SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Although the above "two-extractor" design correlates to the theoretical framework of MSIB, in practice, it is non-trivial to ensure the consistency of two generated subgraphs only with an SKL divergence loss. The main reason is that the input and architectures of two extractors are quite different, leading to the difficulty in output alignment. However, the consistency of two bottleneck subgraphs not only guarantees the informativeness of cross-view MI for abnormality measurement, but also affects the quality of explanations. Considering the significance of preserving consistency, we use a single extractor to generate bottleneck subgraphs for two views. In specific, the bottleneck subgraph extractor first takes the original graph  $G$  as its input and outputs the node probability vector  $\mathbf{p}$  for the bottleneck subgraph extraction of  $G^{(s)}$ . Then, leveraging the node-edge correspondence in DHT, we can directly lift the node probabilities to edge probability vector  $\mathbf{p}^*$  via  $\mathbf{p}_{\mathbb{I}(e_{ij})}^* = \mathbf{p}_i\mathbf{p}_j$  and re-probabilization operation.  $\mathbf{p}^*$  can be used to extract subgraph for the dual hypergraph view. In this way, the generated bottleneck subgraphs in two views can be highly correlated, enhancing the quality of GLAD prediction (MI) and its explanations. Meanwhile, such a "single-extractor" design further simplifies the model architecture by removing extra extractor and loss function (i.e., the  $D_{SKL}$  term in Eq. (3)), reducing the model complexity. Empirical comparison in Sec. 4.4 also validates the effectiveness of this design.
3.4 Cross-view MI Maximization
After bottleneck subgraph extraction, the next step is to maximize the MI  $I(G^{(s)}; G^{*(s)})$  between the bottleneck subgraphs from two views. The estimated MI, in the testing phase, can be used to evaluate the graph-level abnormality. Owing to the discrete and complex nature of graph-structured data, it is difficult to directly estimate the MI between two subgraphs. Alternatively, a feasible solution is to obtain compact representations for two subgraphs, and then, calculate the representation-level MI as a substitute. In SIGNET, we use message passing-based GNN and HGNN with pooling layer (formulated in Appendix C) to learn the subgraph representations  $\mathbf{h}_{G^{(s)}}$  and  $\mathbf{h}_{G^{*(s)}}$  for  $G^{(s)}$  and  $G^{*(s)}$ , respectively. In this case,  $I(G^{(s)}; G^{*(s)})$  can be transferred into a tractable term, i.e., the MI between subgraph representations  $I(\mathbf{h}_{G^{(s)}}; \mathbf{h}_{G^{*(s)}})$ .
After that, the MI term  $I(\mathbf{h}_{G^{(s)}}; \mathbf{h}_{G^{*(s)}})$  can be maximized by using sample-based differentiable MI lower bounds [37], such as Jensen-Shannon (JS) estimator [54], Donsker-Varadhan (DV) estima
tor [55], and Info-NCE estimator [56]. Due to its strong robustness and generalization ability [37, 57], we employ Info-NCE for MI estimation in SIGNET. Specifically, given a batch of graph samples  $\mathcal{B} = \{G_1,\dots ,G_B\}$ , the training loss of SIGNET can be written by:
$$
\begin{array}{l} \mathcal {L} = - \frac {1}{2 | \mathcal {B} |} \sum_ {G _ {i} \in \mathcal {B}} I \left(\mathbf {h} _ {G _ {i} ^ {(s)}}; \mathbf {h} _ {G _ {i} ^ {* (s)}}\right) = - \frac {1}{2 | \mathcal {B} |} \sum_ {G _ {i} \in \mathcal {B}} \left(\ell \left(\mathbf {h} _ {G _ {i} ^ {(s)}}, \mathbf {h} _ {G _ {i} ^ {* (s)}}\right) + \ell \left(\mathbf {h} _ {G _ {i} ^ {* (s)}}, \mathbf {h} _ {G _ {i} ^ {(s)}}\right)\right), \\ \ell \left(\mathbf {h} _ {G _ {i} ^ {(s)}}, \mathbf {h} _ {G _ {i} ^ {* (s)}}\right) = \log \frac {\exp \left(f _ {k} \left(\mathbf {h} _ {G _ {i} ^ {(s)} , \mathbf {h} _ {G _ {i} ^ {* (s)}}}\right) / \tau\right)}{\sum_ {G _ {j} \in \mathcal {B} \backslash G _ {i}} \exp \left(f _ {k} \left(\mathbf {h} _ {G _ {i} ^ {(s)} , \mathbf {h} _ {G _ {j} ^ {(s)}}}\right) / \tau\right)}, \tag {5} \\ \end{array}
$$
where  $f_{k}(\cdot ,\cdot)$  is the cosine similarity function,  $\tau$  is the temperature hyper-parameter, and  $\ell (\mathbf{h}_{G_i^{*(s)}},\mathbf{h}_{G_i^{(s)}})$  is calculated following  $\ell (\mathbf{h}_{G_i^{(s)}},\mathbf{h}_{G_i^{*(s)}})$ .
3.5 Self-Interpretable GLAD Inference
In this subsection, we introduce the self-interpretable GLAD inference protocol with SIGNET (marked in red in Fig. 2(a)) that is composed of two parts: anomaly scoring and explanation.
Anomaly scoring. By minimizing Eq. (5) on training data, the cross-view matching patterns of normal samples are well captured, leading to a higher MI for normal data; on the contrary, the anomalies with anomalous attributal and structural characteristics tend to violate the matching patterns, resulting in their lower cross-view MI in our model. Leveraging this property, during inference, the negative of MI can indicate the abnormality of testing data. For a testing sample  $G_{i}$ , its anomaly score  $s_i$  can be calculated by  $s_i = -I(\mathbf{h}_{G_i^{(s)}};\mathbf{h}_{G_i^{*(s)}})$ , where the MI is estimated by Info-NCE.
Explanation. In SIGNET, the bottleneck subgraph extractor is able to pinpoint the key substructure of the input graph under the guidance of MSIB framework. The learned bottleneck subgraphs are the most discriminative components of graph samples and are highly related to the anomaly scores. Therefore, we can directly regard the bottleneck subgraphs as the explanations of anomaly detection results. In specific, the node probabilities  $\mathbf{p}$  and edge probabilities  $\mathbf{p}^*$  can indicate the significance of nodes and edges, respectively. In practical inference, we can pick the nodes/edges with top-k probabilities or use a threshold-based strategy to acquire a fixed-size explanation subgraph  $G^{(es)}$ .
More discussion about methodology, including the pseudo-code algorithm of SIGNET, the comparison between SIGNET and existing method, and the complexity analysis of SIGNET, is illustrated in Appendix E.
In this section, extensive experiments are conducted to answer three research questions:
- RQ1: Can SIGNET provide informative explanations for the detection results?
- RQ2: How effective is SIGNET on identifying anomalous graph samples?
- RQ3: What are the contributions of the core designs in SIGNET model?
Datasets. For the explainable GLAD task, we introduce 6 datasets with ground-truth explanations, including three synthetic datasets and three real-world datasets. Details are demonstrated below. We also verify the anomaly detection performance of SIGNET on 10 TU datasets [58], following the setting in [4]. Detailed statistics and visualization of datasets are demonstrated in Appendix F.1.
- BM-MT, BM-MN, and BM-MS are three synthetic dataset created by following [13, 19]. Each graph is composed of one base (Tree, Ladder, or Wheel) and one or more motifs that decide the abnormality of the graph. For BM-MT (motif type), each normal graph has a house motif and each anomaly has a 5-cycle motif. For BM-MN (motif number), each normal graph has 1 or 2 house motifs and each anomaly has 3 or 4 house motifs. For BM-MS (motif size), each normal graph has a cycle motif with  $3 \sim 5$  nodes and each anomaly has a cycle motif with  $6 \sim 9$  nodes. The ground-truth explanations are defined as the nodes/edges within motifs.
Table 1: Explanation performance in terms of NX-AUC and EX-AUC (in percent, mean ± std). The best and runner-up results are highlighted with bold and underline, respectively.
Dataset	Metric	OCGIN-GE	GLocalKD-GE	OCGTL-GE	OCGIN-PG	GLocalKD-PG	OCGTL-PG	SIGNET
BM-MT	NX-AUC	48.26±3.18	49.67±0.88	45.79±2.53	-	-	-	78.41±6.88
EX-AUC	52.03±4.32	49.11±2.77	49.80±2.88	64.08±12.23	74.59±7.66	72.72±10.19	77.69±13.14
BM-MN	NX-AUC	46.25±4.60	49.10±0.71	40.53±3.18	-	-	-	76.57±6.62
EX-AUC	60.02±9.20	50.17±3.14	56.34±3.10	54.01±8.01	78.68±8.33	74.36±12.78	83.45±9.33
BM-MS	NX-AUC	52.43±1.70	50.43±0.62	53.44±1.15	-	-	-	76.42±6.81
EX-AUC	54.31±9.61	49.10±2.29	66.87±1.44	43.67±12.66	82.53±8.56	77.45±10.93	79.48±9.97
MNIST-0	NX-AUC	49.48±0.58	50.11±0.64	38.87±3.21	-	-	-	70.38±5.64
EX-AUC	50.85±4.77	49.75±0.55	41.42±2.40	39.53±1.51	54.69±1.78	59.25±4.68	72.78±7.25
MNIST-1	NX-AUC	48.21±2.01	49.50±0.50	47.04±1.66	-	-	-	68.44±3.07
EX-AUC	48.60±3.28	49.78±0.26	45.24±1.11	47.98±4.24	49.24±1.95	57.93±8.54	74.83±5.24
MUTAG	NX-AUC	48.99±1.50	49.70±1.11	49.31±4.94	-	-	-	75.60±8.94
EX-AUC	51.92±9.05	47.65±1.19	45.80±2.81	46.22±7.90	70.47±5.26	65.03±16.90	78.05±9.19
- MNIST-0 and MNIST-1 are two GLAD datasets derived from MNIST-75sp superpixel dataset [59]. Following [60], we consider a specific class (i.e., digit 0 or 1) as the normal class, and regard the samples belonging to other classes as anomalies. The ground-truth explanations are the nodes/edges with nonzero pixel values.
- MUTAG is a molecular property prediction dataset [61]. We set nonmutagenic molecules as normal samples and mutagenic molecules as anomalies. Following [20], -NO2 and -NH2 in mutagenic molecules are viewed as ground-truth explanations.
Baselines. Considering their competitive performance, we consider three state-of-the-art deep GLAD methods, i.e., OCGIN [7], GLocalKD [4], and OCGTL [8], as baselines. To provide explanations for them, we integrate two mainstream post-hoc GNN explainers, i.e., GNNExplainer [19] (GE for short) and PGExplainer [20] (PG for short) into the deep GLAD methods. For GLAD tasks, we further introduce the baselines composed of a graph kernel (i.e., Weisfeiler-Lehman kernel (WL) [62] or Propagation kernel (PK) [63]) and a detector (i.e., iForest (iF) [64] or one-class SVM (OCSVM) [65]).
Metrics and Implementation. For interpretation evaluation, we report explanation ROC-AUCs at node level (NX-AUC) and edge level (EX-AUC) respectively, similar to [19, 20]. For GLAD performance, we report the ROC-AUC w.r.t. anomaly scores and labels (AD-AUC) [7]. We repeat 5 times for all experiments and record the average performance. In SIGNET, we use GIN [2] and Hyper-Conv [30] as the GNN and HGNN encoders. The bottleneck subgraph extractor is selected from GIN [2] and MLP. We perform grid search to pick the key hyper-parameters in SIGNET and baselines. More details of implementation and infrastructures are in Appendix F. Our code is available at https://github.com/yixinliu233/SIGNET.
4.2 Explainability Results (RQ1)
Quantitative evaluation. In Table 1, we report the node-level and edge-level explanation AUC [22] on 6 datasets. Note that PGExplainer [20] can only provide edge-level explanations natively. We have the following observations: 1) SIGNET achieves SOTA performance in almost all scenarios. Compared to the best baselines, the average performance gains of SIGNET are  $27.89\%$  in NX-AUC and  $8.99\%$  in EX-AUC. The superior performance verifies the significance of learning to interpret and detect with a unified model. 2) The post-hoc explainers are not compatible with all GLAD models. For instance, PGExplainer works relevantly well with GLocalKD but cannot provide informative explanations for OCGIN. The GNNExplainer, unfortunately, exhibits poor performance in most scenarios. 3) SIGNET has larger performance gains on real-world datasets, which illustrates the potential of SIGNET in explaining real-world GLAD tasks. 4) Despite its superior performance, the stability of SIGNET is rele
vantly average. Especially on the synthetic datasets, we can find that the standard deviations of
Figure 3: Visualization of explanation results w.r.t. node and edge probabilities.
Table 2: Anomaly detection performance in terms of AD-AUC (in percent, mean ± std). The best and runner-up results are highlighted with bold and underline, respectively.
Dataset	PK-OCSVM	PK-iF	WL-OCSVM	WL-iF	OCGIN	GLocalKD	OCGTL	SIGNET
BM-MT	52.58±0.35	45.30±1.60	53.36±0.46	50.30±0.40	73.33±6.18	74.94±5.12	93.61±0.20	95.89±2.75
BM-MN	97.13±0.19	56.80±4.65	76.60±0.77	49.90±0.20	59.35±2.81	77.51±3.08	99.49±0.05	93.41±1.66
BM-MS	79.34±0.52	57.00±6.15	56.19±0.42	51.30±0.03	58.00±3.44	65.03±2.36	92.01±0.82	94.01±4.88
MNIST-0	48.89±2.19	59.58±1.50	67.19±2.69	59.27±2.43	69.54±2.61	82.29±1.65	80.68±3.14	83.25±2.17
MNIST-1	43.45±1.32	78.97±5.54	63.64±2.60	65.05±3.21	98.25±0.61	93.04±0.65	97.98±0.36	90.12±5.21
MUTAG	53.30±1.29	43.60±1.72	49.64±2.89	49.07±0.32	57.59±3.36	65.09±5.94	63.41±2.60	87.72±3.48
PROTEINS-F	50.49±4.92	60.70±2.55	51.35±4.35	61.36±2.54	70.89±2.44	77.30±5.15	76.51±1.55	75.22±3.91
ENZYMES	53.67±2.66	51.30±2.01	55.24±2.66	51.60±3.81	58.75±5.98	61.39±8.81	62.06±3.36	62.96±4.22
AIDS	50.79±4.30	51.84±2.87	50.12±3.43	61.13±0.71	78.16±3.05	93.27±4.19	99.40±0.57	97.27±1.17
DHFR	47.91±3.76	52.11±3.96	50.24±3.13	50.29±2.77	49.23±3.05	56.71±3.57	59.90±2.96	74.01±4.69
BZR	46.85±5.31	55.32±6.18	50.56±5.87	52.46±3.30	65.91±1.47	69.42±7.78	63.94±8.89	81.44±9.23
COX2	50.27±7.91	50.05±2.06	49.86±7.43	50.27±0.34	53.58±5.05	59.37±12.67	55.23±5.68	71.46±4.64
DD	48.30±3.98	71.32±2.41	47.99±4.09	70.31±1.09	72.27±1.83	80.12±5.24	79.48±2.02	72.72±3.91
NCI1	49.90±1.18	50.58±1.38	50.63±1.22	50.74±1.70	71.98±1.21	68.48±2.39	73.44±0.97	74.89±2.07
IMDB-B	50.75±3.10	50.80±3.17	54.08±5.19	50.20±0.40	60.19±8.90	52.09±3.41	64.05±3.32	66.48±3.49
REDDIT-B	45.68±2.24	46.72±3.42	49.31±2.33	48.26±0.32	75.93±8.65	77.85±2.62	86.81±2.10	82.78±1.11
Avg. Rank	6.6	6.1	6.2	6.4	4.1	2.8	2.1	1.7
NX-AUC and EX-AUC are large. We speculate that the instability is due to the lack of labels that provide reliable supervisory signals for anomaly detection explanations.
Qualitative evaluation. To better understand the behavior of SIGNET, we visualize the explanations (i.e., node and edge probabilities) in Fig. 3. We can witness that SIGNET can assign larger probabilities for the nodes and edges within the discriminative motifs, providing valid explanations for the GLAD predictions. In contrast, the probabilities of base subgraphs are uniformly small, indicating that SIGNET is able to ignore the unrelated substructure. However, we can still witness some irrelevant nodes included in the explanations in the anomaly sample of BM-MN dataset, which indicates that SIGNET may generate noisy explanations in some special cases.
4.3 Anomaly Detection Results (RQ2)
To investigate the anomaly detection performance of SIGNET, we conduct experiments on 16 datasets and summarize the results in Table 2. The following observations can be concluded: 1) SIGNET outperforms all baselines on 10 datasets and achieves competitive performance on the rest. The main reason is that SIGNET can capture graph patterns at node and edge level with two distinct views and concentrate on the key substructure during anomaly scoring. 2) The deep GLAD methods generally perform better than kernel-based methods, indicating that GNN-based deep learning models are effective in identifying anomalous graphs. To sum up, SIGNET can not only accurately detect anomalies but also provide informative interpretations for the predictions.
4.4 Ablation Study (RQ3)
We perform ablation studies to evaluate the effectiveness of core designs in SIGNET, i.e., dual hypergraph-based view construction, single extractor, and InfoNCE MI estimator. We replace these components with alternative designs, and the results are illustrated in Table 3. We consider two strategies for view construction: augmentation-based view construction [46] (Aug. View) and structural
property-based view construction [50] (Str. View). As we discussed in Sec. 3.2, perturbation-based graph augmentation is not appropriate for GLAD tasks, leading to its poor performance. The structural property-based strategy has better performance on BM-BT but still performs weakly on MNIST-0. In contrast, dual hypergraph-based view construction is a better strategy that jointly considers node-level and edge-level information, contributing to optimal detection and explanation performance. We also test the performance of SIGNET with two extractors (2E) and discuss the contribution of SKL divergence. We can find that even with the help of  $D_{SKL}$ , the two-extractor version still underperforms
Table 3: Performance of SIGNET and its variants.
Variant	BM-MS	MNIST-0
NX-AUC	EX-AUC	AD-AUC	NX-AUC	EX-AUC	AD-AUC
Aug. View	66.87	64.48	63.26	41.82	39.25	48.75
Str. View	70.36	68.27	92.79	51.73	50.40	55.29
2E w DSLK	70.71	65.23	92.44	59.22	57.33	81.77
2E w/o DSLK	73.54	37.15	87.84	57.92	43.65	80.97
JS MI Est.	70.35	68.96	75.21	59.06	63.85	71.78
DV MI Est.	72.53	74.16	74.87	60.90	70.28	70.31
SIGNET	76.42	79.48	94.01	70.38	72.78	83.25
the original SIGNET with one extractor. Meanwhile, we can witness that compared to JS [54] and DV [55] MI estimators, Info-NCE estimator can lead to superior performance, especially for anomaly detection.
This paper presents a novel and practical research problem, explainable graph-level anomaly detection (GLAD). Based on the information bottleneck principle, we deduce the framework multi-view subgraph information bottleneck (MSIB) to address the explainable GLAD problem. We develop a new method termed SIGNET by instantiating MSIB framework with advanced neural modules. Extensive experiments verify the effectiveness of SIGNET in identifying anomalies and providing explanations. A limitation of our paper is that we mainly focus on purely unsupervised GLAD scenarios where ground-truth labels are entirely unavailable. As a result, for few-shot or semi-supervised GLAD scenarios [44] where a few labels are accessible, SIGNET cannot directly leverage them for model training and self-interpretation. We leave the exploration of supervised/semi-supervised self-interpretable GLAD problems in future works.
S. Pan was partially supported by an Australian Research Council (ARC) Future Fellowship (FT210100097). F. Li was supported by the National Natural Scientific Foundation of China (No. 62202388) and the National Key Research and Development Program of China (No. 2022YFF1000100).
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