Message Passing Neural Networks (MPNN) Molecules are represented as directed graphs, where the atoms are the nodes and the bonds are the edges of the graph. More formally, let x = (V, E) denote a directed graph where v i ∈ V are the atoms, and e ij ∈ E are the edges of the graph. The network maintains hidden states h t eij for each directed edge, where t is the layer index. At each step, the hidden representations aggregate information from neighboring nodes and edges, and captures a larger neighborhood of atoms. Iteratively, the hidden states are updated as: