有向随机图特征向量的局部化和泛值性
Localization and universality of eigenvectors in directed random graphs
摘要：
虽然随机图的谱性质一直是网络理论的一个长期关注的焦点，但有向图的右特征向量的性质迄今为止还没有一个精确的解析处理。我们提出了一个一般理论的统计右特征向量分量的有向随机图与一个规定的度分布和随机加权链接。我们得到了逆参与比的精确解析表达式，并证明了具有小平均度的有向随机图的右特征向量是局部化的。值得注意的是，如果度分布的四阶矩是有限的，那么局部化转变的临界平均度与度波动无关，这与由度波动控制的无向图中的局部化不同。我们还表明，在高连接限制的右特征向量分量的分布是完全由度波动。对于离域特征向量，我们恢复的普遍结果，从标准的随机矩阵理论是独立的度分布，而本地化的特征向量的特征向量分布依赖于度分布。
英文摘要：
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.
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