to heterogeneous transmission delays, synchronization to periodic trajectories is destroyed by arbitrarily small transmission delays. Furthermore, we demonstrate that for d-regular topologies, uniform transmission perturbations can easily shift the system to synchronize with an unexpected, entirely new frequency. These findings expose a previously unidentified vulnerability in classical robust synchronization, demonstrating that transmission dynamics necessitate fundamental structural modifications to networked reference generators.

Gal Barkai, Irinel-Constantin Morărescu

Université de Lorraine, CNRS, CRAN, F-54000 Nancy, France

Multi-Agent Coordination Multi-Agent Control Robust Control Multi-Agent Consensus

Ask PDF

Abstract

Multi-Agent Consensus

Explore similar work

Grounded Laplacians of Directed Signed Matrix-Weighted Networks: Spectral Properties and Applications to Non-Trivial Consensus

Robustness and Leadership in Markov-switching Consensus Networks

How network perturbations distort agreement trajectories in LTI multi-agent systems