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$Anomaly$ $detection$ problems (also called $change$-$point$ $detection$problems) have been studied in data mining, statistics and computer scienceover the last several decades in applications such as medical conditionmonitoring and weather change detection. In recent days, however, anomalydetection 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 vianetworks. Notions of local and non-local curvatures of higher-dimensionalgeometric shapes and topological spaces play a $fundamental$ role in physicsand mathematics in characterizing anomalous behaviours of these higherdimensional entities. However, using curvature measures to detect anomalies innetworks is not yet very common. To this end, a main goal in this paper toformulate and analyze curvature analysis methods to provide the foundations ofsystematic approaches to find $critical$ $components$ and $detect$ $anomalies$in networks. For this purpose, we use two measures of network curvatures whichdepend on non-trivial global properties, such as distributions of geodesics andhigher-order correlations among nodes, of the given network. Based on thesemeasures, we precisely formulate several computational problems related toanomaly detection in static or dynamic networks, and provide non-trivialcomputational complexity results for these problems. This paper must $not$ beviewed as delivering the final word on appropriateness and suitability ofspecific curvature measures. Instead, it is our hope that this paper willstimulate and motivate further theoretical or empirical research concerning theexciting interplay between notions of curvatures from network and non-networkdomains, a $much$ desired goal in our opinion.

Why did the shape of your network change? (On detecting network anomalies via non-local curvatures)