论文标题
特征值光谱和定向复杂网络的稳定性
Eigenvalue spectra and stability of directed complex networks
论文作者
论文摘要
量化大型随机矩阵的特征值光谱使人们可以理解有助于具有许多相互作用组件的动态系统稳定性的因素。这项工作探讨了组件之间的相互作用网络对特征值频谱的影响。我们基于以前的结果，通常仅考虑到网络的平均程度，通过允许非平凡的网络程度异质性。我们得出了一般加权和定向网络的邻接矩阵的特征值光谱的封闭形式表达式。使用这些结果，这些结果适用于任何大型良好连接的复杂网络，然后在随机矩阵理论中得出校正（由于非零网络异质性）的紧凑公式（由于非零网络异质性）。具体而言，我们得出了Wigner Semi-Circle定律，Girko Circle Law和Elliptic Law和任何离群特征值的修改版本。我们还为定向Barabasi-Albert网络的特征值密度提供了令人惊讶的整洁分析表达。因此，我们能够对网络异质性对复杂动态系统稳定性的影响进行一般推论。
Quantifying the eigenvalue spectra of large random matrices allows one to understand the factors that contribute to the stability of dynamical systems with many interacting components. This work explores the effect that the interaction network between components has on the eigenvalue spectrum. We build upon previous results, which usually only take into account the mean degree of the network, by allowing for non-trivial network degree heterogeneity. We derive closed-form expressions for the eigenvalue spectrum of the adjacency matrix of a general weighted and directed network. Using these results, which are valid for any large well-connected complex network, we then derive compact formulae for the corrections (due to non-zero network heterogeneity) to well-known results in random matrix theory. Specifically, we derive modified versions of the Wigner semi-circle law, the Girko circle law and the elliptic law and any outlier eigenvalues. We also derive a surprisingly neat analytical expression for the eigenvalue density of an directed Barabasi-Albert network. We are thus able to make general deductions about the effect of network heterogeneity on the stability of complex dynamic systems.
