Sparse approximation of complex networks

This paper considers the problem of recovering a sparse approximation A∈Rn×n of an unknown (exact) adjacency matrix Atrue for a network from a corrupted version of a communicability matrix C=exp⁡(Atrue)+N∈Rn×n, where N denotes an unknown “noise matrix”. We consider two methods for determining an approximation A of Atrue: (i) a Newton method with soft-thresholding and line search, and (ii) a proximal gradient method with line search. These methods are applied to compute the solution of the minimization problemarg⁡minA∈Rn×n{‖exp⁡(A)−C‖F2+μ‖vec(A)‖1}, where μ>0 is a regularization parameter that controls the amount of shrinkage. We discuss the effect of μ on the computed solution, conditions for convergence, and the rate of convergence of the methods. Computed examples illustrate their performance when applied to directed and undirected networks.