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Graph data represents relationships, connections, or affinities. Normal relationships produce repeated, and so common, substructures in graph data. We present techniques for discovering anomalous substructures in graphs, for example small cliques, nodes with unusual neighborhoods, or small unusual subgraphs, using extensions of spectral graph techniques commonly used for clustering. Although not all anomalous structure represents terrorist or criminal activity, it is plausible that all terrorist or criminal activity creates anomalous substructure in graph data. Using our techniques, unusual regions of a graph can be selected for deeper analysis.