SlowGuess/NeurIPS_2023_OCR huggingface dataset:
where  $Y_{f}[e]$  refers to the original prediction of event  $e$ ,  $\beta$  is the regularization coefficient and  $K$  is a constraint on the explanation size. We then adjust Eq. 4 to incorporate temporal motifs.
The first term in Eq. 4 can be estimated with the cross-entropy between the original prediction and the output of base model  $f$  given  $\mathcal{G}_{\mathrm{exp}}^e$  as Eq. 1, where  $\mathcal{G}_{\mathrm{exp}}^e$  is induced by  $\mathcal{M}_{\mathrm{exp}}$ . Since temporal motifs are essential building blocks of the surrounding subgraph and we have access to the posterior distribution of  $\mathcal{M}_{\mathrm{exp}}$  conditioned on  $\mathcal{M}$  with importance scores, we propose to formulate the second term in Eq. 4 as the mutual information between the original motif set  $\mathcal{M}$  and the selected motif subset  $\mathcal{M}_{\mathrm{exp}}$ . We utilize a variational approximation  $\mathbb{Q}(\mathcal{M}_{\mathrm{exp}})$  to replace its marginal distribution  $\mathbb{P}(\mathcal{M}_{\mathrm{exp}})$  and obtain the upper bound of  $I(\mathcal{M}, \mathcal{M}_{\mathrm{exp}})$  with Kullback-Leibler divergence:
$$
I (\mathcal {M}, \mathcal {M} _ {\exp}) \leq \mathbb {E} _ {\mathcal {M}} D _ {\mathrm {K L}} \left(\mathbb {P} _ {\phi} \left(\mathcal {M} _ {\exp} \mid \mathcal {M}\right); \mathbb {Q} \left(\mathcal {M} _ {\exp}\right)\right) \tag {5}
$$
where  $\phi$  involve learnable parameters in Motif Encoder (Eq. 3) and the MLP for importance scores.
Choice of Prior Distribution. Different choices of  $\mathbb{Q}(\mathcal{M}_{\mathrm{exp}})$  in Eq. 5 may lead to different inductive bias. We consider two practical prior distributions for  $\mathbb{Q}(\mathcal{M}_{\mathrm{exp}})$ : uniform and empirical. In the uniform setting [42, 66],  $\mathbb{Q}(\mathcal{M}_{\mathrm{exp}})$  is the product of Bernoulli distributions with probability  $p \in [0,1]$ , that is, each motif shares the same probability  $p$  being in the explanation. The KL divergence thus becomes  $D_{\mathrm{KL}}(\mathbb{P}_{\phi}(\mathcal{M}_{\mathrm{exp}}|\mathcal{M}); \mathbb{Q}(\mathcal{M}_{\mathrm{exp}})) = \sum_{I_i \in \mathcal{M}} p_{I_i} \log \frac{p_{I_i}}{p} + (1 - p_{I_i}) \log \frac{1 - p_{I_i}}{1 - p}$ . Here  $p$  is a hyperparameter that controls both the randomness level in the prior distribution and the prior belief about the explanation volume (i.e., the proportion of motifs that are important for the prediction).
However, uniform distribution ignores the effect of the null model, which is a better indication of randomness in the field of temporal graphs. To tackle this challenge, we further propose to leverage the null model to define empirical prior distribution for  $\mathbb{Q}(\mathcal{M}_{\mathrm{exp}})$ . A null model is essentially a randomized version of the empirical network, generated by shuffling or randomizing certain properties while preserving some structural aspects of the original graph. Following prior works on the null model [67, 22], we utilize the common null model in this work, where the event order is randomly shuffled. The null model shares the same degree spectrum and time-shuffled event orders with the input graph [53] (see more details in Appendix D.1). We categorize the motif instances in  $\mathcal{M}$  by their equivalence relation defined in Definition B. Let  $(U_{1},\dots ,U_{T})$  denote  $T$  equivalence classes of temporal motifs and  $(q_{1},\dots ,q_{T})$  is the sequence of normalized class probabilities occurring in  $\mathcal{M}_{\mathrm{exp}}$  with  $q_{i} = \sum_{I_{j}\in U_{i}}p_{I_{j}} / \sum_{I_{j}\in \mathcal{M}}p_{I_{j}}$ , where  $p_{I_j}$  is the importance score of the motif instance  $I_{j}$ . Correspondingly, we have  $(m_1,\dots ,m_T)$  denoting the sequence of normalized class probabilities in the null model. The prior belief about the average probability of a motif being important for prediction is fixed as  $p$ . Thus minimizing Eq. 5 is equivalent to the following equation.
$$
\min  _ {\phi} D _ {\mathrm {K L}} \left(\mathbb {P} _ {\phi} \left(\mathcal {M} _ {\exp} \mid \mathcal {M}\right); \mathbb {Q} \left(\mathcal {M} _ {\exp}\right)\right) \Leftrightarrow \min  _ {\phi} (1 - s) \log \frac {1 - s}{1 - p} + s \sum_ {i = 1} ^ {T} q _ {i} \log \frac {s q _ {i}}{p m _ {i}}, \tag {6}
$$
where  $s$  is computed by  $s = \sum_{I_j\in \mathcal{M}}p_{I_j} / |\mathcal{M}|$ , which measures the sparsity of the generated explanation. Combing Eq. 1 and Eq. 6 leads to the following overall optimization objective:
$$
\min  _ {\phi} \mathbb {E} _ {e \in \mathcal {E} (t)} \sum_ {c = 0, 1} - \mathbb {1} \left(Y _ {f} [ e ] = c\right) \log \left(f \left(\mathcal {G} _ {\exp} ^ {e}\right) [ e ]\right) + \beta \left((1 - s) \log \frac {1 - s}{1 - p} + s \sum_ {i = 1} ^ {T} q _ {i} \log \frac {s q _ {i}}{p m _ {i}}\right). \tag {7}
$$
Eq. 7 aims at optimizing the explanation accuracy with the least amount of information. It learns to identify the most interaction-related temporal motifs and push their importance scores close to 1, leading to deterministic existences of certain motifs in the target explanation. Meanwhile, the interaction-irrelevant components are assigned smaller importance scores to balance the trade-off in Eq. 7. TempME shares spirits with perturbation-based explanations [68, 69], where "interpretable components [68]" corresponds to temporal motifs and the "reference" is the null model.
Complexity. A brute-force implementation of the sampling algorithm (Alg. 1) has the time complexity  $\mathcal{O}(Cl)$ . Following Liu et al. [52], we create a  $2l$ -digit to represent a temporal motif with  $l$  events, where each pair of digits is an event between the node represented by the first digit and the node represented by the second digit. We utilize these  $2l$ -digits to classify the temporal motifs by their equivalence relations, thus resulting in a complexity of  $\mathcal{O}(C)$ . An acceleration strategy with the tree-structured sampling and detailed complexity analysis are given in Appendix D.3.
Dataset. We evaluate the effectiveness of TempME on six real-world temporal graph datasets, Wikipedia, Reddit, Enron, UCI, Can.Parl., and US Legis [70, 71, 72] that cover a wide range of domains. Wikipedia and Reddit are bipartite networks with rich interaction attributes. Enron and UCI are social networks without any interaction attributes. Can.Parl. and US Legis are two political networks with a single attribute. Detailed dataset statistics are given in Appendix E.1.
Base Model. The proposed TempME can be employed to explain any temporal graph neural network (TGNN) that augments local message passing. We adopt three state-of-the-art temporal graph neural networks as the base model: TGAT [3], TGN [4], and GraphMixer [5]. TGN and GraphMixer achieve high performance with only one layer due to their powerful expressivity or memory module. TGAT typically contains 2-3 layers to achieve the best performance. Following previous training setting [23, 1, 6], we randomly sample an equal amount of negative links and consider event prediction as a binary classification problem. All models are trained in an inductive setting [6, 1].
Baselines. Assuming the base model contains  $L$  layers and we aim at explaining the prediction on the event  $e$ , we first extract  $L$ -hop neighboring historical events as the computational graph  $\mathcal{G}(e)$ . For baselines, we first compare with two self-interpretable techniques, Attention (ATTN) and Gradient-based Explanation (Grad-CAM [37]). For ATTN, we extract the attention weights in TGAT and TGN and take the average across heads and layers as the importance scores for events. For Grad-CAM, we calculate the gradient of the loss function w.r.t. event features and take the norms as the importance scores. Explanation  $\mathcal{G}_{\mathrm{exp}}^{e}$  is generated by ranking events in  $\mathcal{G}(e)$  and selecting a subset of explanatory events with the highest importance scores. We further compare with learning-based approaches, GNNExplainer [13], PGExplainer [14] and TGNNExplainer [23], following the baseline setting in prior work [23]. The former two are proposed to explain static GNNs while TGNNExplainer is a current state-of-the-art model specifically designed for temporal GNNs.
Configuration. Standard fixed splits [73, 72] are applied on each datasets. Following previous studies on network motifs [52, 22, 35, 56], we have empirically found that temporal motifs with at most 3 nodes and 3 events are sufficiently expressive for the explanation task (Fig. 4). We use GINE [74], a modified version of GIN [62] that incorporates edge features in the aggregation function, as theMESSAGEPASSING function and mean-pooling as the READOUT function by default.
5.2 Explanation Performance
Evaluation Metrics. To evaluate the explanation performance, we report Fidelity and Sparsity following TGNNExplainer [23]. Let  $\mathcal{G}_{\mathrm{exp}}^e$  and  $\mathcal{G}$  denote the explanation for event  $e$  and the original temporal graph, respectively. Fidelity measures how valid and faithful the explanations are to the model's original prediction. If the original prediction is positive, then an explanation leading to an increase in the model's prediction logit is considered to be more faithful and valid and vice versa. Fidelity is defined as  $\mathrm{Fid}(\mathcal{G},\mathcal{G}_{\mathrm{exp}}^e) = \mathbb{1}(Y_f[e] = 1)(f(\mathcal{G}_{\mathrm{exp}}^e)[e] - f(\mathcal{G})[e]) + \mathbb{1}(Y_f[e] = 0)(f(\mathcal{G})[e] - f(\mathcal{G}_{\mathrm{exp}}^e)[e])$ . Sparsity is defined as Sparsity =  $|\mathcal{G}_{\mathrm{exp}}^e| / |\mathcal{G}(e)|$ , where  $\mathcal{G}(e)$  denotes the computational graph of event  $e$ . An ideal explanation should be compact and succinct, therefore,
Table 1: ACC-AUC of TempME and baselines over six datasets and three base models. The AUC values are computed over 16 sparsity levels between 0 and 0.3 at the interval of 0.02. The best result is in bold and second best is underlined.
Wikipedia	Reddit	UCI	Enron	USLegis	Can.Parl.
TGAT	Random	70.91±1.03	81.97±0.92	54.51±0.52	48.94±1.28	54.24±1.34	51.66±2.26
ATTN	77.31±0.01	86.80±0.01	27.25±0.01	68.28±0.01	62.24±0.00	79.92±0.01
Grad-CAM	83.11±0.01	90.29±0.01	26.06±0.01	19.93±0.01	78.98±0.01	50.42±0.01
GNNExplainer	84.34±0.16	89.44±0.56	62.38±0.46	77.82±0.88	89.42±0.50	80.59±0.58
PGExplainer	84.26±0.78	92.31±0.92	59.47±1.68	62.37±3.82	91.42±0.94	75.92±1.12
TGNNExplainer	85.74±0.56	95.73±0.36	68.26±2.62	82.02±1.94	90.37±0.84	80.67±1.49
TempME	85.81±0.53	96.69±0.38	76.47±0.80	81.85±0.26	96.10±0.20	84.48±0.97
TGN	Random	91.90±1.42	91.42±1.94	87.15±2.23	82.72±2.24	72.31±2.64	76.43±1.65
ATTN	93.28±0.01	93.81±0.01	83.24±0.01	83.57±0.01	75.62±0.01	79.38±0.01
Grad-CAM	93.46±0.01	92.60±0.01	87.51±0.01	81.12±0.01	81.46±0.01	77.19±0.01
GNNExplainer	95.62±0.53	95.50±0.35	94.68±0.42	88.61±0.50	82.91±0.46	83.32±0.64
PGExplainer	94.28±0.84	94.42±0.36	92.39±0.85	88.34±1.24	90.62±0.75	88.46±1.42
TGNNExplainer	93.51±0.98	96.21±0.47	94.24±0.52	90.32±0.82	90.40±0.83	84.70±1.19
TempME	95.80±0.42	98.66±0.80	96.34±0.30	92.64±0.27	94.37±0.88	90.63±0.72
GraphMixer	Random	77.31±2.37	85.08±0.72	53.56±1.27	64.07±0.86	85.54±0.93	87.79±0.51
Grad-CAM	76.63±0.01	84.44±0.41	82.64±0.01	72.50±0.01	88.98±0.01	85.80±0.01
GNNExplainer	89.21±0.63	95.10±0.36	61.02±0.37	74.23±0.13	89.67±0.35	92.28±0.10
PGExplainer	85.19±1.24	92.46±0.42	63.76±1.06	75.39±0.43	92.37±0.10	90.63±0.32
TGNNExplainer	87.69±0.86	95.82±0.73	80.47±0.87	81.87±0.45	93.04±0.45	93.78±0.74
TempME	90.15±0.30	95.05±0.19	87.06±0.12	79.69±0.33	95.00±0.16	95.98±0.21
higher fidelity with lower Sparsity denotes a better explanation performance. Besides, we further adopt the ACC-AUC metric, which is the AUC value of the proportion of generated explanations that have the same predicted label by the base model over sparsity levels from 0 to 0.3.
(a) TGAT
(b) TGN
Figure 3: Fidelity-Sparsity Curves on Wikipedia dataset with different base models
(c) GraphMixer
Results. Table 1 shows the explanation performance of TempME and other baselines w.r.t. ACC-AUC. TempME outperforms baselines on different datasets and base models in general. Notably, TempME achieves state-of-the-art performance in explaining TGN with strong ACC-AUC results ( $\geq 90\%$ ) over all six datasets. Specifically, the effectiveness of TempME is consistent across datasets with and without attributes, whereas the performance of baseline models exhibits considerable variation. For example, ATTN and Grad-CAM work well on datasets with rich attributes, e.g., Wikipedia and Reddit, while may yield poor performances on unattributed datasets. Therefore, events with large gradients or attention values are not sufficient to explain the decision-making logic of the base model.
Figure 3 demonstrates the Fidelity-Sparsity curves of TempME and compared baselines on Wikipedia with different base models. From Figure 3, we observe that TempME surpasses the baselines in terms of explanation fidelity, especially with a low sparsity level. In addition, it reveals that the optimal sparsity level varies among different base models. For TGAT, increasing sparsity initially diminishes and later enhances the general fidelity. Conversely, for TGN and GraphMixer, increasing sparsity consistently improves fidelity. These findings indicate that TGAT gives priority to a narrow subset (e.g.,  $1\%$ ) of historical events, while TGN and GraphMixer rely on a wider range of historical events.
Cohesiveness. To evaluate the cohesive level of the explanations, we propose the following metric:
$$
\text {C o h e s i v e n e s s} = \frac {1}{\left| \mathcal {G} _ {\exp} ^ {e} \right| ^ {2} - \left| \mathcal {G} _ {\exp} ^ {e} \right|} \sum_ {e _ {i} \in \mathcal {G} _ {\exp} ^ {e}} \sum_ {e _ {j} \in \mathcal {G} _ {\exp} ^ {e}; e _ {i} \neq e _ {j}} \cos \left(\frac {\left| t _ {i} - t _ {j} \right|}{\Delta T}\right) \mathbb {1} \left(e _ {i} \sim e _ {j}\right), \tag {8}
$$
where  $\Delta T$  means the time duration in the computational graph  $\mathcal{G}(e)$ ,  $\mathbb{1}(e_i\sim e_j)$  indicates whether  $e_i$  is spatially adjacent to  $e_j$ . Meanwhile, temporally proximate event pairs are assigned with larger weights of  $\cos (|t_i - t_j| / \Delta T)$ . A higher level of cohesiveness indicates a more cohesive explanation. From Table 2, we observe that ATTN and Grad-CAM excel in generating cohesive explanations compared to learning-based explainers, e.g., GNNExplainer, TGNNExplainer. However, TempME still surpasses all baselines and achieves the highest cohesiveness levels, primarily due to its ability to extract and utilize self-connected motifs, allowing it to generate explanations that are both coherent and cohesive.
Efficiency Evaluation. We empirically investigate the efficiency of TempME in terms of the inference time for generating one explanation and report the results for Wikipedia and Reddit on TGAT in Table 3, where the averages are calculated across all test events. GNNExplainer and TGNNExplainer optimize explanations individually for each instance, making them less efficient. Notably, TGNNExplainer is particularly time-consuming due to its reliance on the MCTS algorithm. In contrast, TempME trains a generative model using historical events, which allows for
Table 2: Cohesiveness evaluation on Reddit and UCI with TGAT.
Reddit	UCI
ATTN	0.0502	0.0708
Grad-CAM	0.0422	0.0722
GNNExplainer	0.0270	0.0337
PGExplainer	0.0233	0.0332
TGNNExplainer	0.0397	0.0538
TempME	0.0574	0.0749
generalization to future unseen events. As a result, TempME demonstrates high efficiency and fast inference.
Table 3: Inference time (seconds) of one explanation for TGAT
Wikipedia	Reddit
Random	0.02±0.06	0.03±0.09
ATTN	0.02±0.00	0.04±0.00
Grad-CAM	0.03±0.00	0.04±0.00
GNNExplainer	8.24±0.26	10.44±0.67
PGExplainer	0.08±0.01	0.08±0.01
TGNNExplainer	26.87±3.71	83.70±16.24
TempME	0.13±0.02	0.15±0.02
Table 4: Link prediction results (Average Precision) of base models with Motif Embedding (ME)
UCI	Enron	USLegis	Can.Parl.
TGAT	76.28	65.68	72.35	65.18
TGAT+ME	83.65(↑7.37)	68.37(↑2.69)	95.31(↑22.96)	76.35(↑11.17)
TGN	75.82	76.40	77.28	64.23
TGN+ME	77.46(↑1.64)	75.62(↓0.78)	83.90(↑6.62)	79.46(↑15.23)
GraphMixer	89.13	69.42	66.71	76.98
GraphMixer+ME	90.11(↑0.98)	70.13(↑0.71)	81.42(↑14.71)	79.33(↑2.35)
Motif-enhanced Link Prediction. The extracted motifs can not only be used to generate explanations but also boost the performance of TGNNs. Let  $m_I$  denote the motif embedding generated by the Temporal Motif Encoder (Eq. 3) and  $\mathcal{M}$  is the temporal motif set around the node of interest. We aggregate all these motif embeddings using  $\sum_{I\in \mathcal{M}}m_I / |\mathcal{M}|$  and concatenate it with the node representations before the final MLP layer in the base model. The performance of base models on link prediction with and without Motif Embeddings (ME) is shown in Table 4. Motif Embedding provides augmenting information for link prediction and generally improves the performance of base models. Notably, TGAT achieves a substantial boost, with an Average Precision of  $95.31\%$  on USLegis, surpassing the performance of state-of-the-art models on USLegis [72, 73]. More results are given in Appendix E.3.
Ablation Studies. We analyze the hyperparameter sensitivity and the effect of prior distributions used in TempME, including the number of temporal motifs  $C$ , the number of events in the motifs  $l$  and the prior belief about the explanation volume  $p$ . The results are illustrated in Figure 4.
Firstly, when using smaller motifs (e.g.,  $l = 2$ ), TempME achieves comparable explanation accuracy when a sufficient number of motifs are sampled. However, the accuracy plateaus with fewer temporal motifs when  $l = 3$  or  $l = 4$ . Unfortunately, there are only three equivalence classes for temporal motifs with only two events, limiting the diversity of perspectives in explanations. Following previous analysis on temporal motifs [52, 22,
(a) Number of Motifs  $C$
(b) Prior belif of the explanation volume  $p$
Figure 4: (a) Hyperparameter sensitivity of number of temporal motifs  $C$  and motif length  $l$ . (b) Comparison between uniform and empirical prior distribution in terms of ACC-AUC over sparsity levels from 0 to 0.3.
[24], we suggest considering temporal motifs with up to 3 events in the explanation task for the sake of algorithmic efficiency. Secondly, TempME achieves the highest ACC-AUC when the prior belief of the explanation volume is in the range of [0.3, 0.5]. Notably, TempME performs better with empirical prior distribution when  $p$  is relatively small, resulting in sparser and more compact explanations. This improvement can be attributed to the incorporation of the null model, which highlights temporal motifs that differ significantly in frequency from the null model. Figure 4 (b) verifies the rationality and effectiveness of the empirical prior distribution in TempME. Additional insight into the role of the null model in explanation generation can be found in the explanation visualizations in Appendix E.5.
It is worth noting that when  $p$  is close to 1, uniform prior distribution leads to deterministic existences of all temporal motifs while empirical prior distribution pushes the generated explanations towards the null model, which forms the reason for the ACC-AUC difference of empirical and uniform as  $p$  approaches 1.
We further conduct ablation studies on the main components of
Table 5: Ablation studies on the main components of TempME in terms of the explanation ACC-AUC on Wikipedia.
TGAT	TGN	GraphMixer
TempME	85.81±0.53	95.80±0.42	90.15±0.30
Temporal Motif Encoder
w/ GCN	83.26±0.70	94.62±0.34	88.62±0.95
w/ GAT	84.37±0.63	95.46±0.73	88.59±0.84
w/ Adaptive Pooling	85.24±0.46	95.73±0.27	90.15±0.27
w/o Event Anonymization	81.47±1.14	93.77±0.52	88.32±0.93
w/o Time Encoding	79.42±0.82	92.64±0.70	87.90±0.86
Loss function
w/ uniform	85.60±0.75	94.60±0.38	87.46±0.71
TempME. We report the explanation ACC-AUC on Wikipedia in Table 5. Specifically, we first replace the GINE convolution with GCN and GAT and replace the mean-pooling with adaptive pooling in the Temporal Motif Encoder. Then we iteratively remove event anonymization and time encoding in the creation of event features before they are fed into the Temporal Motif Encoder (Eq. 3). Results in Table 5 demonstrate that all the above variants lead to performance degradation. Moreover, the Time Encoding results in a more severe performance drop across three base models. We further evaluate the effectiveness of empirical prior distribution by comparing it with uniform prior distribution. In both prior distributions, the prior belief on the explanation size  $p$  is set to 0.3. We report the best results in Table 5. We can observe that the empirical prior distribution gains a performance boost across three base models, demonstrating the importance of the null model in identifying the most significant motifs.
In this work, we present TempME, a novel explanation framework for temporal graph neural networks. Utilizing the power tool of temporal motifs and the information bottleneck principle, TempME is capable of identifying the historical events that are the most contributing to the predictions made by TGNNs. The success of TempME bridges the gap in explainability research on temporal GNNs and points out worth-exploring directions for future research. For instance, TempME can be deployed to analyze the predictive behavior of different models, screen effective models that can capture important patterns, and online services to improve the reliability of temporal predictions.
By enabling the generation of explainable predictions and insights, temporal GNNs can enhance decision-making processes in critical domains such as healthcare, finance, and social networks. Improved interpretability can foster trust and accountability, making temporal GNNs more accessible to end-users and policymakers. However, it is crucial to ensure that the explanations provided by the models are fair, unbiased, and transparent. Moreover, ethical considerations, such as privacy preservation, should be addressed to protect individuals' sensitive information during the analysis of temporal graphs.
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The main notations used throughout this paper are summarized in Table 6.