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arXiv · 2605.11728

Spectral Sensitivity of Directed Weighted Networks: Why Weakening Edges May Trigger Synchronization

Xinyu Wu·Xizhi Liu·Chenyao Zhang·Tianping Chen·Wenlian Lu
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Abstract

Synchronization in dynamical systems on directed weighted networks is often associated with stronger coupling and denser interactions. This paper shows that the opposite can also occur: weakening selected edges may increase the generalized algebraic connectivity, denoted by $κ$, and in some nonlinear systems this spectral improvement is accompanied by a transition from nonsynchronization to synchronization. To explain this effect, we develop a perturbation-based spectral sensitivity framework for directed weighted networks. We derive an explicit first-order formula for the response of $κ$ to edge-weight perturbations and show that it decomposes into a directed cut-energy term and a stationary redistribution term. This decomposition clarifies how asymmetric flow structure and invariant-mass redistribution jointly determine the synchronization role of each edge. Based on this theory, we design sensitivity-guided algorithms for edge weakening, edge deletion, negative-edge insertion, and edge strengthening. Experiments on synthetic and real networks show that these methods identify critical edges whose modification yields substantial gains in $κ$. Simulations of first- and second-order nonlinear consensus dynamics further show markedly faster convergence and, in some cases, a transition from incoherence to synchronization. The results provide a local spectral mechanism by which reducing or reallocating coupling can enhance synchronization-related performance.

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Xinyu Wu, Xizhi Liu, Chenyao Zhang, Tianping Chen, Wenlian Lu. 2026-05-12. Spectral Sensitivity of Directed Weighted Networks: Why Weakening Edges May Trigger Synchronization. https://arxiv.org/abs/2605.11728

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