We show that the problem of recovering the topology and admittance of an electrical network from power and voltage data at all vertices is often ill-posed, and sometimes it even has multiple solutions. We reformulate the problem to seek for a sparse network, i.e., with few edges, which fits the data up to a given tolerance. We propose an algorithm to solve this reformulated problem. It combines, in an iterative procedure, the resolution of non-negative linear regression problems, and techniques of spectral graph sparsification. The algorithm is based on original results bounding the fitting error of a sparse approximation of a network. We illustrate our techniques with several experimental results in which we are able to recover a sparse network.