  Graph convolutional networks (GCNs) have well-documented performance in
various graph learning tasks, but their analysis is still at its infancy. Graph
scattering transforms (GSTs) offer training-free deep GCN models that extract
features from graph data, and are amenable to generalization and stability
analyses. The price paid by GSTs is exponential complexity in space and time
that increases with the number of layers. This discourages deployment of GSTs
when a deep architecture is needed. The present work addresses the complexity
limitation of GSTs by introducing an efficient so-termed pruned (p)GST
approach. The resultant pruning algorithm is guided by a
graph-spectrum-inspired criterion, and retains informative scattering features
on-the-fly while bypassing the exponential complexity associated with GSTs.
Stability of the novel pGSTs is also established when the input graph data or
the network structure are perturbed. Furthermore, the sensitivity of pGST to
random and localized signal perturbations is investigated analytically and
experimentally. Numerical tests showcase that pGST performs comparably to the
baseline GST at considerable computational savings. Furthermore, pGST achieves
comparable performance to state-of-the-art GCNs in graph and 3D point cloud
classification tasks. Upon analyzing the pGST pruning patterns, it is shown
that graph data in different domains call for different network architectures,
and that the pruning algorithm may be employed to guide the design choices for
contemporary GCNs.
