The spectral graph theory provides an algebraical approach to investigate the characteristics of weighted networks using the eigenvalues and eigenvectors of a matrix (e.g., normalized Laplacian matrix) that represents the structure of the network. However, it is difficult for large-scale and complex networks (e.g., social network) to represent their structure as a matrix correctly. If there is a universality that the eigenvalues are independent of the detailed structure in large-scale and complex network, we can avoid the difficulty. In this paper, we clarify the Wigner's Semicircle Law for weighted networks as such a universality. The law indicates that the eigenvalues of the normalized Laplacian matrix for weighted networks can be calculated from the a few network statistics (the average degree, the average link weight, and the square average link weight) when the weighted networks satisfy the sufficient condition of the node degrees and the link weights.

中文翻译：

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加权随机网络的 Wigner 半圆定律

谱图理论提供了一种代数方法来使用表示网络结构的矩阵（例如，归一化拉普拉斯矩阵）的特征值和特征向量来研究加权网络的特征。然而，大规模复杂的网络（例如社交网络）很难将其结构正确地表示为矩阵。如果在大规模复杂网络中存在特征值独立于详细结构的普遍性，我们就可以避免这个困难。在本文中，我们阐明了加权网络的 Wigner 半圆定律作为这种普遍性。该定律表明，加权网络的归一化拉普拉斯矩阵的特征值可以从几个网络统计数据（平均度、平均链路权重、