Background: Distributed networks of interacting agents, or multi-agent systems, are fundamental to modeling complex phenomena in engineering, social science, and biology. The introduction of antagonistic (negative) interactions, which creates "signed networks," poses significant challenges to system stability and the achievement of collective agreement. While consensus on signed networks has been studied, ensuring stability, particularly through decentralized mechanisms, remains a critical open problem. Unaddressed instabilities can lead to unbounded system states or oscillations, precluding any functional collective behavior.

Methods: This article introduces a novel framework called Local Node Compensation (LNC) to enhance the stability of distributed systems on signed, undirected graphs. The proposed method involves a decentralized control protocol where individual nodes adjust their dynamics based on locally available information. We leverage principles from algebraic graph theory, particularly the spectral properties of the signed Laplacian matrix, to analyze the system. The stability of the network under the LNC protocol is formally proven using Lyapunov stability theory and analysis of the system's eigenvalues.

Results: Our theoretical analysis demonstrates that the LNC protocol guarantees system stability under well-defined conditions. The method effectively shifts the eigenvalues of the signed Laplacian, preventing the instabilities that can arise from unbalanced network structures. We present extensive numerical simulations on various network topologies, including both structurally balanced and unbalanced graphs. The results validate our theoretical findings, showing that the LNC method successfully stabilizes networks that are otherwise unstable and improves the convergence performance compared to standard protocols.

Conclusion: The Local Node Compensation framework offers a robust, scalable, and fully decentralized solution for ensuring stability in signed networks. This method overcomes key limitations of existing approaches and has significant implications for applications requiring coordinated control in the presence of antagonistic interactions, such as in opinion dynamics, robotic swarms, and distributed computing.