论文标题
特征向量的定位和普遍性在有向随机图中
Localization and universality of eigenvectors in directed random graphs
论文作者
论文摘要
尽管随机图的光谱特性一直是网络理论的长期重点，但到目前为止，有向图的正确特征向量的特性已经避开了精确的分析处理。我们为有针对性的随机图中的右特征向量成分的统计数据提供了一个通用理论，该图具有规定的程度分布和随机加权链接。我们获得了反向参与率的精确分析表达式，并表明定向平均程度较小的定向随机图的正确特征向量是本地化的。值得注意的是，如果度分分布的第四刻是有限的，则定位转换的临界平均度与程度波动无关，该度波动与在无向图中的定位不同，该图由程度波动控制。我们还表明，在高连通性中，右特征向量组件的分布仅由程度波动决定。对于离域特征向量，我们从独立于度分布的标准随机矩阵理论中恢复了通用结果，而对于局部特征向量，特征向量分布取决于程度分布。
Although the spectral properties of random graphs have been a long-standing focus of network theory, the properties of right eigenvectors of directed graphs have so far eluded an exact analytic treatment. We present a general theory for the statistics of the right eigenvector components in directed random graphs with a prescribed degree distribution and with randomly weighted links. We obtain exact analytic expressions for the inverse participation ratio and show that right eigenvectors of directed random graphs with a small average degree are localized. Remarkably, if the fourth moment of the degree distribution is finite, then the critical mean degree of the localization transition is independent of the degree fluctuations, which is different from localization in undirected graphs that is governed by degree fluctuations. We also show that in the high connectivity limit the distribution of the right eigenvector components is solely determined by the degree fluctuations. For delocalized eigenvectors, we recover the universal results from standard random matrix theory that are independent of the degree distribution, while for localized eigenvectors the eigenvector distribution depends on the degree distribution.
