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

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.In this paper we seek to address research questions of the following generic nature:"Given a static or dynamic network, identify the critical components of the network that "encode" significant non-trivial global properties of the network".To identify critical components, one first needs to provide details for following four specific items:(i) network model selection,(ii) network evolution rule for dynamic networks, (iii) definition of elementary critical components, and (iv) network property selection (i.e., the global properties of the network to be investigated).The specific details for these items for this paper are as follows:(i) Network model selection: Our network model will be undirected graphs.(ii) Network evolution rule for dynamic networks: Our dynamic networks follow the time series model and are given as a sequence of networks over discrete time steps, where each network is obtained from the previous one in the sequence by adding and/or deleting some nodes and/or edges.(iii) Critical component definition: Individual edges are elementary members of critical components.(iv) Network property selection: The network measure for this paper will be appro...

Detecting network anomalies using Forman–Ricci curvature and a case study for human brain networks

“…Informally, the higher-dimensional complex in the topological association is obtained by “gluing” nodes, edges, cycles and other sub-graphs of the given graph. There are many alternate ways such topological associations can be performed 12 – 14 , 41 , 42 . Our topological association is similar to that used in 12 and is described as follows.…”

“…There are many alternate ways such topological associations can be performed 12 – 14 , 41 , 42 . Our topological association is similar to that used in 12 and is described as follows. For , we topologically associate a q -simplex with a -clique , i.e., 0-simplexes, 1-simplexes and 2-simplexes are associated with nodes, edges and 3-cycles (triangles), respectively.…”

“…In this paper, we will use the two terms “graph” and “network” interchangeably . We provide a formal definition of the network anomaly detection problem following a mathematical framework similar to what is described in 12 . To identify critical components of a temporal network, one first needs to provide details for the following four specific items: (i) the network model under consideration, (ii) a definition of the elementary components of the network, and (iii) how the network changes over time, (iv) the property of the network that will be used to identify critical components.…”

“…In another direction and historically much before the discretizations of Ricci curvature were formulated, Gromov and others defined a notion of network curvature (the ”Gromov-hyperbolic” curvature) via geodesic triangles that captures properties of the set of exact and approximate geodesics of the entire network 26 , 32 . There is a large body of research works dealing with various aspects of this measure, e.g., see 12 , 33 – 37 . The Gromov-hyperbolic curvature is a global measure in the sense that it assigns a scalar value to the entire graph, and therefore is not directly suitable for comparing components of two graphs.…”