3): Convolutions with directed message passing. GNNs generally use a series of convolution, or messagepassing, functions to iteratively update nodes' feature vectors, from which a final property prediction can be made. The challenge here lies in deriving a convolution function that ensures that Equation ( ) and Equation ( ) are respected. In other words, to preserve Equation ( ), feature vectors corresponding to crystal sites (which will ultimately be used to predict the vacancy formation energies, ∆H p0 and ∆H pn+1 ) cannot be influenced by feature vectors corresponding to image nodes (which will be used to predict the image energies, ∆H p {0...n} ). To ensure Equation ( ), the converse must be true and the image energies clearly must depend on the path to which they belong, and therefore the endpoints and their vacancy formation energies. These considerations are also shown schematically in Figure .