A Study on the Effect of Triads on the Wigner’s Semicircle Law of Networks

Abstract

Spectral graph theory is widely used to analyze network characteristics. In spectral graph theory, the network structure is represented with a matrix, and its eigenvalues and eigenvectors are used to clarify the characteristics of the network. However, it is difficult to accurately represent the structure of a social network with a matrix. We derived the Wigner’s semicircle law that appears in the universality for the eigenvalue distribution of the normalized Laplacian matrix representing the structure of social networks, and proposed the analysis method to apply the spectral graph theory to social networks using the Wigner’s semicircle law. In previous works, we assume that nodes in a network are connected independently. However, in actual social networks, there are dependent structures called triads where link connections cannot be independent. For example, a triad is generated when a person makes a new friend via the introduction by its friend. In this paper, we experimentally investigate the effect of triads on the Wigner’s semicircle law, and clarify how effectively the Wigner’s semicircle law can be used for the analysis of networks with triads.