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The spectral graph theory provides an algebraical approach to investigate thecharacteristics of weighted networks using the eigenvalues and eigenvectors ofa matrix (e.g., normalized Laplacian matrix) that represents the structure ofthe network. However, it is difficult for large-scale and complex networks(e.g., social network) to represent their structure as a matrix correctly. Ifthere is a universality that the eigenvalues are independent of the detailedstructure in large-scale and complex network, we can avoid the difficulty. Inthis paper, we clarify the Wigner's Semicircle Law for weighted networks assuch a universality. The law indicates that the eigenvalues of the normalizedLaplacian matrix for weighted networks can be calculated from the a few networkstatistics (the average degree, the average link weight, and the square averagelink weight) when the weighted networks satisfy the sufficient condition of thenode degrees and the link weights.

The Wigner's Semicircle Law of Weighted Random Networks