Estimating the structure of social networks from incomplete sets of observed information by using compressed sensing

For complex large scale networks, like social networks, it is typically impossible to observe complete information about their topology structure or link weight directly. A recent proposal, the network resonance method, can estimate the eigenvalues and eigenvectors of the Laplacian matrix for representing network structure, by using the resonance phenomena of oscillation dynamics on networks. However, it is generally not possible to observe all the eigenvalues and eigenvectors. In practice, the observed values must be assumed to include some non-negligible errors. This paper uses compressed sensing to create a new method of reconstructing the original Laplacian matrix from some of its eigenvalues and eigenvectors. Since almost all node pairs in social networks have no link, we can expect that compressed sensing will be effective. The estimated Laplacian matrix of a social network enables to us to know its structure and link weights.