In studying biological systems, identifying the underlying gene regulatory networks from data has been important and will continue to affect the study of gene regulatory networks. We consider the problem of reconstructing the network structure from observed data, and in turn uncovering the underlying mechanisms responsible for the observed behaviors. A key challenge inherent in the network reconstruction problem comes from the necessity to deal with noisy and partial measurements. In previous work , we have proposed a method based on compressive sensing (CS) for reconstructing a sparse network structure, without any a priori information of connectivity, based on the time-series gene expression data. In this paper, we extend our previous work to consider a more general problem in which there might be hidden nodes which affect system dynamics. Then, we ask whether it is still possible to reconstruct the graph structure reliably when the dynamics of a certain node is corrupted by arbitrarily large errors and in addition, all the measurements are contaminated by measurement noise. We show that we can infer the graph structure by solving a two-stage convex optimization problem and demonstrate our studies with numerical example to illustrate its performance.