The Wigner's Semicircle Law of Weighted Random Networks

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.