  In this paper we introduce DYMAG: a message passing paradigm for GNNs built
on the expressive power of continuous, multiscale graph-dynamics. Standard
discrete-time message passing algorithms implicitly make use of simplistic
graph dynamics and aggregation schemes which limit their ability to capture
fundamental graph topological properties. By contrast, DYMAG makes use of
complex graph dynamics based on the heat and wave equation as well as a more
complex equation which admits chaotic solutions. The continuous nature of the
dynamics are leveraged to generate multiscale (dynamic-time snapshot)
representations which we prove are linked to various graph topological and
spectral properties. We demonstrate experimentally that DYMAG achieves superior
performance in recovering the generating parameters of Erd\"os-Renyi and
stochastic block model random graphs and the persistent homology of synthetic
graphs and citation network. Since the behavior of proteins and biomolecules is
sensitive to graph topology and exhibits important structure at multiple
scales, we find that DYMAG outperforms other methods at predicting salient
features of various biomolecules.
