Node Anomaly Detection (NAD) has gained significant attention in the deep learning community due to its diverse applications in real-world scenarios. Existing NAD methods primarily embed graphs within a single Euclidean space, while overlooking the potential of non-Euclidean spaces. Besides, to address the prevalent issue of limited supervision in real NAD tasks, previous methods tend to leverage synthetic data to collect auxiliary information, which is not an effective solution as shown in our experiments. To overcome these challenges, we introduce a novel SpaceGNN model designed for NAD tasks with extremely limited labels. Specifically, we provide deeper insights into a task-relevant framework by empirically analyzing the benefits of different spaces for node representations, based on which, we design a Learnable Space Projection function that effectively encodes nodes into suitable spaces. Besides, we introduce the concept of weighted homogeneity, which we empirically and theoretically validate as an effective coefficient during information propagation. This concept inspires the design of the Distance Aware Propagation module. Furthermore, we propose the Multiple Space Ensemble module, which extracts comprehensive information for NAD under conditions of extremely limited supervision. Our findings indicate that this module is more beneficial than data augmentation techniques for NAD. Extensive experiments conducted on 9 real datasets confirm the superiority of SpaceGNN, which outperforms the best rival by an average of 8.55% in AUC and 4.31% in F1 scores. Our code is available at https://github.com/xydong127/SpaceGNN.

Graph Anomaly Detection (GAD) in heterogeneous networks presents unique challenges due to node and edge heterogeneity. Existing Graph Neural Network (GNN) methods primarily focus on homogeneous GAD and thus fail to address three key issues: (C1) Capturing abnormal signal and rich semantics across diverse meta-paths; (C2) Retaining high-frequency content in HIN dimension alignment; and (C3) Learning effectively from difficult anomaly samples with class imbalance. To overcome these, we propose ChiGAD, a spectral GNN framework based on a novel Chi-Square filter, inspired by the wavelet effectiveness in diverse domains. Specifically, ChiGAD consists of: (1) Multi-Graph Chi-Square Filter, which captures anomalous information via applying dedicated Chi-Square filters to each meta-path graph; (2) Interactive Meta-Graph Convolution, which aligns features while preserving high-frequency information and incorporates heterogeneous messages by a unified Chi-Square Filter; and (3) Contribution-Informed Cross-Entropy Loss, which prioritizes difficult anomalies to address class imbalance. Extensive experiments on public and industrial datasets show that ChiGAD outperforms state-of-the-art models on multiple metrics. Additionally, its homogeneous variant, ChiGNN, excels on seven GAD datasets, validating the effectiveness of Chi-Square filters. Our code is available at https://github.com/HsipingLi/ChiGAD.

The smoothing issue in graph learning leads to indistinguishable node representations, posing significant challenges for graph-related tasks. However, our experiments reveal that this problem can uncover underlying properties of node anomaly detection (NAD) that previous research has missed. We introduce Individual Smoothing Patterns (ISP) and Neighborhood Smoothing Patterns (NSP), which indicate that the representations of anomalous nodes are harder to smooth than those of normal ones. In addition, we explore the theoretical implications of these patterns, demonstrating the potential benefits of ISP and NSP for NAD tasks. Motivated by these findings, we propose SmoothGNN, a novel unsupervised NAD framework. First, we design a learning component to explicitly capture ISP for detecting node anomalies. Second, we design a spectral graph neural network to implicitly learn ISP to enhance detection. Third, we design an effective coefficient based on our findings that NSP can serve as coefficients for node representations, aiding in the identification of anomalous nodes. Furthermore, we devise a novel anomaly measure to calculate loss functions and anomalous scores for nodes, reflecting the properties of NAD using ISP and NSP. Extensive experiments on 9 real datasets show that SmoothGNN outperforms the best rival by an average of 14.66% in AUC and 7.28% in Average Precision, with 75x running time speedup, validating the effectiveness and efficiency of our framework. Our code is available at https://github.com/xydong127/SmoothGNN.

The spread of rumors on social media, particularly during significant events like the US elections and the COVID-19 pandemic, poses a serious threat to social stability and public health. Current rumor detection methods primarily rely on propagation graphs to improve the model performance. However, the effectiveness of these methods is often compromised by noisy and irrelevant structures in the propagation process. To tackle this issue, techniques such as weight adjustment and data augmentation have been proposed. However, they depend heavily on rich original propagation structures, limiting their effectiveness in handling rumors that lack sufficient propagation information, especially in the early stages of dissemination. In this work, we introduce the Key Propagation Graph Generator (KPG), a novel reinforcement learning-based framework, that generates contextually coherent and informative propagation patterns for events with insufficient topology information and identifies significant substructures in events with redundant and noisy propagation structures. KPG comprises two key components: the Candidate Response Generator (CRG) and the Ending Node Selector (ENS). CRG learns latent variable distributions from refined propagation patterns to eliminate noise and generate new candidates for ENS, while ENS identifies the most influential substructures in propagation graphs and provides training data for CRG. Furthermore, we develop an end-to-end framework that utilizes rewards derived from a pre-trained graph neural network to guide the training process. The resulting key propagation graphs are then employed in downstream rumor detection tasks. Extensive experiments conducted on four datasets demonstrate that KPG outperforms current state-of-the-art methods.

Graph-level anomaly detection has gained significant attention as it finds applications in various domains, such as cancer diagnosis and enzyme prediction. However, existing methods fail to capture the spectral properties of graph anomalies, resulting in unexplainable framework design and unsatisfying performance. In this paper, we re-investigate the spectral differences between anomalous and normal graphs. Our main observation shows a significant disparity in the accumulated spectral energy between these two classes. Moreover, we prove that the accumulated spectral energy of the graph signal can be represented by its Rayleigh Quotient, indicating that the Rayleigh Quotient is a driving factor behind the anomalous properties of graphs. Motivated by this, we propose Rayleigh Quotient Graph Neural Network (RQGNN), the first spectral GNN that explores the inherent spectral features of anomalous graphs for graph-level anomaly detection. Specifically, we introduce a novel framework with two components: the Rayleigh Quotient learning component (RQL) and Chebyshev Wavelet GNN with RQ-pooling (CWGNN-RQ). RQL explicitly captures the Rayleigh Quotient of graphs and CWGNN-RQ implicitly explores the spectral space of graphs. Extensive experiments on 10 real-world datasets show that RQGNN outperforms the best rival by 6.74% in Macro-F1 score and 1.44% in AUC, demonstrating the effectiveness of our framework. Our code is available at https://github.com/xydong127/RQGNN.

Node embedding learns low-dimensional vectors for nodes in the graph. Recent state-of-the-art embedding approaches take Personalized PageRank (PPR) as the proximity measure and factorize the PPR matrix or its adaptation to generate embeddings. However, little previous work analyzes what information is encoded by these approaches, and how the information correlates with their superb performance in downstream tasks. In this work, we first show that state-of-the-art embedding approaches that factorize a PPR-related matrix can be unified into a closed-form framework. Then, we study whether the embeddings generated by this strategy can be inverted to better recover the graph topology information than random-walk based embeddings. To achieve this, we propose two methods for recovering graph topology via PPR-based embeddings, including the analytical method and the optimization method. Extensive experimental results demonstrate that the embeddings generated by factorizing a PPR-related matrix maintain more topological information, such as common edges and community structures, than that generated by random walks, paving a new way to systematically comprehend why PPR-based node embedding approaches outperform random walk-based alternatives in various downstream tasks. To the best of our knowledge, this is the first work that focuses on the interpretability of PPR-based node embedding approaches.

Given a graph$\mathcal {G}$with a set of attributes, theattributed network embedding (ANE)aims to learn low-dimensional representations of nodes that preserve both graph topology and node attribute proximity. ANE is shown to be more effective than plain network embedding methods (using only graph topology) on many graph mining tasks. However, existing ANE solutions still provide inferior performance on tasks like node classification and link prediction, as will be shown in our experiments. The key issue is that when combining graph topology and attribute information, most existing solutions take attributes with equal importance, while in real scenarios, different attribute exerts distinct influence over the network due to the heterogeneous nature among attributes. Motivated by this, we presentLATAM, a learning-based framework for ANE via trainable proximity matrix factorization. To capture the node-attribute relationships, we first construct the attribute-augmented graph by adding attribute nodes (resp. edges) to the original graph. Then, we define the attribute-augmented random walk and proximity on the attribute-augmented graph, where the weights of different attributes can be learned automatically by our designed loss functions so that more indicative attributes tend to have higher weights, imposing a higher impact on the node connectivity. To achieve this, we incorporate a differentiable SVD to back-propagate gradients of attribute weights in an end-to-end process. To scale our LATAM to large graphs, we further propose sampling techniques to learn attribute weights and an efficient attribute-augmented push algorithm to compute the proximity matrix. Extensive experiments on 8 public attributed networks against 11 existing methods show the effectiveness of our LATAM.

Abstract Active learning for graph neural networks (GNNs) aims to select B nodes to label for the best possible GNN performance. Carefully selected labeled nodes can help improve GNN performance and hence motivates a line of research works. Unfortunately, existing methods still provide inferior GNN performance or cannot scale to large networks.Motivated by these limitations, in this paper, we present FICOM, an effective and scalable GNN active learning framework. Firstly, we formulate the node selection as an optimization problem where we consider the importance of a node from (i) the importance of a node during the feature propagation with a connection to the personalized PageRank (PPR), and (ii) the diversity of a node brings in the embedding space generated by feature propagation. We show that the defined problem is submodular, and a greedy solution can provide a $$(1-1/e)$$ ( 1 - 1 / e ) -approximate solution.However, a standard greedy solution requires getting the node with the maximum marginal gain of the objective score in each iteration, which incurs a prohibitive running cost and cannot scale to large datasets. As our main contribution, we present FICOM, an efficient and scalable solution that provides $$(1-1/e)$$ ( 1 - 1 / e ) -approximation guarantee and scales to graphs with millions of nodes on a single machine. The main idea is that we adaptively maintain the lower- and upper-bound of the marginal gain for each node v. In each iteration, we can first derive a small subset of candidate nodes and then compute the exact score for this subset of candidate nodes so that we can find the node with the maximum marginal gain efficiently. Extensive experiments on six benchmark datasets using four GNNs, including GCN, SGC, APPNP, and GCNII, show that our FICOM consistently outperforms existing active learning approaches on semi-supervised node classification tasks using different GNNs. Moreover, our solution can finish within 5 h on a million-node graph.

In online social networks, users with similar interests tend to come together, forming social communities. Nowadays, user-defined communities become a prominent part of online social platforms as people who have joined such communities tend to be more active in social networks. Therefore, recommending explicit communities to users provides great potential to advance online services.

Subset embedding is the task to learn low-dimensional representations for a subset of nodes according to the graph topology. It has applications when we focus on a subset of users, e.g., young adults, and aim to make better recommendations for these target users. In real-world scenarios, graphs are dynamically changing. Thus, it is more desirable to dynamically maintain the subset embeddings to reflect graph updates. The state-of-the-art methods, e.g., DynPPE, still adopt a hashing-based method, while hashing-based solutions are shown to be less effective than matrix factorization (MF)-based methods in existing studies. At the same time, MF-based methods in the literature are too expensive to update the embedding when the graph changes, making them inapplicable on dynamic graphs. Motivated by this, we present Tree-SVD, an efficient and effective MF-based method for dynamic subset embedding. If we simply maintain the whole proximity matrix, then we need to re-do the MF, e.g., truncated Singular Value Decomposition (SVD), on the whole matrix after graph updates, which is prohibitive. To tackle this issue, our main idea is to do hierarchical SVD (HSVD) on the proximity matrix of the given subset, which vertically divides the proximity matrix into multiple sub-matrices, and then repeatedly do SVD on sub-matrices and merge the intermediate results to obtain the final embedding. We first present Tree-SVD, which combines a sparse randomized SVD with an HSVD. Our theoretical analysis shows that our Tree-SVD gains the efficiency of sparse randomized SVD and the flexibility of the HSVD with theoretical guarantees. To further reduce update costs, we present a lazy-update strategy. In this strategy, we only update sub-matrices that changes remarkably in terms of the Frobenius norm. We present theoretical analysis to show the guarantees with our lazy-update strategy. Extensive experiments show the efficiency and effectiveness of Tree-SVD on node classification and link prediction tasks.

Node embedding learns a low-dimensional representation for each node in the graph. Recent progress on node embedding shows that proximity matrix factorization methods gain superb performance and scale to large graphs with millions of nodes. Existing approaches first define a proximity matrix and then learn the embeddings that fit the proximity by matrix factorization. Most existing matrix factorization methods adopt the same proximity for different tasks, while it is observed that different tasks and datasets may require different proximity, limiting their representation power.