  Graph neural networks (GNNs) have become pivotal tools for processing
graph-structured data, leveraging the message passing scheme as their core
mechanism. However, traditional GNNs often grapple with issues such as
instability, over-smoothing, and over-squashing, which can degrade performance
and create a trade-off dilemma. In this paper, we introduce a discriminatively
trained, multi-layer Deep Scattering Message Passing (DSMP) neural network
designed to overcome these challenges. By harnessing spectral transformation,
the DSMP model aggregates neighboring nodes with global information, thereby
enhancing the precision and accuracy of graph signal processing. We provide
theoretical proofs demonstrating the DSMP's effectiveness in mitigating these
issues under specific conditions. Additionally, we support our claims with
empirical evidence and thorough frequency analysis, showcasing the DSMP's
superior ability to address instability, over-smoothing, and over-squashing.
