Unifying Label-inputted Graph Neural Networks with Deep Equilibrium Models

For node classification, Graph Neural Networks (GNN) assign predefined labels to graph nodes according to node features propagated along the graph structure. Apart from the traditional end-to-end manner inherited from deep learning, many subsequent works input assigned labels into GNNs to improve their classification performance. Such label-inputted GNNs (LGNN) combine the advantages of learnable feature propagation and long-range label propagation, producing state-of-the-art performance on various benchmarks. However, the theoretical foundations of LGNNs are not well-established, and the combination is with seam because the long-range propagation is memory-consuming for optimization. To this end, this work interprets LGNNs with the theory of Implicit GNN (IGNN), which outputs a fixed state point of iterating its network infinite times and optimizes the infinite-range propagation with constant memory consumption. Besides, previous contributions to LGNNs inspire us to overcome the heavy computation in training IGNN by iterating the network only once but starting from historical states, which are randomly masked in forward-pass to implicitly guarantee the existence and uniqueness of the fixed point. Our improvements to IGNNs are network agnostic: for the first time, they are extended with complex networks and applied to large-scale graphs. Experiments on two synthetic and six real-world datasets verify the advantages of our method in terms of long-range dependencies capturing, label transitions modelling, accuracy, scalability, efficiency, and well-posedness.