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 a polynomial algorithm for estimating the so-called modified Gromov–Hausdorff (mGH) distance, a relaxation of the standard Gromov–Hausdorff (GH) distance with similar topological properties. We implement the algorithm for the case of compact metric spaces induced by unweighted graphs as part of Python library 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.

格罗莫夫-豪斯多夫距离测量可表示为紧凑度量空间的对象之间的形状差异，例如点云、流形或图形。计算任何 Gromov-Hausdorff 距离都相当于解决 NP 难优化问题，认为该概念对于应用来说不切实际。在本文中，我们提出了一种多项式算法，用于估计所谓的修正格罗莫夫-豪斯多夫（mGH）距离，这是具有相似拓扑特性的标准格罗莫夫-豪斯多夫（GH）距离的松弛。我们在 Python 库scikit-tda中实现了由未加权图引起的紧凑度量空间情况的算法，并展示了其在现实世界和合成网络上的性能。该算法在大多数具有无标度特性的图上精确地找到 mGH 距离。我们使用计算出的 mGH 距离成功检测现实世界社交和计算机网络中的异常值。