Method For Efficiently Orthogonalizing The Eigenvectors Of The Laplacian Matrix To Estimate Social Network Structure

The structure of social networks or human relationships is difficult to understand since we cannot observed their links and link weights directly. The network resonance method was proposed to obtain information on the unknown Laplacian matrix representing the social network structure. This method extracts information on the eigenvalues and eigenvectors of the Laplacian matrix by observing user dynamics on social networks. The original Laplacian matrix can be reconstructed if all eigenvalues and eigenvectors are known. However, the network resonance method has a problem: the information available about eigenvectors is limited to the absolute value of each element. Therefore, to determine the Laplacian matrix, it is necessary to determine the signs of each element of all eigenvectors. However, sign determination incurs the computation cost of the order of O(2(n)) for each of n eigenvectors. This paper proposes a method to determine the signs of each eigenvector element efficiently. The main idea of the method is to generate n(2) - n different sign determination problems for n eigenvectors and to solve them in parallel. All that is required is to obtain n different eigenvectors determined in the shortest time from the n(2) - n different sign determination problems. Since the ratio of the number of sign determinations completed in the method is 1/(n - 1), its efficiency rises with the number of network users. In addition, simulations on networks generated by the BA model show that proposed method offers sign determination in polynomial time.