Anomaly detection problems (also called change-point detection problems) have been studied in data mining, statistics and computer science over the last several decades (mostly in non-network context ) in applications such as medical condition monitoring, weather change detection and speech recognition. In recent days, however, anomaly detection problems have become increasing more relevant in the context of network science since useful insights for many complex systems in biology, finance and social science are often obtained by representing them via networks. Notions of local and non-local curvatures of higher-dimensional geometric shapes and topological spaces play a fundamental role in physics and mathematics in characterizing anomalous behaviours of these higher dimensional entities. However, using curvature measures to detect anomalies in networks is not yet very common. To this end, a main goal in this paper to formulate and analyze curvature analysis methods to provide the foundations of systematic approaches to find critical components and detect anomalies in networks. For this purpose, we use two measures of network curvatures which depend on non-trivial global properties, such as distributions of geodesics and higher-order correlations among nodes, of the given network. Based on these measures, we precisely formulate several computational problems related to anomaly detection in static or dynamic networks, and provide non-trivial computational complexity results for these problems. This paper must not be viewed as delivering the final word on appropriateness and suitability of specific curvature measures. Instead, it is our hope that this paper will stimulate and motivate further theoretical or empirical research concerning the exciting interplay between notions of curvatures from network and non-network domains, a much desired goal in our opinion.

中文翻译：

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为什么您的网络形态发生了变化？（通过非局部曲率检测网络异常）

在过去的几十年中，异常检测问题（也称为变化点检测问题）在数据挖掘、统计和计算机科学中（主要是在非网络环境中）在医疗状况监测、天气变化检测和语音识别等应用中得到了研究。 . 然而，最近几天，异常检测问题在网络科学的背景下变得越来越重要，因为生物学、金融和社会科学中许多复杂系统的有用见解通常是通过网络表示它们来获得的。高维几何形状和拓扑空间的局部和非局部曲率的概念在物理和数学中在表征这些高维实体的异常行为方面发挥着基本作用。然而，使用曲率测量来检测网络中的异常情况还不是很常见。为此，本文的主要目标是制定和分析曲率分析方法，为查找网络中的关键组件和检测异常提供系统方法的基础。为此，我们使用两种网络曲率的度量，这些度量取决于给定网络的非平凡全局属性，例如测地线的分布和节点之间的高阶相关性。基于这些措施，我们精确地制定了几个与静态或动态网络中的异常检测相关的计算问题，并为这些问题提供了非平凡的计算复杂度结果。不得将本文视为对特定曲率测量的适当性和适用性的最终确定。反而，