Distances and Isomorphism between Networks and the Stability of Network Invariants

Abstract

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. CONTENTS1. Introduction 1.1. Contributions and organization of the paper 1.2. Related literature 1.3. Notation and basic terminology 2. Networks, isomorphism, and network distances 2.1. The second network distance 2.2. Special families: dissimilarity networks and directed metric spaces 2.3. Two families of examples: the directed circles 2.3.1. The general directed circle 2.3.2. The directed circles with finite reversibility 3. The case of compact networks 3.1. Compact networks and finite sampling 3.1.1. Proofs of result in §3.1 3.2. Compact networks and weak isomorphism 4. Invariants of networks 4.1. Quantitative stability of invariants of networks 5. Computational aspects 5.1. The complexity of computing d N

The Gromov-Wasserstein distance between networks and stable network invariants

“…Remark 19. Theorem 18 is in the same spirit as related results for gauged measure spaces [Stu12] and for networks without measure equipped with a Gromov-Hausdorff-type network distance [CM17]. The "tripod structure" X ← Z → Y described above is much more difficult to obtain in the setting of [CM17].…”

Graph Distance from the Topological View of Non-backtracking Cycles

“…In this work, we have focused on the problem of deriving a notion of graph distance for complex networks based on the length spectrum function. We add to the repertoire of distance measures [45,26,4,8,39,43,12,13] the Truncated Non-Backtracking Spectral Distance (TNBSD): a principled, interpretable, efficient, and effective measure that takes advantage of the fact that the non-backtracking cycles of a graph can be interpreted as its free homotopy classes. TNBSD is principled because it is backed by the theory of the length spectrum, which characterizes the 2-core of a graph up to isomorphism; it is interpretable because we can study its behavior in the presence of structural features such as hubs and triangles, and we can use the resulting geometric features of the eigenvalue distribution to our advantage; it is efficient because it takes no more time than computing a few of the largest eigenvalues of the non-backtracking matrix; and we have presented extensive experimental evidence to show that it is effective at discriminating between complex networks in various contexts, including visualization, clustering, sampling, and anomaly detection.…”

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