Structural equation models (SEMs) provide a statistical description of directed networks. The networks modeled by SEMs may have signed edge weights, a property that is pertinent to represent the activating and inhibitory interactions characteristic of biological systems, as well as the collaborative and antagonist behaviors found in social networks, among other applications. They may also have cyclic paths, accommodating the presence of protein stabilizing loops, or the feedback in decision making processes. Starting from the mathematical description of a linear SEM, this paper aims to identify the topology, edge directions, and edge weights of the underlying network. It is established that perturbation data is essential for this purpose, otherwise directional ambiguities cannot be resolved. It is also proved that the required amount of data is significantly reduced when the network topology is assumed to be sparse; that is, when the number of incoming edges per node is much smaller than the network size. Identifying a dynamic network with step changes across time is also considered, but it is left as an open problem to be addressed in an extended version of this paper.