Signal processing on graphs is based on the properties of an elementary operator generally associated with a notion of random walk / diffusion process. One limitation of these approaches is that the operator is systematically isotropic, a property that is passed on to any notion of filtering based on it. In multi-dimensional signal processing (images, video, etc), on the other hand, non-isotropic filters (or even filters that only take one direction into account) are used extensively, which greatly increases the possibilities. These non-isotropic filters are, in particular, the basic element of convolutional neural networks, which would likely have poorer performance with isotropic filters alone (i.e. impulse response with circular/spherical symmetry). The isotropy of the filters is also currently considered to be a major obstacle to the expressiveness of convolutional neural networks on graphs, which could be overcome using non-isotropic signal processing constructions on graphs. In addition to homogeneous graphs, operators used for signal processing or neural networks on bipartite or more generally heterogeneous graphs also have this property of isotropy where the neighbours of a node are treated identically. Although this time there is no obvious link with classical approaches, the notion of anisotropic or directional operator also seems relevant here to differentiate processing according to the multiple facets that can contribute to a given relationship.

To approach the concept of directionality in graphs, we will rely on the fact that a graph can often be viewed as a discretization of a Riemannian manifold. We will also examine extensions to bipartite graphs, which share similarities with a relationship between two manifolds, as well as heterogeneous graphs composed of multiple relations. Applications to graph neural networks will be explored to investigate the flexibility gained through directionality.