Wigner's Semicircle Law of Weighted Random Networks

Abstract

Spectral graph theory provides an algebraic 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 to accurately represent the structures of large-scale and complex networks (e.g., social network) as a matrix. This difficulty can be avoided if there is a universality, such that the eigenvalues are independent of the detailed structure in large-scale and complex network. In this paper, we clarify Wigner's Semicircle Law for weighted networks as such a universality. The law indicates that the eigenvalues of the normalized Laplacian matrix of weighted networks can be calculated from a few network statistics (the average degree, average link weight, and square average link weight) when the weighted networks satisfy a sufficient condition of the node degrees and the link weights.