论文标题
密集随机网络的光谱密度和Wigner半圆形的崩溃
The spectral density of dense random networks and the breakdown of the Wigner semicircle law
论文作者
论文摘要
尽管长期研究了随机网络的光谱，但是网络拓扑对网络光谱的密集限制的影响仍然很少了解。通过考虑具有四个不同程度分布的网络的配置模型，我们表明，密集随机网络的邻接矩阵的光谱密度取决于程度波动的强度。特别是，具有指数度分布的密集网络的特征值分布受一个简单方程式的控制，我们从该方程式中揭示了光谱密度的对数奇异性。我们还得出了特征值分布的第四刻与程度分布的方差之间的关系，这导致了对密集随机网络的Wigner Semicircle定律的分解的足够条件。基于相同的关系，我们提出了一个分类方案，该方案是在密度极限下的光谱密度不同的普遍行为。我们的理论发现应导致对图表上定义的模型的平均场所行为的重要见解。
Although the spectra of random networks have been studied for a long time, the influence of network topology on the dense limit of network spectra remains poorly understood. By considering the configuration model of networks with four distinct degree distributions, we show that the spectral density of the adjacency matrices of dense random networks is determined by the strength of the degree fluctuations. In particular, the eigenvalue distribution of dense networks with an exponential degree distribution is governed by a simple equation, from which we uncover a logarithmic singularity in the spectral density. We also derive a relation between the fourth moment of the eigenvalue distribution and the variance of the degree distribution, which leads to a sufficient condition for the breakdown of the Wigner semicircle law for dense random networks. Based on the same relation, we propose a classification scheme of the distinct universal behaviours of the spectral density in the dense limit. Our theoretical findings should lead to important insights on the mean-field behaviour of models defined on graphs.
