Detecting Anomalous Activity on Networks with the Graph Fourier Scan Statistic

We consider the problem of deciding, based on a single noisy measurement at each vertex of a given graph, whether the underlying unknown signal is constant over the graph or there exists a cluster of vertices with anomalous activation. This problem is relevant to several applications such as surveillance, disease outbreak detection, biomedical imaging, environmental monitoring, etc. Since the activations in these problems often tend to be localized to small groups of vertices in the graphs, we model such activity by a class of signals that are supported over a (possibly disconnected) cluster with low cut size relative to its size. We analyze the corresponding generalized likelihood ratio (GLR) statistics and relate it to the problem of finding a sparsest cut in the graph. We develop a tractable relaxation of the GLR statistic based on the combinatorial Laplacian of the graph, which we call the graph Fourier scan statistic, and analyze its properties. We show how its performance as a testing procedure depends directly on the spectrum of the graph, and use this result to explicitly derive its asymptotic properties on a few significant graph topologies. Finally, we demonstrate theoretically and with simulations that the graph Fourier scan statistic can outperform naive testing procedures based on global averaging and vertex-wise thresholding. We also demonstrate the usefulness of the GFSS by analyzing groundwater Arsenic concentrations from a U.S. Geological Survey dataset.