Distances and Isomorphism between Networks and the Stability of Network Invariants

We develop the theoretical foundations of a network distance that has recently been applied to various subfields of topological data analysis, namely persistent homology and hierarchical clustering. While this network distance has previously appeared in the context of finite networks, we extend the setting to that of compact networks. The main challenge in this new setting is the lack of an easy notion of sampling from compact networks; we solve this problem in the process of obtaining our results. The generality of our setting means that we automatically establish results for exotic objects such as directed metric spaces and Finsler manifolds. We identify readily computable network invariants and establish their quantitative stability under this network distance. We also discuss the computational complexity involved in precisely computing this distance, and develop easily-computable lower bounds by using the identified invariants. By constructing a wide range of explicit examples, we show that these lower bounds are effective in distinguishing between networks. Finally, we provide a simple algorithm that computes a lower bound on the distance between two networks in polynomial time and illustrate our metric and invariant constructions on a database of random networks and a database of simulated hippocampal networks