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Why Did the Shape of Your Network Change? (On Detecting Network Anomalies via Non-local Curvatures)

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Abstract

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.

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Notes

  1. A non-trivial property usually refers to a property such that a significant percentage of all possible networks satisfies the property and also a significant percentage of all possible networks does not satisfy the property. A non-local property (also called global property) usually refers to a property that cannot be inferred by simply looking at a local neighborhood of any one node.

  2. \({\tilde{O}}(\cdot )\) is a standard computational complexity notation that omits poly-logarithmic factors.

  3. For faster implementation, in the loop of Step 2 we can do binary search for the least possible \(\kappa \) over the range \(\{1,2, \ldots ,|{\widetilde{E}}|\}\) for which the polytope’s optimal solution value is at most \(\varGamma \), requiring \(\lceil \log _2 (1 + |{\widetilde{E}}|) \rceil \) iterations instead of \(|{\widetilde{E}}|\) iterations. For clarity, we omit such obvious improvements.

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Acknowledgements

We thank Anastasios Sidiropoulos and Nasim Mobasheri for very useful discussions. This research work was partially supported by NSF Grants IIS-1160995 and IIS-1814931.

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  1. Department of Computer Science, University of Illinois at Chicago, Chicago, IL, 60607, USA

    Bhaskar DasGupta & Farzane Yahyanejad

  2. Department of Mathematics, University of Illinois at Chicago, Chicago, IL, 60607, USA

    Mano Vikash Janardhanan

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  1. Bhaskar DasGupta
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DasGupta, B., Janardhanan, M.V. & Yahyanejad, F. Why Did the Shape of Your Network Change? (On Detecting Network Anomalies via Non-local Curvatures). Algorithmica 82, 1741–1783 (2020). https://doi.org/10.1007/s00453-019-00665-7

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