  We provide a characterization of two types of directed homology for
fully-connected, feedforward neural network architectures. These exact
characterizations of the directed homology structure of a neural network
architecture are the first of their kind. We show that the directed flag
homology of deep networks reduces to computing the simplicial homology of the
underlying undirected graph, which is explicitly given by Euler characteristic
computations. We also show that the path homology of these networks is
non-trivial in higher dimensions and depends on the number and size of the
layers within the network. These results provide a foundation for investigating
homological differences between neural network architectures and their realized
structure as implied by their parameters.
