Simplicial Complex Neural Networks

Graph-structured data, where nodes exhibit either pair-wise or high-order relations, are ubiquitous and essential in graph learning. Despite the great achievement made by existing graph learning models, these models use the direct information (edges or hyperedges) from graphs and do not adopt the underlying indirect information (hidden pair-wise or high-order relations). To address this issue, in this paper, we propose a general framework named Simplicial Complex Neural (SCN) network, in which we construct a simplicial complex based on the direct and indirect graph information from a graph so that all information can be employed in the complex network learning. Specifically, we learn representations of simplices by aggregating and integrating information from all the simplices together via layer-by-layer simplicial complex propagation. In consequence, the representations of nodes, edges, and other high-order simplices are obtained simultaneously and can be used for learning purposes. By making use of block matrix properties, we derive the theoretical bound of the simplicial complex filter learnt by the propagation and establish the generalization error bound of the proposed simplicial complex network. We perform extensive experiments on node (0-simplex), edge (1-simplex), and triangle (2-simplex) classifications, and promising results demonstrate the performance of the proposed method is better than that of existing graph and hypergraph network approaches.