Efficient estimation of a Gromov--Hausdorff distance between unweighted graphs

Gromov--Hausdorff distances measure shape difference between the objects representable as compact metric spaces, e.g. point clouds, manifolds, or graphs. Computing any Gromov--Hausdorff distance is equivalent to solving an NP-Hard optimization problem, deeming the notion impractical for applications. In this paper we propose polynomial algorithm for estimating the so-called modified Gromov--Hausdorff (mGH) distance, whose topological equivalence with the standard Gromov--Hausdorff (GH) distance was established in \cite{memoli12} (M\'emoli, F, \textit{Discrete \& Computational Geometry, 48}(2) 416-440, 2012). We implement the algorithm for the case of compact metric spaces induced by unweighted graphs as part of Python library \verb|scikit-tda|, and demonstrate its performance on real-world and synthetic networks. The algorithm finds the mGH distances exactly on most graphs with the scale-free property. We use the computed mGH distances to successfully detect outliers in real-world social and computer networks.