Sparse approximation of complex networks

Abstract

This paper considers the problem of recovering a sparse approximation $A\in R^{n\times n}$ of an unknown (exact) adjacency matrix $A_{\text{true}}$ for a network from a corrupted version of a communicability matrix $C=\exp \left(A_{\text{true}}\right)+N\in R^{n\times n}$, where N denotes an unknown “noise matrix”. We consider two methods for determining an approximation A of $A_{\text{true}}$: $(i)$ a Newton method with soft-thresholding and line search, and $(\mathrm{ii})$ a proximal gradient method with line search. These methods are applied to compute the solution of the minimization problem$\underset{A\in R^{n\times n}}{\arg \min }\{{\Vert \exp \left(A\right)-C\Vert }_F^2+\mu {\Vert \text{vec}\left(A\right)\Vert }_1\},$ where $\mu >0$ 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.