Persistent Homology Lower Bounds on High Order Network Distances

Abstract: High order networks are weighted hypergraphs col- lecting relationships between elements of tuples, not necessarily pairs. Valid metric distances between high order networks have been defined but they are difficult to compute when the number of nodes is large. The goal here is to find tractable approximations of these network distances. The paper does so by mapping high order networks to filtrations of simplicial complexes and showing that the distance between networks can be lower bounded by the difference between the homological features of their respective filtrations. Practical implications are explored by classifying weighted pairwise networks constructed from different gener- ative processes and by comparing the coauthorship networks of engineering and mathematics academic journals. The persistent homology methods succeed in identifying different generative models, in discriminating engineering and mathematics commu- nities, as well as in differentiating engineering communities with different research interests.