How did the shape of your network change? (On detecting anomalies in static and dynamic networks via change of non-local curvatures)
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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 due to several reasons such as lack of preferred geometric interpretation of networks and lack of experimental evidences that may lead to specific desired curvature properties. To this end, our goal in this paper to formulate and analyze curvature analysis methods to provide the foundations of systematic approaches to find critical components and anomaly detection in networks. For this purpose, we use two measures of network curvatures: (i) the Gromov-hyperbolic combinatorial curvature based on the properties of exact and approximate geodesics and higher-order connectivities, and (ii) geometric curvatures based on defining k-complex-based Forman's combinatorial Ricci curvature for elementary components, and using Euler characteristic of the complex that is topologically associated with the given graph. 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 (e.g., exact or approximation algorithms, approximation hardness) for these problems.
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