Information Propagation Analysis of Social Network Using the Universality of Random Matrix

TL;DR

This paper demonstrates that the eigenvalue distribution of normalized Laplacian matrices in social networks follows the Wigner semicircle law, enabling analysis of information spread speed despite uncertain link weights.

Contribution

It introduces a novel approach using the universality of random matrices to analyze social network dynamics without precise relationship data.

Findings

Eigenvalue distribution follows the Wigner semicircle law regardless of link weight distribution.

Information propagation speed varies at most by a factor of 2 with network structure changes.

Spectral graph theory combined with random matrix universality provides robust analysis tools.

Abstract

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.