  The geometry of a graph is encoded in dynamical processes on the graph. Many
graph neural network (GNN) architectures are inspired by such dynamical
systems, typically based on the graph Laplacian. Here, we introduce
Dirac--Bianconi GNNs (DBGNNs), which are based on the topological Dirac
equation recently proposed by Bianconi. Based on the graph Laplacian, we
demonstrate that DBGNNs explore the geometry of the graph in a fundamentally
different way than conventional message passing neural networks (MPNNs). While
regular MPNNs propagate features diffusively, analogous to the heat equation,
DBGNNs allow for coherent long-range propagation. Experimental results showcase
the superior performance of DBGNNs over existing conventional MPNNs for
long-range predictions of power grid stability and peptide properties. This
study highlights the effectiveness of DBGNNs in capturing intricate graph
dynamics, providing notable advancements in GNN architectures.
