用于主稳定性分析的正弧权重设计可对角化有向拉普拉斯算子
Positive Arc-Weight Design Makes Every Directed Laplacian Diagonalizable
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中文总结 AI 辅助
本文针对有向拉普拉斯算子未必可对角化的问题，提出一种正弧权重设计方法，使弱连通有向图的加权入度拉普拉斯算子可对角化，恢复主稳定性函数变分方程的解耦形式，并给出共同弧权重的容许区间计算准则。
中文摘要 AI 辅助
标准主稳定性函数（MSF）公式传统上依赖可对角化的网络拉普拉斯算子，因为可对角化能将变分方程分解为独立方程。然而，有向拉普拉斯算子未必可对角化。本文证明，每个弱连通有向图都存在严格正的弧权重，使其加权入度拉普拉斯算子可对角化。我们的构造首先提取一个弱连通生成有向无环图，该图在每个根强连通分量中恰好有一个源顶点，分配正权重使其余所有顶点的加权入度两两不同；再为原有的向图剩余弧分配一个共同的足够小的正权重。在该扰动下，非零特征值保持两两不同，而零特征值为半单的，重数等于根强连通分量的数量。因此，所得拉普拉斯算子拥有完整的特征向量集，恢复了MSF变分方程的完全解耦形式。我们还给出了基于判别式的准则，用于计算共同弧权重的容许区间。
英文摘要
For directed networks, the Laplacian need not be diagonalizable, so the standard master-stability variational equations cannot in general be fully decoupled into independent eigenmodes. We prove that this obstruction can always be removed by coupling-strength design: every weakly connected digraph admits a strictly positive weighting of its existing arcs for which the weighted in-degree Laplacian is diagonalizable. The construction uses a spanning directed acyclic subgraph with one source in each root strongly connected component, assigns distinct positive weighted indegrees to its non-source vertices, and then restores all remaining arcs with a common sufficiently small positive weight. The zero eigenvalue remains semisimple and all nonzero eigenvalues remain simple. We also give a discriminant criterion that computes an admissible interval of restoring weights. Thus, any fixed weakly connected directed topology can be positively weighted so that master-stability perturbations admit a complete modal decomposition.
