Structured sparse optimization is an important and challenging problem for analyzing high-dimensional data in a variety of applications such as bioinformatics, medical imaging, social networks, and astronomy. Although a number of structured sparsity models have been explored, such as trees, groups, clusters, and paths, connected subgraphs have been rarely explored in the current literature. One of the main technical challenges is that there is no structured sparsity-inducing norm that can directly model the space of connected subgraphs, and there is no exact implementation of a projection oracle for connected subgraphs due to its NP-hardness. In this paper, we explore efficient approximate projection oracles for connected subgraphs, and propose two new efficient algorithms, namely, Graph-IHT and Graph-GHTP, to optimize a generic nonlinear objective function subject to connectivity constraint on the support of the variables. Our proposed algorithms enjoy strong guarantees analogous to several current methods for sparsity-constrained optimization, such as Projected Gradient Descent (PGD), Approximate Model Iterative Hard Thresholding (AM-IHT), and Gradient Hard Thresholding Pursuit (GHTP) with respect to convergence rate and approximation accuracy. We apply our proposed algorithms to optimize several well-known graph scan statistics in several applications of connected subgraph detection as a case study, and the experimental results demonstrate that our proposed algorithms outperform state-of-the-art methods.
Event detection in social media is an important but challenging problem. Most existing approaches are based on burst detection, topic modeling, or clustering techniques, which cannot naturally model the implicit heterogeneous network structure in social media. As a result, only limited information, such as terms and geographic locations, can be used. This paper presents Non-Parametric Heterogeneous Graph Scan (NPHGS), a new approach that considers the entire heterogeneous network for event detection: we first model the network as a "sensor" network, in which each node senses its "neighborhood environment" and reports an empirical p-value measuring its current level of anomalousness for each time interval (e.g., hour or day). Then, we efficiently maximize a nonparametric scan statistic over connected subgraphs to identify the most anomalous network clusters. Finally, the event represented by each cluster is summarized with information such as type of event, geographical locations, time, and participants. As a case study, we consider two applications using Twitter data, civil unrest event detection and rare disease outbreak detection, and present empirical evaluations illustrating the effectiveness and efficiency of our proposed approach. We present GraphScan, a novel method for detecting arbitrarily shaped connected clusters in graph or network data. Given a graph structure, data observed at each node, and a score function defining the anomalousness of a set of nodes, GraphScan can efficiently and exactly identify the most anomalous (highest-scoring) connected subgraph. Kulldorff’s spatial scan, which searches over circles consisting of a center location and its k − 1 nearest neighbors, has been extended to include connectivity constraints by FlexScan. However, FlexScan performs an exhaustive search over connected subsets and is computationally infeasible for k > 30. Alternatively, the upper level set (ULS) scan scales well to large graphs but is not guaranteed to find the highest-scoring subset. We demonstrate that GraphScan is able to scale to graphs an order of magnitude larger than FlexScan, while guaranteeing that the highest-scoring subgraph will be identified. We evaluate GraphScan, Kulldorff’s spatial scan (searching over circles)... We explore scalable and accurate dynamic pattern detection methods in graph-based data sets. We apply our proposed Dynamic Subset Scan method to the task of detecting, tracking, and source-tracing contaminant plumes spreading through a water distribution system equipped with noisy, binary sensors. While static patterns affect the same subset of data over a period of time, dynamic patterns may affect different subsets of the data at each time step. These dynamic patterns require a new approach to define and optimize penalized likelihood ratio statistics in the subset scan framework, as well as new computational techniques that scale to large, real-world networks. To address the first concern, we develop new subset scan methods that allow the detected subset of nodes to change over time, while incorporating temporal consistency constraints to reward patterns that do not dramatically change between adjacent time steps. Second, our Additive Graph Scan algorithm allows our novel scan statistic to process small graphs (500 nodes) in 4.1 seconds on average while maintaining an approximation ratio over 99 compared to an exact optimization method, and to scale to large graphs with over 12,000 nodes in 30 minutes on average. Evaluation results across multiple detection, tracking, and source-tracing tasks demonstrate substantial performance gains achieved by the Dynamic Subset Scan approach. The fused lasso was proposed recently to enable recovery of high-dimensional patterns which are piece-wise constant on a graph, by penalizing the ‘1-norm of dierences of measurements at vertices that share an edge. While there have been some attempts at coming up with ecient algorithms for solving the fused lasso optimization, a theoretical analysis of its performance is mostly lacking except for the simple linear graph topology. In this paper, we investigate sparsistency of fused lasso for general graph structures, i.e. its ability to correctly recover the exact support of piece-wise constant graphstructured patterns asymptotically (for largescale graphs). To emphasize this distinction over previous work, we will refer to it as Edge Lasso. We focus on the (structured) normal means setting, and our results provide necessary and sucient conditions on the graph properties as well as the signal-to-noise ratio needed to ensure sparsistency. We examplify our results using simple graph-structured patterns, and demonstrate that in some cases fused lasso is sparsistent at very weak signal-to-noise ratios (scaling as p (logn)=jAj, where n is the number of vertices in the graph and A is the smallest set of vertices with constant activation). In other cases, it performs no better than thresholding the dierence We consider the change-point detection problem of deciding, based on noisy measurements, whether an unknown signal over a given graph is constant or is instead piecewise constant over two connected induced subgraphs of relatively low cut size. We analyze the corresponding generalized likelihood ratio (GLR) statistics and relate it to the problem of finding a sparsest cut in a graph. We develop a tractable relaxation of the GLR statistic based on the combinatorial Laplacian of the graph, which we call the spectral 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 few significant graph topologies. Finally, we demonstrate both theoretically and by simulations that the spectral scan statistic can outperform naive testing procedures based on edge thresholding and @math testing. With the fast growth of smart devices and social networks, a lot of computing systems collect data that record different types of activities. An important computational challenge is to analyze these data, extract patterns, and understand activity trends. We consider the problem of mining activity networks to identify interesting events, such as a big concert or a demonstration in a city, or a trending keyword in a user community in a social network. We define an event to be a subset of nodes in the network that are close to each other and have high activity levels. We formalize the problem of event detection using two graph-theoretic formulations. The first one captures the compactness of an event using the sum of distances among all pairs of the event nodes. We show that this formulation can be mapped to the maxcut problem, and thus, it can be solved by applying standard semidefinite programming techniques. The second formulation captures compactness using a minimum-distance tree. This formulation leads to the prize-collecting Steiner-tree problem, which we solve by adapting existing approximation algorithms. For the two problems we introduce, we also propose efficient and effective greedy approaches and we prove performance guarantees for one of them. We experiment with the proposed algorithms on real datasets from a public bicycling system and a geolocation-enabled social network dataset collected from twitter. The results show that our methods are able to detect meaningful events.