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 M\'emoli F, 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.

翻译：Gromov-Hausdorff距离用于度量可表示为紧致度量空间（如点云、流形或图）的物体之间的形状差异。计算任意Gromov-Hausdorff距离等价于求解NP难优化问题，这导致该概念在实际应用中缺乏可行性。本文提出一种多项式算法，用于估计所谓的修正Gromov-Hausdorff（mGH）距离——其与标准Gromov-Hausdorff（GH）距离的拓扑等价性已在Mémoli F（2012）的研究中得到证明。我们将该算法实现为Python库$\verb|scikit-tda|$的组成部分，专门处理由无权图诱导的紧致度量空间情形，并在真实世界网络与合成网络上验证了其性能。该算法能在多数具有无标度特性的图上精确计算mGH距离。我们利用计算得到的mGH距离，成功检测了真实世界社交网络与计算机网络中的异常节点。