Truncated Affinity Maximization: One-class Homophily Modeling for Graph Anomaly Detection

We reveal a one-class homophily phenomenon, which is one prevalent property we find empirically in real-world graph anomaly detection (GAD) datasets, i.e., normal nodes tend to have strong connection/affinity with each other, while the homophily in abnormal nodes is significantly weaker than normal nodes. However, this anomaly-discriminative property is ignored by existing GAD methods that are typically built using a conventional anomaly detection objective, such as data reconstruction. In this work, we explore this property to introduce a novel unsupervised anomaly scoring measure for GAD, local node affinity, that assigns a larger anomaly score to nodes that are less affiliated with their neighbors, with the affinity defined as similarity on node attributes/representations. We further propose Truncated Affinity Maximization (TAM) that learns tailored node representations for our anomaly measure by maximizing the local affinity of nodes to their neighbors. Optimizing on the original graph structure can be biased by nonhomophily edges (i.e., edges connecting normal and abnormal nodes). Thus, TAM is instead optimized on truncated graphs where non-homophily edges are removed iteratively to mitigate this bias. The learned representations result in significantly stronger local affinity for normal nodes than abnormal nodes. Extensive empirical results on 10 real-world GAD datasets show that TAM substantially outperforms seven competing models, achieving over 10% increase in AUROC/AUPRC compared to the best contenders on challenging datasets. Our code is available at https://github.com/mala-lab/TAM-master/.

Paper

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Peer review

Reviewer oFzr5/10 · confidence 3/52023-06-20

Summary

This paper proposed an unsupervised algorithm for detecting abnormal nodes in a graph. Based on the one-class homophily assumption and the overwhelming presence of normal nodes in a graph, this paper uses the truncated affinity maximization to enable stronger local affinity for normal nodes than abnormal ones. To eliminate the training bias brought by the non-homophily edges, Normal Structure-preserved Graph Truncation is proposed to remove non-homophily edges iteratively. The proposed method achieves competitive performance on datasets with synthetic and real anomalies.

Strengths

1. The paper is well-structured and easy to follow. 2. The proposed method TAM has strong motivation. The unsupervised setting is close to realistic scenario that normal nodes and anomalies exist in a graph and their labels are unknown. 3. Extensive experiments prove that TAM is a strong baseline for unsupervised anomaly node detection

Weaknesses

1. I suggest adding some supervised graph anomaly detection method on section 2, such as CARE-GNN [1], PC-GNN [2] and Fraudre [3]. Although the supervised setting is far from reality, these algorithms also proposed some ideas for eliminating the negative impact of heterophily edges on anomaly detection. 2. Is TAM wide-applicable on different GNNs? How does TAM perform on other GNN backbones other than GCN? 3. In table 1, why baselines and TAM perform so poor regarding AUPRC. It would be better to draw the roc curve and precision-recall curve. 4. It is difficult to understand Figure 2 (b), what are $c_1$, $c_2$, $c_k$? I suggest reorganizing this illustration. 5. I suggest placing the limitation in a separate paragraph or subsection. Besides the limitations discussed in Conclusion, are there any other shortcoming of TAM? [1] Enhancing graph neural network-based fraud detectors against camouflaged fraudsters [2] Pick and choose: a GNN-based imbalanced learning approach for fraud detection [3] Fraudre: Fraud detection dual-resistant to graph inconsistency and imbalance

Questions

Please refer to Weaknesses.

Rating

5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

3 good

Contribution

2 fair

Limitations

yes

Reviewer zqNH6/10 · confidence 4/52023-07-03

Summary

This paper studies the problem of graph anomaly detection and the authors proposed a novel method based on an identified property named one-class homophily. It is observed that normal nodes have strong connections with each other while abnormal nodes have weaker connections. Existing GAD methods overlook this property. The authors propose a novel unsupervised anomaly scoring measure, i.e., local node affinity, that considers the similarity of nodes to their neighbors. They introduce Truncated Affinity Maximization (TAM) to learn tailored node representations for this measure. TAM optimizes on truncated graphs to mitigate bias from non-homophily edges. Experimental results on several real-world graphs with both manually injected and real anomalies show that TAM outperforms several competing models, achieving over 10% improvement in AUROC/AUPRC on challenging datasets.

Strengths

+ This paper focuses on an interesting and important problem. Graph anomaly detection is of value in both research and practice, especially since it is beneficial for various applications. + The proposed method is technically sound. The performance in several data with both manually injected and real-world anomalies is promising. + The experimental results show that the proposed method achieves consistent performance improvements over the baselines, which demonstrates its superiority. Also, a comprehensive ablation study to show the effectiveness of different components.

Weaknesses

- There is no theoretical analysis to justify the rationale of the proposed method. - The proposed method has not been tested on large-scale graphs to show its efficiency. - The design of some components may need more investigation empirically (see details below).

Questions

- Theoretical analysis. Although the proposed method has shown its effectiveness using comprehensive experiments, i.e., comparing seven SOTA on real-world graphs with both manually injected and real anomalies, there is no theoretical analysis to justify the rationale of the proposed method. - Efficiency test. The largest graph used in the experiments contains ~4k nodes and ~175k edges, which are relatively not very large. To better demonstrate the efficiency of the method, larger graph benchmarks may be needed, e.g., OGB. - Component design and impact. Some components may need more investigation empirically, especially the graph truncation strategy as well as the impact of different K. In detail, (1) in section 4.2, the raw graph and (random) edge drop have been compared. A straightforward question is what is the performance of simple similarity-based methods, e.g., removing some less similar edges? (2) Since there are K truncated graphs and each one relies on the previous one, the number of edges will be less and less. Is it possible that for larger K, the graph will contain quite some isolated small subgraphs? If so, is there any negative impact on the performance? Some minor comments: - Figure 4(b) exhibits a variation in performance trends across different graphs. Particularly, on the Reddit and YelpChi datasets, the AUPRC decreases as the value of K increases. Is there an explanation for this contrasting behavior?

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

3 good

Presentation

3 good

Contribution

3 good

Limitations

No potential negative societal impact or the authors adequately addressed the limitations.

Reviewer uzaW7/10 · confidence 5/52023-07-07

Summary

In this paper, the authors introduced a novel unsupervised anomaly scoring measure (local node affinity) for GAD, and further proposed a Truncated Affinity Maximization (TAM) for GAD. TAM learns tailored node representations for our anomaly measure by maximizing the local affinity of nodes to their neighbors, and is optimized on truncated graphs where non-homophily edges are removed iteratively to mitigate this bias. The authors also conduct a series of experiments to evaluate the anomaly detection performance of TAM.

Strengths

1. This paper is well-motivated and easy to follow. 2. The idea of exploring the one-class homophily characteristic for designing a GAD method is interesting. 3. The experimental section provides comprehensive evaluations of the proposed method from different perspectives, and demonstrates the effectiveness of the proposed method.

Weaknesses

1. In the related work section, the authors should discuss the connections and differences between the proposed methods and other existing works, as well as the necessity for this work. 2. As the authors claimed, one-class homogeneity does not hold in all cases. Therefore algorithms designed on this basis would be relatively limited. 3. The authors propose to optimize TAM on truncated graphs to avoid the negative impact of non-homophily edges. How the author guarantees to obtain ideal truncated graphs with the removal of non-homophily edges iteratively. Besides, does it have some criteria to judge whether the truncation map obtained is ideal or not?

Questions

See weakness.

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

5: You are absolutely certain about your assessment. You are very familiar with the related work and checked the math/other details carefully.

Soundness

3 good

Presentation

3 good

Contribution

3 good

Limitations

See weakness.

Reviewer NSnj4/10 · confidence 4/52023-07-09

Summary

Graph anomaly detection aims to identify abnormal nodes in a given graph. The manuscript argues that the existing graph anomaly detection datasets have one-class homophily where the homophily of normal nodes is much stronger than abnormal nodes. To utilize the one-class homophily phenomenon in graph anomaly detection, a new similarity-based anomaly scoring measure, named local node affinity, is proposed. The proposed method, TAM, learns the parameters of multiple graph neural networks called LAMNets by maximizing the sum of local node affinity on the original graph. The iterative graph truncation technique (NSGT) removes the links between normal and abnormal nodes in each step, and the resulting graphs are fed into LAMNets. During the message passing, LAMNet uses the truncated adjacency matrix. The local node affinity-based anomaly score is measured on the original graph using node representations generated by LAMNets. Experimental results show that TAM achieves high AUROC and AUPRC values compared to seven graph anomaly detection methods on six benchmark datasets.

Strengths

1. NSGT introduces randomness by removing the non-homophily edges in a probabilistic approach. To utilize such randomness, TAM uses an ensemble strategy by performing NSGT multiple times and feeding the resulting graphs to multiple LAMNets. Table 2 in Appendix C shows that the ensemble strategy improves the performance of TAM. 2. The manuscript provides empirical justifications for the major claims. For instance, the homophily distribution in Figure 1(a) supports the one-class homophily phenomenon. Figure 3(a) and Figure 3(b) show that the Euclidean distance between the nodes connected by a non-homophily edge tends to be greater than that between the nodes connected by a homophily edge. 3. Ablation studies with various variants of TAM and hyperparameter sensitivity analyses show that each component of TAM is effective.

Weaknesses

1. It is not guaranteed that the one-class homophily property holds for all graph anomaly detection datasets. For instance, both normal and abnormal nodes in the YelpChi-RUR dataset are homophilic, as shown in Figure 1 (d) in Appendix A. 2. The performance of NSGT might depend on the quality of the node attributes. NSGT performs the graph truncation by considering the node attributes of the original graph. As mentioned in lines 170-171, if the original node features contain many irrelevant attributes, NSGT might not work well. Furthermore, features of abnormal nodes can be intentionally camouflaged [1, 2]. [1] Liu et al., Alleviating the Inconsistency Problem of Applying Graph Neural Network to Fraud Detection, SIGIR 2020. [2] Dou et al., Enhancing Graph Neural Network-based Fraud Detectors against Camouflaged Fraudsters, CIKM 2020. - One suggestion is to utilize the node representation vectors calculated by LAMNets. In addition, an attribute selection strategy can be introduced to remove irrelevant or redundant node attributes. 3. While this manuscript assumes an unsupervised learning setting, one of my concerns is that most state-of-the-art methods assume a semi-supervised learning setting. Indeed, in many real practices, some supervision can be provided for graph anomaly detection, e.g., labeling undesirable nodes or patterns. Given this, the benefit of the proposed method can be limited. 4. Existing studies [3] convert YelpChi by merging different relations into a single relation. Why YelpChi-RUR and Amazon-UPU are used instead of the full datasets? Instead of selecting one particular relation, the authors can utilize all relations and treat them as a single relation. [3] Chen et al., GCCAD: Graph Contrastive Learning for Anomaly Detection, TKDE 2022. 5. The statistics of Amazon and YelpChi differ from the actual ones. For Amazon, the number of nodes is 10,224, and the number of anomalies is 693 (6.78%). For YelpChi, the number of nodes is 23,831, and the number of anomalies is 1,217 (5.11%). 6. Minor Comments: - Lines 206-207 are confusing. According to lines 206-207, the truncated adjacency matrix instead of the original adjacency matrix is used to optimize LAMNets. However, the local affinity is calculated based on the original adjacency matrix during optimization, as stated in lines 261-262. - In lines 215-221, it is not specified how to handle the case where $d_{i, max}$ is less than $d_{mean}$. - In lines 116-117, "such degree" should be modified to "such as degree".

Questions

1. Why does NSGT utilize raw attributes instead of node representations learned from LAMNet? 2. Is there any experimental result on large-scale graph anomaly detection datasets? 3. Isolated nodes can appear after performing NSGT. How does TAM-T compute the anomaly scores of such isolated nodes?

Rating

4: Borderline reject: Technically solid paper where reasons to reject, e.g., limited evaluation, outweigh reasons to accept, e.g., good evaluation. Please use sparingly.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

2 fair

Presentation

2 fair

Contribution

2 fair

Limitations

1. As the TAM model works solely on the assumption of one-class homophily, it might not perform effectively on the graphs where abnormal nodes are homophilic, e.g., YelpChi-RUR. 2. Since NSGT directly utilizes the raw node features to truncate the graph, the quality of the graph truncation might be affected by the quality of the raw features. However, there can be many irrelevant or redundant attributes in the original features. 3. Due to the definition of local node affinity measure, TAM cannot handle isolated nodes in the graph even though the node features are available.

Reviewer uzaW2023-08-13

Rebuttal feedback

I have read the authors' rebuttal carefully, and their response addresses my concerns, e.g., one-class homogeneity and obtaining ideal truncated graphs. Based on that, I would like to increase my rating.

Authorsrebuttal2023-08-14

Discussion

We're very pleased to hear that our rebuttal helps address your concerns. Thanks a lot for the positive comments and the increased rating!

Reviewer zqNH2023-08-18

Thanks for the rebuttal

Thanks for the rebuttal especially the added results. I read the authors' responses as well as other reviews and responses. Most of my concerns have been addressed. I would like to increase my rating.

Authorsrebuttal2023-08-19

Thanks for the increased rating

It's great that our rebuttal has addressed most of your concerns. Thank you very much for the increased rating and your support on our work!

Area Chair g6rY2023-08-18

Please read the rebuttals and add a comment to acknowledge

Dear Reviewers, Many thanks for your time and efforts on this paper. As the end of discussion is coming soon, please read the rebuttal to see if your concerns/questions are properly resolved and add a comment to acknowledge that you have read the rebuttal. Many Thanks.

Program Chairsdecision2023-09-21

Decision

Accept (poster)

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