聚类异质性对稀疏网络谱密度的影响
Effects of clustering heterogeneity on the spectral density of sparse networks
AI总结：
本文推导了任意单边和三角形分布的稀疏网络谱密度精确方程，发现异质网络中三角形度序列波动增加使谱密度更对称，且传统大度近似在复杂模体网络中失效，理论结果获数值证实。
摘要：
我们推导出了精确方程，用于计算具有任意单边和三角形数量分布的稀疏网络的谱密度。这些方程能够系统地研究聚类对网络邻接矩阵谱性质的影响。在异质网络中，我们证明随着三角形度序列波动的增加，谱密度变得更加对称。这一现象可以通过三角形度分布大方差网络的低聚类系数来解释。在规则聚类网络的同质情况下，我们发现微扰和非微扰近似均无法预测高连通性极限下的谱密度。这表明传统的大度近似在研究具有更复杂模体的网络谱性质时可能无效。我们的理论结果得到了有限邻接矩阵数值对角化的完全证实。
英文摘要：
We derive exact equations for the spectral density of sparse networks with an arbitrary distribution of the number of single edges and triangles per node. These equations enable a systematic investigation of the effect of clustering on the spectral properties of the network adjacency matrix. In the case of heterogeneous networks, we demonstrate that the spectral density becomes more symmetric as the fluctuations in the triangle-degree sequence increase. This phenomenon is explained by the small clustering coefficient of networks with a large variance of the triangle-degree distribution. In the homogeneous case of regular clustered networks, we find that both perturbative and non-perturbative approximations fail to predict the spectral density in the high-connectivity limit. This suggests that traditional large-degree approximations may be ineffective in studying the spectral properties of networks with more complex motifs. Our theoretical results are fully confirmed by numerical diagonalizations of finite adjacency matrices.
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