无权图之间修正Gromov-Hausdorff距离的高效估计
Efficient estimation of the modified Gromov-Hausdorff distance between unweighted graphs
AI总结：
本文提出一种多项式时间算法估计修正Gromov-Hausdorff距离，并在scikit-tda中实现，实验表明其能在多数无标度图上精确计算并有效检测网络异常值。
摘要：
Gromov-Hausdorff距离度量可表示为紧度量空间的对象（例如点云、流形或图）之间的形状差异。计算任何Gromov-Hausdorff距离都等价于求解一个NP-Hard优化问题，这使得该概念在实际应用中不切实际。本文提出了一种多项式算法，用于估计所谓的修正Gromov-Hausdorff（mGH）距离，该距离与标准Gromov-Hausdorff（GH）距离的拓扑等价性已在Mémoli F, 2012中建立。我们将该算法实现为Python库scikit-tda的一部分，用于处理由无权图诱导的紧度量空间，并在真实世界和合成网络上展示了其性能。该算法在大多数具有无标度性质的图上精确求出mGH距离。我们使用计算得到的mGH距离成功检测了真实世界社交网络和计算机网络中的异常值。
英文摘要：
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 Mémoli 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.
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