This paper introduces a novel cost sensitive weighted samples approach to a cascade of Graph Neural Networks for learning from imbalanced data in the graph structured input domain. This is shown to be very effective in addressing the effects of imbalanced data distribution on learning systems. The proposed idea is based on a weighting mechanism which forces the network to encode misclassified graphs (or nodes) more strongly. We evaluate the approach through an application to the well known Web spam detection problem, and demonstrate that the general-ization performance is improved as a result. Indeed the results obtained reported in this paper are the best reported so far for both datasets.