Important data mining problems such as nearest-neighbor search and clustering admit theoretical guarantees when restricted to objects embedded in a metric space. Graphs are ubiquitous, and clustering and classification over graphs arise in diverse areas, including, e.g., image processing and social networks. Unfortunately, popular distance scores used in these applications, that scale over large graphs, are not metrics and thus come with no guarantees. Classic graph distances such as, e.g., the chemical distance and the Chartrand-Kubiki-Shultz distance are arguably natural and intuitive, and are indeed also metrics, but they are intractable: as such, their computation does not scale to large graphs. We define a broad family of graph distances, that includes both the chemical and the Chartrand-Kubiki-Shultz distances, and prove that these are all metrics. Crucially, we show that our family includes metrics that are tractable. Moreover, we extend these distances by incorporating auxiliary node attributes, which is important in practice, while maintaining both the metric property and tractability.

中文翻译：

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一族易处理的图形指标

重要的数据挖掘问题（例如最近邻居搜索和聚类）在限于度量空间中嵌入的对象时，可以提供理论上的保证。图无处不在，并且图的聚类和分类出现在不同的领域，包括图像处理和社交网络。不幸的是，这些应用程序中使用的流行距离得分（覆盖大图）不是度量指标，因此无法保证。经典的图形距离（例如化学距离和Chartrand-Kubiki-Shultz距离）可以说是自然而直观的，并且确实是度量标准，但是它们是棘手的：因此，它们的计算不能缩放到大图。我们定义了一系列图形距离，包括化学距离和Chartrand-Kubiki-Shultz距离，并证明这些都是指标。至关重要的是，我们证明了我们的家庭包括易于处理的指标。此外，我们通过合并辅助节点属性来扩展这些距离，这在实践中很重要，同时又保持了度量属性和易处理性。