Persistent path homology of directed networks

Abstract

While standard persistent homology has been successful in extracting information from metric datasets, its applicability to more general data, e.g. directed networks, is hindered by its natural insensitivity to asymmetry. We extend a construction of homology of digraphs due to Grigoryan, Lin, Muranov and Yau to the persistent framework. The result, which we call persistent path homology or PPH, encodes a rich level of detail about the asymmetric structure of the input directed network. For example, we prove that PPH identifies a class of directed cyclic networks as directed analogues of the circle. In general, PPH produces signatures that differ from natural extensions of Rips or Cech persistence to the directed setting, but we prove that PPH agrees with Cech persistence on symmetric spaces. Additionally, we prove that PPH agrees with Cech persistence on directed networks satisfying a local condition that we call squarefreeness. We prove stability of PPH by utilizing a separate theory of homotopy of digraphs that is compatible with path homology. Finally, we study computational aspects of PPH, and derive an algorithm showing that over field coeficients, computing PPH requires the same worst case running time as standard persistent homology.