论文标题
分解方程的分析解决方案的异质随机图：光谱和定位属性
Analytic solution of the resolvent equations for heterogeneous random graphs: spectral and localization properties
论文作者
论文摘要
异质随机图的光谱和定位特性取决于分辨分布方程，到目前为止，它们抵抗了分析处理。我们通过分析求解具有高度连通性极限的随机图的分解方程，从中我们对程度波动对光谱密度，逆参与率和局部密度的分布的影响进行了详尽的分析。我们表明，所有特征向量均已扩展，并且当程度分布的方差足够大时，光谱密度表现出对数或幂律差异。我们通过表明状态在光谱中心的局部密度的分布显示由度分布的方差确定的幂律尾部，从而阐明了这种奇异行为。在弱度波动的状态下，光谱密度具有有限的支持，从而促进了大型复合系统在随机图上的稳定性。
The spectral and localization properties of heterogeneous random graphs are determined by the resolvent distributional equations, which have so far resisted an analytic treatment. We solve analytically the resolvent equations of random graphs with an arbitrary degree distribution in the high-connectivity limit, from which we perform a thorough analysis of the impact of degree fluctuations on the spectral density, the inverse participation ratio, and the distribution of the local density of states. We show that all eigenvectors are extended and that the spectral density exhibits a logarithmic or a power-law divergence when the variance of the degree distribution is large enough. We elucidate this singular behaviour by showing that the distribution of the local density of states at the center of the spectrum displays a power-law tail determined by the variance of the degree distribution. In the regime of weak degree fluctuations the spectral density has a finite support, which promotes the stability of large complex systems on random graphs.
