  Recently, neural network based approaches have achieved significant
improvement for solving large, complex, graph-structured problems. However,
their bottlenecks still need to be addressed, and the advantages of multi-scale
information and deep architectures have not been sufficiently exploited. In
this paper, we theoretically analyze how existing Graph Convolutional Networks
(GCNs) have limited expressive power due to the constraint of the activation
functions and their architectures. We generalize spectral graph convolution and
deep GCN in block Krylov subspace forms and devise two architectures, both with
the potential to be scaled deeper but each making use of the multi-scale
information in different ways. We further show that the equivalence of these
two architectures can be established under certain conditions. On several node
classification tasks, with or without the help of validation, the two new
architectures achieve better performance compared to many state-of-the-art
methods.
