The Metric Space of Networks

Abstract

We study the question of reconstructing a weighted, directed network up to isomorphism from its motifs. In order to tackle this question we first relax the usual (strong) notion of graph isomorphism to obtain a relaxation that we call weak isomorphism. Then we identify a definition of distance on the space of all networks that is compatible with weak isomorphism. This global approach comes equipped with notions such as completeness, compactness, curves, and geodesics, which we explore throughout this paper. Furthermore, it admits global-to-local inference in the following sense: we prove that two networks are weakly isomorphic if and only if all their motif sets are identical, thus answering the network reconstruction question. Further exploiting the additional structure imposed by our network distance, we prove that two networks are weakly isomorphic if and only if certain essential associated structures---the skeleta of the respective networks---are strongly isomorphic.

This paper has been withdrawn by the authors. This paper has been superseded by v3 of arXiv:1708.04727 (merged from arXiv:1708.04727 (v2), arXiv:1804.02820, and arXiv:1711.04211), which in turn has appeared in the Journal of Applied and Computational Topology

References in corpus (3)

• Hierarchical Quasi-Clustering Methods for Asymmetric Networks
• Lossless Representation of Graphs using Distributions
• Barycenters in uniformly convex geodesic spaces

Cited by in corpus (3)

• Sampling random graph homomorphisms and applications to network data analysis
• Graph Distance from the Topological View of Non-backtracking Cycles
• Persistent Homotopy Groups of Metric Spaces