Spectral graph theory gives an algebraic approach to the analysis of 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 matrices with the feature of social networks. As such a random matrix, we use the 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 a result, we show that the worst-case speed of the information propagation changes up to twice if the structure (i.e., relationship among people) of a social network changes.

This letter presents a novel tensor voting mechanism — analytic tensor voting (ATV), to get rid of the difficulties in original tensor voting, especially the efficiency. One of the main advantages is its explicit voting formulations, which benefit the completion of tensor voting theory and computational efficiency. Firstly, new decaying function was designed following the basic spirit of decaying function in original tensor voting (OTV). Secondly, analytic stick tensor voting (ASTV) was formulated using the new decaying function. Thirdly, analytic plate and ball tensor voting (APTV, ABTV) were formulated through controllable stick tensor construction and tensorial integration. These make the each voting of tensor can be computed by several non-iterative matrix operations, improving the efficiency of tensor voting remarkably. Experimental results validate the effectiveness of proposed method.