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Near-optimal Anomaly Detection in Graphs using Lovasz Extended Scan Statistic
James L Sharpnack · Akshay Krishnamurthy · Aarti Singh

Sun Dec 08 02:00 PM -- 06:00 PM (PST) @ Harrah's Special Events Center, 2nd Floor
The detection of anomalous activity in graphs is a statistical problem that arises in many applications, such as network surveillance, disease outbreak detection, and activity monitoring in social networks. Beyond its wide applicability, graph structured anomaly detection serves as a case study in the difficulty of balancing computational complexity with statistical power. In this work, we develop from first principles the generalized likelihood ratio test for determining if there is a well connected region of activation over the vertices in the graph in Gaussian noise. Because this test is computationally infeasible, we provide a relaxation, called the Lov\'asz extended scan statistic (LESS) that uses submodularity to approximate the intractable generalized likelihood ratio. We demonstrate a connection between LESS and maximum a-posteriori inference in Markov random fields, which provides us with a poly-time algorithm for LESS. Using electrical network theory, we are able to control type 1 error for LESS and prove conditions under which LESS is risk consistent. Finally, we consider specific graph models, the torus, $k$-nearest neighbor graphs, and $\epsilon$-random graphs. We show that on these graphs our results provide near-optimal performance by matching our results to known lower bounds.

Author Information

James L Sharpnack (CMU)
Akshay Krishnamurthy (Microsoft Research)
Aarti Singh (CMU)

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