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.

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异构随机图求解方程的解析解：光谱和定位特性

异构随机图的光谱和局部化特性由解析分布方程确定，迄今为止，这些方程一直抵制解析处理。我们解析求解具有任意度数分布的随机图在高连通性极限下的解析方程，从中我们对度数波动对谱密度、逆参与比和局部分布的影响进行了深入分析。状态密度。我们证明了所有特征向量都被扩展并且当度分布的方差足够大时，谱密度表现出对数或幂律发散。我们通过显示频谱中心的局部状态密度分布显示出由度分布的方差确定的幂律尾部来阐明这种奇异行为。在弱度波动的情况下，谱密度具有有限的支持，这促进了随机图上大型复杂系统的稳定性。