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The scattering transform is a multilayered wavelet-based deep learningarchitecture that acts as a model of convolutional neural networks. Recently,several works have introduced generalizations of the scattering transform fornon-Euclidean settings such as graphs. Our work builds upon these constructionsby introducing windowed and non-windowed geometric scattering transforms forgraphs based upon a very general class of asymmetric wavelets. We show thatthese asymmetric graph scattering transforms have many of the same theoreticalguarantees as their symmetric counterparts. As a result, the proposedconstruction unifies and extends known theoretical results for many of theexisting graph scattering architectures. In doing so, this work helps bridgethe gap between geometric scattering and other graph neural networks byintroducing a large family of networks with provable stability and invarianceguarantees. These results lay the groundwork for future deep learningarchitectures for graph-structured data that have learned filters and alsoprovably have desirable theoretical properties.

Understanding Graph Neural Networks with Generalized Geometric Scattering Transforms