When working with network datasets, the theoretical framework of detection theory for Euclidean vector spaces no longer applies. Nevertheless, it is desirable to determine the detectability of small, anomalous graphs embedded into background networks with known statistical properties. Casting the problem of subgraph detection in a signal processing context, this article provides a framework and empirical results that elucidate a detection theory" for graph-valued data. Its focus is the detection of anomalies in unweighted, undirected graphs through L1 properties of the eigenvectors of the graph's so-called modularity matrix. This metric is observed to have relatively low variance for certain categories of randomly-generated graphs, and to reveal the presence of an anomalous subgraph with reasonable reliability when the anomaly is not well-correlated with stronger portions of the background graph. An analysis of subgraphs in real network datasets confirms the efficacy of this approach."

We propose a generalized framework for block-structured nonconvex optimization, which can be applied to structured subgraph detection in interdependent networks, such as multi-layer networks, temporal networks, networks of networks, and many others. Specifically, we design an effective, efficient, and parallelizable projection algorithm, namely Graph Block-structured Gradient Projection (GBGP), to optimize a general non-linear function subject to graph-structured constraints. We prove that our algorithm: 1) runs in nearly-linear time on the network size; 2) enjoys a theoretical approximation guarantee. Moreover, we demonstrate how our framework can be applied to two very practical applications and conduct comprehensive experiments to show the effectiveness and efficiency of our proposed algorithm.

When working with network datasets, the theoretical framework of detection theory for Euclidean vector spaces no longer applies. Nevertheless, it is desirable to determine the detectability of small, anomalous graphs embedded into background networks with known statistical properties. Casting the problem of subgraph detection in a signal processing context, this article provides a framework and empirical results that elucidate a detection theory" for graph-valued data. Its focus is the detection of anomalies in unweighted, undirected graphs through L1 properties of the eigenvectors of the graph's so-called modularity matrix. This metric is observed to have relatively low variance for certain categories of randomly-generated graphs, and to reveal the presence of an anomalous subgraph with reasonable reliability when the anomaly is not well-correlated with stronger portions of the background graph. An analysis of subgraphs in real network datasets confirms the efficacy of this approach."

Non-parametric graph scan (NPGS) statistics are used to detect anomalous connected subgraphs on graphs, and have a wide variety of applications, such as disease outbreak detection, road traffic congestion detection, and event detection in social media. In contrast to traditional parametric scan statistics (e.g., the Kulldorff statistic), NPGS statistics are free of distributional assumptions and can be applied to heterogeneous graph data. In this paper, we make a number of contributions to the computational study of NPGS statistics. First, we present a novel reformulation of the problem as a sequence of Budget Price-Collecting Steiner Tree (B-PCST) sub-problems. Second, we show that this reformulated problem is NP-hard for a large class of nonparametric statistic functions. Third, we further develop efficient exact and approximate algorithms for a special category of graphs in which the anomalous subgraphs can be reformulated in a fixed tree topology. Finally, using extensive experiments we demonstrate the performance of our proposed algorithms in two real-world application domains (water pollution detection in water sensor networks and spatial event detection in social media networks) and contrast against state-of-the-art connected subgraph detection methods.

When working with network datasets, the theoretical framework of detection theory for Euclidean vector spaces no longer applies. Nevertheless, it is desirable to determine the detectability of small, anomalous graphs embedded into background networks with known statistical properties. Casting the problem of subgraph detection in a signal processing context, this article provides a framework and empirical results that elucidate a detection theory" for graph-valued data. Its focus is the detection of anomalies in unweighted, undirected graphs through L1 properties of the eigenvectors of the graph’s so-called modularity matrix. This metric is observed to have relatively low variance for certain categories of randomly-generated graphs, and to reveal the presence of an anomalous subgraph with reasonable reliability when the anomaly is not well-correlated with stronger portions of the background graph. An analysis of subgraphs in real network datasets confirms the efficacy of this approach."