  We consider a Graph Neural Network (GNN) non-Markovian modeling framework to
identify coarse-grained dynamical systems on graphs. Our main idea is to
systematically determine the GNN architecture by inspecting how the leading
term of the Mori-Zwanzig memory term depends on the coarse-grained interaction
coefficients that encode the graph topology. Based on this analysis, we found
that the appropriate GNN architecture that will account for $K$-hop dynamical
interactions has to employ a Message Passing (MP) mechanism with at least $2K$
steps. We also deduce that the memory length required for an accurate closure
model decreases as a function of the interaction strength under the assumption
that the interaction strength exhibits a power law that decays as a function of
the hop distance. Supporting numerical demonstrations on two examples, a
heterogeneous Kuramoto oscillator model and a power system, suggest that the
proposed GNN architecture can predict the coarse-grained dynamics under fixed
and time-varying graph topologies.
