  Spectral graph theory gives an algebraical approach to analyze the dynamics
of a network by using the matrix that represents the network structure.
However, it is not easy for social networks to apply the spectral graph theory
because the matrix elements cannot be given exactly to represent the structure
of a social network. The matrix element should be set on the basis of the
relationship between persons, but the relationship cannot be quantified
accurately from obtainable data (e.g., call history and chat history). To get
around this problem, we utilize the universality of random matrix with the
feature of social networks. As such random matrix, we use normalized Laplacian
matrix for a network where link weights are randomly given. In this paper, we
first clarify that the universality (i.e., the Wigner semicircle law) of the
normalized Laplacian matrix appears in the eigenvalue frequency distribution
regardless of the link weight distribution. Then, we analyze the information
propagation speed by using the spectral graph theory and the universality of
the normalized Laplacian matrix. As the results, we show that the worst-case
speed of the information propagation changes at most 2 if the structure (i.e.,
relationship among people) of a social network changes.
