Anomaly detection on graphs has recently attracted considerable attention due to its broad range of high-impact applications, including cybersecurity, financial transactions, and recommendation systems. Although many efforts have thus far been made, how to effectively handle the high inconsistency between users' behavior and labels, a fundamental issue in anomaly detection, has not yet received sufficient concern. Moreover, the inconsistency problem is hard to investigate and even deteriorates the performance of anomaly detectors. To this end, we propose a novel graph self-supervised learning framework, Capsule Graph Infomax (termed CapsGI), to overcome the inconsistency of anomaly detection. Inspired by the recent advances of capsules on images, we explore another possibility of reforming the node embedding by capsule ideas to represent the unique node's properties. Concretely, by disentangling heterogeneous factors underlying each node representation, we can establish node capsules such that their representation can reflect intrinsic node properties. To strengthen the connection among normal nodes, CapsGI further represents the part-whole contrastive learning between lower-level capsules (part) and higher-level capsules (whole) by explicitly considering the context graph relations. Extensive experiments on multiple real-world datasets demonstrate that our model significantly outperforms state-of-the-art models.

This paper studies the problem of efficiently computing a maximum independent set from a large graph, a fundamental problem in graph analysis. Due to the hardness results of computing an exact maximum independent set or an approximate maximum independent set with accuracy guarantee, the existing algorithms resort to heuristic techniques for approximately computing a maximum independent set with good performance in practice but no accuracy guarantee theoretically. Observing that the existing techniques have various limits, in this paper, we aim to develop efficient algorithms (with linear or near-linear time complexity) that can generate a high-quality (large-size) independent set from a graph in practice. In particular, firstly we develop a Reducing-Peeling framework which iteratively reduces the graph size by applying reduction rules on vertices with very low degrees (Reducing) and temporarily removing the vertex with the highest degree (Peeling) if the reduction rules cannot be applied. Secondly, based on our framework we design two baseline algorithms, BDOne and BDTwo, by utilizing the existing reduction rules for handling degree-one and degree-two vertices, respectively. Both algorithms can generate higher-quality (larger-size) independent sets than the existing algorithms. Thirdly, we propose a linear-time algorithm, LinearTime, and a near-linear time algorithm, NearLinear, by designing new reduction rules and developing techniques for efficiently and incrementally applying reduction rules. In practice, LinearTime takes similar time and space to BDOne but computes a higher quality independent set, similar in size to that of an independent set generated by BDTwo. Moreover, in practice NearLinear has a good chance to generate a maximum independent set and it often generates near-maximum independent sets. Fourthly, we extend our techniques to accelerate the existing iterated local search algorithms. Extensive empirical studies show that all our algorithms output much larger independent sets than the existing linear-time algorithms while having a similar running time, as well as achieve significant speedup against the existing iterated local search algorithms.

We study the problem of structural graph clustering, a fundamental problem in managing and analyzing graph data. Given an undirected unweighted graph, structural graph clustering is to assign vertices to clusters, and to identify the sets of hub vertices and outlier vertices as well, such that vertices in the same cluster are densely connected to each other while vertices in different clusters are loosely connected. In this paper, we develop a new two-step paradigm for scalable structural graph clustering based on our three observations. Then, we present a pSCAN approach, within the paradigm, aiming to reduce the number of structural similarity computations, and propose optimization techniques to speed up checking whether two vertices are structure-similar. pSCAN outputs exactly the same clusters as the existing approaches SCAN and SCAN++, and we prove that pSCAN is worst-case optimal. Moreover, we propose efficient techniques for updating the clusters when the input graph dynamically changes, and we also extend our techniques to other similarity measures, e.g., Jaccard similarity. Performance studies on large real and synthetic graphs demonstrate the efficiency of our new approach and our dynamic cluster maintenance techniques. Noticeably, for the twitter graph with 1 billion edges, our approach takes 25 minutes while the state-of-the-art approach cannot finish even after 24 hours.

In this paper, we study the problem of structural graph clustering, a fundamental problem in managing and analyzing graph data. Given a large graph G = (V, E), structural graph clustering is to assign vertices in V to clusters and to identify the sets of hub vertices and outlier vertices as well, such that vertices in the same cluster are densely connected to each other while vertices in different clusters are loosely connected to each other. Firstly, we prove that the existing SCAN approach is worst-case optimal. Nevertheless, it is still not scalable to large graphs due to exhaustively computing structural similarity for every pair of adjacent vertices. Secondly, we make three observations about structural graph clustering, which present opportunities for further optimization. Based on these observations, in this paper we develop a new two-step paradigm for scalable structural graph clustering. Thirdly, following this paradigm, we present a new approach aiming to reduce the number of structural similarity computations. Moreover, we propose optimization techniques to speed up checking whether two vertices are structure-similar to each other. Finally, we conduct extensive performance studies on large real and synthetic graphs, which demonstrate that our new approach outperforms the state-of-the-art approaches by over one order of magnitude. Noticeably, for the twitter graph with 1 billion edges, our approach takes 25 minutes while the state-of-the-art approach cannot finish even after 24 hours.