  Persistent homology, a technique from computational topology, has recently
shown strong empirical performance in the context of graph classification.
Being able to capture long range graph properties via higher-order topological
features, such as cycles of arbitrary length, in combination with multi-scale
topological descriptors, has improved predictive performance for data sets with
prominent topological structures, such as molecules. At the same time, the
theoretical properties of persistent homology have not been formally assessed
in this context. This paper intends to bridge the gap between computational
topology and graph machine learning by providing a brief introduction to
persistent homology in the context of graphs, as well as a theoretical
discussion and empirical analysis of its expressivity for graph learning tasks.
