We consider estimation of multiple high-dimensional Gaussian graphical models corresponding to a single set of nodes under several distinct conditions. We assume that most aspects of the networks are shared, but that there are some structured differences between them. Specifically, the network differences are generated from node perturbations: a few nodes are perturbed across networks, and most or all edges stemming from such nodes differ between networks. This corresponds to a simple model for the mechanism underlying many cancers, in which the gene regulatory network is disrupted due to the aberrant activity of a few specific genes. We propose to solve this problem using the structured joint graphical lasso, a convex optimization problem that is based upon the use of a novel symmetric overlap norm penalty, which we solve using an alternating directions method of multipliers algorithm. Our proposal is illustrated on synthetic data and on an application to brain cancer gene expression data.