Personalized PageRank Vectors are widely used as fundamental graph-learning tools for detecting anomalous spammers, learning graph embeddings, and training graph neural networks. The well-known local FwdPush algorithm[5] approximates PPVs and has a sublinear rate of O(1 over αε). A recent study [51] found that when high precision is required, FwdPush is similar to the power iteration method, and its run time is pessimistically bounded by O(m over α log 1 over ε). This paper looks closely at calculating PPVs for both directed and undirected graphs. By leveraging the linear invariant property, we show that FwdPush is a variant of Gauss-Seidel and propose a Successive Over-Relaxation based method, FwdPushSOR to speed it up by slightly modifying FwdPush. Additionally, we prove FwdPush has local linear convergence rate O(vol (S) over α log 1 over ε) enjoying advantages of two existing bounds. We also design a new local heuristic push method that reduces the number of operations by 10-50 percent compared to FwdPush. For undirected graphs, we propose two momentum-based acceleration methods that can be expressed as one-line updates and speed up non-acceleration methods by O (1 / √ α). Our experiments on six real-world graph datasets confirm the efficiency of FwdPushSOR and the acceleration methods for directed and undirected graphs, respectively.

Tracking a targeted subset of nodes in an evolving graph is important for many real-world applications. Existing methods typically focus on identifying anomalous edges or finding anomaly graph snapshots in a stream way. However, edge-oriented methods cannot quantify how individual nodes change over time while others need to maintain representations of the whole graph all the time, thus computationally inefficient.

Dynamic graph representation learning is a task to learn node embeddings over dynamic networks, and has many important applications, including knowledge graphs, citation networks to social networks. Graphs of this type are usually large-scale but only a small subset of vertices are related in downstream tasks. Current methods are too expensive to this setting as the complexity is at best linear-dependent on both the number of nodes and edges.

In this paper, we aim to develop stochastic hard thresholding algorithms for the important problem of AUC maximization in imbalanced classification. The main challenge is the pairwise loss involved in AUC maximization. We overcome this obstacle by reformulating the U-statistics objective function as an empirical risk minimization (ERM), from which a stochastic hard thresholding algorithm (SHT-AUC) is developed. To our best knowledge, this is the first attempt to provide stochastic hard thresholding algorithms for AUC maximization with a per-iteration cost O(bd) where d and b are the dimension of the data and the minibatch size, respectively. We show that the proposed algorithm enjoys the linear convergence rate up to a tolerance error. In particular, we show, if the data is generated from the Gaussian distribution, then its convergence becomes slower as the data gets more imbalanced. We conduct extensive experiments to show the efficiency and effectiveness of the proposed algorithms.

The Area Under the ROC Curve (AUC) is a widely used performance measure for imbalanced classification arising from many application domains where high-dimensional sparse data is abundant. In such cases, each d dimensional sample has only k non-zero features with k ≪ d, and data arrives sequentially in a streaming form. Current online AUC optimization algorithms have high per-iteration cost O(d) and usually produce non-sparse solutions in general, and hence are not suitable for handling the data challenge mentioned above. In this paper, we aim to directly optimize the AUC score for high-dimensional sparse datasets under online learning setting and propose a new algorithm, FTRL-AUC. Our proposed algorithm can process data in an online fashion with a much cheaper per-iteration cost O(k), making it amenable for high-dimensional sparse streaming data analysis. Our new algorithmic design critically depends on a novel reformulation of the U-statistics AUC objective function as the empirical saddle point reformulation, and the innovative introduction of the “lazy update” rule so that the per-iteration complexity is dramatically reduced from O(d) to O(k). Furthermore, FTRL-AUC can inherently capture sparsity more effectively by applying a generalized Follow-The-Regularized-Leader (FTRL) framework. Experiments on real-world datasets demonstrate that FTRL-AUC significantly improves both run time and model sparsity while achieving competitive AUC scores compared with the state-of-the-art methods. Comparison with the online learning method for logistic loss demonstrates that FTRL-AUC achieves higher AUC scores especially when datasets are imbalanced. Experiments on real-world datasets demonstrate that FTRL-AUC significantly improves both run time and model sparsity while achieving competitive AUC scores compared with the state-of-the-art methods. Comparison with the online learning method for logistic loss demonstrates that FTRL-AUC achieves higher AUC scores especially when datasets are imbalanced.

The motives and means of explicit state censorship have been well studied, both quantitatively and qualitatively. Self-censorship by media outlets, however, has not received nearly as much attention, mostly because it is difficult to systematically detect. We develop a novel approach to identify news media self-censorship by using social media as a sensor. We develop a hypothesis testing framework to identify and evaluate censored clusters of keywords and a near-linear-time algorithm (called GraphDPD) to identify the highest scoring clusters as indicators of censorship. We evaluate the accuracy of our framework, versus other state-of-the-art algorithms, using both semi-synthetic and real-world data from Mexico and Venezuela during Year 2014. These tests demonstrate the capacity of our framework to identify self-censorship, and provide an indicator of broader media freedom. The results of this study lay the foundation for detection, study, and policy-response to self-censorship.

Online learning algorithms update models via one sample per iteration, thus efficient to process large-scale datasets and useful to detect malicious events for social benefits, such as disease outbreak and traffic congestion on the fly. However, existing algorithms for graph-structured models focused on the offline setting and the least square loss, incapable for online setting, while methods designed for online setting cannot be directly applied to the problem of complex (usually non-convex) graph-structured sparsity model. To address these limitations, in this paper we propose a new algorithm for graph-structured sparsity constraint problems under online setting, which we call GraphDA. The key part in GraphDA is to project both averaging gradient (in dual space) and primal variables (in primal space) onto lower dimensional subspaces, thus capturing the graph-structured sparsity effectively. Furthermore, the objective functions assumed here are generally convex so as to handle different losses for online learning settings. To the best of our knowledge, GraphDA is the first online learning algorithm for graph-structure constrained optimization problems. To validate our method, we conduct extensive experiments on both benchmark graph and real-world graph datasets. Our experiment results show that, compared to other baseline methods, GraphDA not only improves classification performance, but also successfully captures graph-structured features more effectively, hence stronger interpretability.

The area of anomaly detection has recently been expanded in the graph-based data. Anomalous vertices are often exhibited as a connected subgraph. Few works, however, have focused on connected anomalous subgraph detection because of the challenge of optimizing graph functionals under connectivity constraints. We employ Non-Parametric Graph Scan (NPGS) statistics for detecting anomalies within graph-based data. Based on the NPGS statistics, we proposed an efficient approximate approach to the connected anomalous subgraph detection problem that provides provable guarantees on performance and quality. In particular, we first decompose the problem into a sequence of subproblems, each of which can be reduced to a Budget Price-Collecting Steiner Tree (BPCST) problem, and then 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. Our method has a wide variety of applications, such as disease outbreak detection, road traffic congestion detection, and event detection in social media, because the NPGS statistics is free of distribution assumptions and can be applied to heterogeneous graph data.

Detection of interesting (e.g., coherent or anomalous) clusters has been studied extensively on plain or univariate networks, with various applications. Recently, algorithms have been extended to networks with multiple attributes for each node in the real-world. In a multi-attributed network, often, a cluster of nodes is only interesting for a subset (subspace) of attributes, andthis type of clusters is called subspace clusters. However, in the current literature, few methods are capable of detecting subspace clusters, which involves concurrent feature selection and network cluster detection. These relevant methods are mostly heuristic-driven and customized for specific application scenarios. In this work, we present a generic and theoretical framework for detection of interesting subspace clusters in large multi-attributed networks. Specifically, we propose a subspace graph-structured matching pursuit algorithm, namely, SG-Pursuit, to address a broad class of such problems for different scorefunctions (e.g., coherence or anomalous functions) and topology constraints (e.g., connected subgraphs and dense subgraphs). We prove that our algorithm 1) runs in nearly-linear time on the network size and the total number of attributes and 2) enjoys rigorous guarantees (geometrical convergence rate and tight error bound) analogous to those of the state-of-the-art algorithms for sparse feature selection problems and subgraph detection problems. As a case study, we specialize SG-Pursuit to optimizea number of well-known score functions for two typical tasks, including detection of coherent dense and anomalous connected subspace clusters in real-world networks. Empirical evidence demonstrates that our proposed generic algorithm SG-Pursuit is superior over state-of-the-art methods that are designed specifically for these two tasks.

Spatial event detection is an important and challenging problem. Unlike traditional event detection that focuses on the timing of global urgent event, the task of spatial event detection is to detect the spatial regions (e.g. clusters of neighboring cities) where urgent events occur. In this paper, we focus on the problem of spatial event detection using textual information in social media. We observe that, when a spatial event occurs, the topics relevant to the event are often discussed more coherently in cities near the event location than those far away. In order to capture this pattern, we propose a new method called Graph Topic Scan Statistic (Graph-TSS) that corresponds to a generalized log-likelihood ratio test based on topic modeling. We first demonstrate that the detection of spatial event regions under Graph-TSS is NP-hard due to a reduction from classical node-weighted prize-collecting Steiner tree problem (NW-PCST). We then design an efficient algorithm that approximately maximizes the graph topic scan statistic over spatial regions of arbitrary form. As a case study, we consider three applications using Twitter data, including Argentina civil unrest event detection, Chile earthquake detection, and United States influenza disease outbreak detection. Empirical evidence demonstrates that the proposed Graph-TSS performs superior over state-of-the-art methods on both running time and accuracy.

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.