Label Propagation (LPA) and Graph Convolutional Neural Networks (GCN) are both message passing algorithms on graphs. Both solve the task of node classification but LPA propagates node label information across the edges of the graph, while GCN propagates and transforms node feature information. However, while conceptually similar, theoretical relation between LPA and GCN has not yet been investigated. Here we study the relationship between LPA and GCN in terms of two aspects: (1) feature/label smoothing where we analyze how the feature/label of one node is spread over its neighbors; And, (2) feature/label influence of how much the initial feature/label of one node influences the final feature/label of another node. Based on our theoretical analysis, we propose an end-to-end model that unifies GCN and LPA for node classification. In our unified model, edge weights are learnable, and the LPA serves as regularization to assist the GCN in learning proper edge weights that lead to improved classification performance. Our model can also be seen as learning attention weights based on node labels, which is more task-oriented than existing feature-based attention models. In a number of experiments on real-world graphs, our model shows superiority over state-of-the-art GCN-based methods in terms of node classification accuracy.
The paper proposed an unified model for Label Propagation (LPA) and Graph Convolutional Neural Networks (GCN). It is shown how it is possible to infer the relationship between LPA and GCN in terms of label or feature smoothing (how label/feature does propagate over the neighbors) and label or feature influence over the other nodes. The results are given in terms of two theorems (whose proofs are in an appendix) which essentially state that the total label influence of nodes with a particular label “l” on a specific node is proportional to the probability that node is labelled as “l” by LPA. In practice, LPA acts as a regularizer to learn transformation matrices and edge weights simultaneously in GCN. By means of a simple joint loss (eq.8), the regularized training show that transductive learning with the joint model surpasses GCN/GNN baselines. 
While the proof of the first theorem is reasonable and seems correct (Taylor+ Schwarz inequality + L-Lip), the second left me a little puzzled, in particular wrt eq 16 and 17. Is it possible to add a graphical explanation or idea of the proof? Why “y” has to be reset at each iteration?
Unfortunately, the improvement brought by the framework is marginal, even if it seems general. 
The other limitation is that we have transductive training, but the authors are well aware about this.