In this paper, we devise a novel lightweight Graph Convolutional Network (GCN) design dubbed as Multi-Rate Magnitude Pruning (MRMP) that jointly trains network topology and weights. Our method is variational and proceeds by aligning the weight distribution of the learned networks with an a priori distribution. In the one hand, this allows implementing any fixed pruning rate, and also enhancing the generalization performances of the designed lightweight GCNs. In the other hand, MRMP achieves a joint training of multiple GCNs, on top of shared weights, in order to extrapolate accurate networks at any targeted pruning rate without retraining their weights. Extensive experiments conducted on the challenging task of skeleton-based recognition show a substantial gain of our lightweight GCNs particularly at very high pruning regimes.