  Recovery of signals with elements defined on the nodes of a graph, from
compressive measurements is an important problem, which can arise in various
domains such as sensor networks, image reconstruction and group testing. In
some scenarios, the graph may not be accurately known, and there may exist a
few edge additions or deletions relative to a ground truth graph. Such
perturbations, even if small in number, significantly affect the Graph Fourier
Transform (GFT). This impedes recovery of signals which may have sparse
representations in the GFT bases of the ground truth graph. We present an
algorithm which simultaneously recovers the signal from the compressive
measurements and also corrects the graph perturbations. We analyze some
important theoretical properties of the algorithm. Our approach to correction
for graph perturbations is based on model selection techniques such as
cross-validation in compressed sensing. We validate our algorithm on signals
which have a sparse representation in the GFT bases of many commonly used
graphs in the network science literature. An application to compressive image
reconstruction is also presented, where graph perturbations are modeled as
undesirable graph edges linking pixels with significant intensity difference.
In all experiments, our algorithm clearly outperforms baseline techniques which
either ignore the perturbations or use first order approximations to the
perturbations in the GFT bases.
