  This work develops a flexible and mathematically sound framework for the
design and analysis of graph scattering networks with variable branching ratios
and generic functional calculus filters. Spectrally-agnostic stability
guarantees for node- and graph-level perturbations are derived; the vertex-set
non-preserving case is treated by utilizing recently developed
mathematical-physics based tools. Energy propagation through the network layers
is investigated and related to truncation stability. New methods of graph-level
feature aggregation are introduced and stability of the resulting composite
scattering architectures is established. Finally, scattering transforms are
extended to edge- and higher order tensorial input. Theoretical results are
complemented by numerical investigations: Suitably chosen cattering networks
conforming to the developed theory perform better than traditional
graph-wavelet based scattering approaches in social network graph
classification tasks and significantly outperform other graph-based learning
approaches to regression of quantum-chemical energies on QM7.
