  The existing definitions of graph convolution, either from spatial or
spectral perspectives, are inflexible and not unified. Defining a general
convolution operator in the graph domain is challenging due to the lack of
canonical coordinates, the presence of irregular structures, and the properties
of graph symmetries. In this work, we propose a novel and general graph
convolution framework by parameterizing the kernels as continuous functions of
pseudo-coordinates derived via graph positional encoding. We name this
Continuous Kernel Graph Convolution (CKGConv). Theoretically, we demonstrate
that CKGConv is flexible and expressive. CKGConv encompasses many existing
graph convolutions, and exhibits a stronger expressiveness, as powerful as
graph transformers in terms of distinguishing non-isomorphic graphs.
Empirically, we show that CKGConv-based Networks outperform existing graph
convolutional networks and perform comparably to the best graph transformers
across a variety of graph datasets. The code and models are publicly available
at https://github.com/networkslab/CKGConv.
