SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$\sigma(\alpha) = 1$  and  $\sigma(\beta) = 0$  after training. In that case,  $X_{j}^{\prime \prime} = X_{j}$ , which means  $X_{j}$  is not changed over steps 2 and 3 of the Mee layer. Therefore,  $X_{0}^{i} = G N_{intra}^{i,0}(\mathcal{G}, X_{0}^{i-1})$ , this is equivalent to a simple GNN layer that  $X^{i} = G N N_{i}(\mathcal{G}, X^{i-1})$  as  $X_{0}^{i}$  is the features of the original graph and  $G N_{intra}$  is a GNN layer. Therefore, MeGraph degenerates to standard message-passing GNNs in this case.
E Implementation and Training Details
We use PyTorch [45] and Deep Graph Library (DGL) [54] to implement our method.
We implement S-EdgePool using DGL, extending from the original implementation of EdgePool in the Pytorch Geometric library (PYG) [19]. We did Constant optimization over the implementation to speed up the training and inference of the pooling. We further use Taichi-Lang [29] to speed up the dynamic node clustering process of S-EdgePool. The practical running time of MeGraph model with height  $h > 1$  after optimization is about  $2h$  times as the  $h = 1$  baseline. This is still slower than the theoretical computational complexity due to the constant in the implementation and the difficulty of paralleling the sequential visitation of edges (according to their scores) in the EdgePool and S-EdgePool. This process could be further speed up by implementing the operations with the CUDA library. We provide the practical running time for  $h > 1$  and  $h = 1$  in GNN benchmark and OGB-G datasets in Tables 7 and 8. The speed is slower than the theoretical one partially due to the pooling ratio of edges being larger than the nodes, making the number of edges decrease slowly over the hierarchy. To further speed up, we could use the variants of MeGraph introduced in App. C.5 by skipping some computation modules.
We run all our experiments on V100 GPUs and M40 GPUs. For training the neural networks, we use Adam [33] as the optimizer. We report the hyper-parameters of the Megraph in Table 9.
For models using GFuN layer as the core GN block, we find it benefits from using layer norms [4]. However, for models using GCN layer as the core GN block, we find it performs best when using batch norms [31].
F.1 Experimental Protocol
We evaluate MeGraph on public real-world graph benchmarks. To fairly compare MeGraph with the baselines, we use the following experimental protocols. We first report the public baseline results and our reproduced standard GCN's results. We then replace GCN layers with GFuN layers (which is equivalent to MeGraph  $(h = 1)$ ) to serve as another baseline. We tune the hyper-parameters (such as learning rate, dropout rate and the readout global pooling method, etc.) of MeGraph  $(h = 1)$  and choose the best configurations. We then run other diversely configured MeGraph candidates by tuning
Table 9: Hyper-parameters of the standard version of MeGraph for each dataset. It is worth noting that the total number of GNN layers is equals to one plus the number of Mee layers as  $n + 1$ .
Hyper-parameters	Synthetic Datasets	Graph Theory Benchmark	LRGB Benchmark	GNN Benchmark	OGB Benchmark	TU Datasets
Repeated Runs	10	5	4	4	5	1 for each fold
Epochs per run	200 for BA* 500 for Tree*	300 (200 for MCC)	200	200 (100 for MNIST, CIFRA10)	100	100 (200 for ENZYMES)
Learning rate	0.002	0.002 (0.005 for MCC)	0.001	0.001	0.001	0.002
Weight decay	0.0005	0.0005	0	0	0.0005	0.0005
Node hidden dim	64	128	160	144	300	128
Edge hidden dim (for GFuN)	64	128	160	144	300	128
Num Mee layers n	-	-	4	3	4	2
Height h	-	-	9	5	5	3 or 5
Batch size	32	32	128	128	32	128
Input embedding	False	True	True	True	True	True
Global pooling	Mean	Mean Max	Mean Max Sum	Mean	Mean	Mean Max Sum
Dataset split (train:val:test)	8:1:1	8:1:1	Original split	Original split	Original split	10-fold cross validation
other hyper-parameters that only matters for  $h > 1$ , and these hyper-parameters are referred to as the MeGraph hyper-parameters. Detailed configurations are put in Table 9 in App. E.
F.2 Visualization
We plotted the graph hierarchies discovered by MeGraph in the shortest path tasks of Graph Theory Benchmark. In Figure 5, the S-EdgePool with  $\eta_v = 0.3$ ,  $\tau_c = 4$  well preserves the structure of the graph after pooling, while the S-EdgePool with  $\eta_v = 0.3$  (no cluster size limit) sometimes pooled too many nodes together, breaking the graph structure. The former S-EdgePool leads to better performance as indicated in Table 1. We also plot the hierarchy for SBM-generated graphs in Figure 6, indicating that EdgePool can handle graphs that naturally contains clusters of different size.
F.3 Other Real-Wrold Datasets
TU dataset consists of over 120 datasets of varying sizes from a wide range of applications. We choose 10 datasets, 5 of which are molecule datasets (MUTAG, NCI1, PROTEINS, D&D and ENZYMES) and the other 5 are social networks (IMDB-B, IMDB-M, REDDIT-BINARY, REDDIT-MULTI-5K and REDDIT-MULTI-12K). They are all graph classification tasks. For more details of each dataset, please refer to the original work [42].
Our Megraph uses the same network structure and hyper-parameters for the same type of dataset. As shown in Table 10, our Megraph achieves about  $1\%$  absolute gain than the  $h = 1$  Baselines.
F.4 GFuN
We show our GFuN results on real-world datasets compared to our reproduced GCN in Table 11, 12 and 13. Both GCN and GFuN have the same hyper-parameters except the batch norm for GCN and layer norm for GFuN as stated in Appendix E.
Figure 5: The visualization of the resulting graph hierarchy of MeGraph. Each row is a graph hierarchy, and the nodes with the same color in the same graph are in the same cluster and are pooled into a single node in the next hierarchy. The 1st, 3rd hierarchies use S-EdgePool with  $\eta_v = 0.3$ ,  $\tau_c = 4$  and the 2nd and 4th hierarchies use S-EdgePool with  $\eta_v = 0.3$ . The graph in the 1st and 2nd hierarchies is generated by the 'lobster' method, the graph in the 3rd and 4th hierarchies is generated by the 'geo' method. The task is to compute the length of the shortest path between two nodes.
Figure 6: The graph hierarchy visualization, using S-EdgePool with  $\eta_v = 0.3$ ,  $\tau_c = 4$  (left two figures) and Louvain pooling (right two figures) on the same SBM generated graph. The task is the shortest path.
F.5 Synthetic Datasets
Figures 7 and 8 show the influence of the height  $h$  and the number of Mee layers  $n$  for MeGraph model on the BASHape and BACommunity datasets. The trends are similar while less significant on easier datasets BASHape and BACommunity.
F.6 Varying GN block
We vary the aggregation function of the GN block as attention (w/ ATT) and gated function (w/ GATE). We observe similar results as in Sec. 4.2 and 4.3, verifying the robustness of MeGraph over different GN blocks. Results are shown in Tables 16, 17 and 18.
Table 10: Results on Tu Dataset.  $\dagger$  means the results taken from [11] (*: The result of GCN on ENZYMES is 100 epoch).
Model	MUTAG ↑	NCI1 ↑	PROTEINS ↑	D&D ↑	ENZYMES ↑	Average
GCN†	87.20 ±5.11	83.65 ±1.69	75.65 ±3.24	79.12 ±3.07	66.50 ±6.91*	78.42
GIN†	89.40 ±5.60	82.70 ±1.70	76.20 ±2.80	-	-	-
GCN	92.46 ±6.55	82.55 ±0.99	77.82 ±4.52	80.56 ±2.40	74.17 ±5.59	81.51
MeGraph (h=1)	93.01 ±6.83	82.53 ±1.89	81.32 ±4.08	81.32 ±3.17	74.83 ±3.20	82.60
MeGraph	93.07 ±6.71	83.99 ±0.98	81.41 ±3.10	81.24 ±2.39	75.17 ±4.86	82.98
MeGraphbest	94.12 ±5.02	84.40 ±1.11	81.68 ±3.40	82.00 ±2.86	75.17 ±4.86	83.47
Model	IMDB-B ↑	IMDB-M ↑	RE-B ↑	RE-M5K ↑	RE-M12K ↑	Average
GCN	76.00 ±3.44	50.33 ±1.89	91.15 ±1.63	56.47 ±1.54	48.71 ±0.88	64.53
MeGraph (h=1)	68.60 ±3.53	51.33 ±2.23	93.10 ±1.16	57.47 ±2.31	51.56 ±1.06	64.41
MeGraph	72.40 ±2.80	51.27 ±2.71	93.75 ±1.25	57.69 ±2.22	52.03 ±0.86	65.43
MeGraphbest	74.30 ±2.97	52.00 ±2.49	93.75 ±1.25	58.45 ±2.22	52.13 ±1.01	66.13
Table 11: Comparison between GCN and GFuN on GNN benchmark.
Model	ZINC ↓	AQSOL ↓	MNIST ↑	CIFAR10 ↑	PATTERN ↑	CLUSTER ↑
GCN	0.426 ±0.015	1.397 ±0.029	90.140 ±0.140	51.050 ±0.390	84.672 ±0.054	47.541 ±0.940
GFuN	0.364 ±0.003	1.386 ±0.024	95.560 ±0.190	61.060 ±0.500	84.845 ±0.021	58.178 ±0.079
MeGraph (h=1)	0.323 ±0.002	1.075 ±0.007	97.570 ±0.168	69.890 ±0.209	84.845 ±0.021	58.178 ±0.079
MeGraph	0.260 ±0.005	1.002 ±0.021	97.860 ±0.098	69.925 ±0.631	86.507 ±0.067	68.603 ±0.101
Table 12: Comparison between GCN and GFuN on OGB-G.
Model	molhiv ↑	molbace ↑	molbbbp ↑	molclintosh ↑	molsider ↑
GCN	75.40 ±1.29	76.01 ±3.31	67.35 ±0.96	89.62 ±2.27	58.08 ±0.78
GFuN	78.54 ±1.14	71.77 ±2.15	67.56 ±1.11	89.77 ±3.48	58.28 ±0.51
MeGraph (h=1)	78.54 ±1.14	71.77 ±2.15	67.56 ±1.11	89.77 ±3.48	58.28 ±0.51
MeGraph	77.20 ±0.88	78.52 ±2.51	69.57 ±2.33	92.04 ±2.19	59.01 ±1.45
Model	moltox21 ↑	moltoxcast ↑	molesol ↓	molfreesolv ↓	mollipo ↓
GCN	75.11 ±0.41	64.13 ±0.52	1.141 ±0.02	2.407 ±0.15	0.788 ±0.01
GFuN	75.89 ±0.45	64.49 ±0.46	1.079 ±0.02	2.017 ±0.08	0.768 ±0.00
MeGraph (h=1)	75.89 ±0.45	64.49 ±0.46	1.079 ±0.02	2.017 ±0.08	0.768 ±0.00
MeGraph	78.11 ±0.47	67.67 ±0.53	0.886 ±0.02	1.876 ±0.05	0.726 ±0.00
Table 13: Comparison between GCN and GFuN on Tu Dataset.
Model	MUTAG ↑	NCI1 ↑	PROTEINS ↑	D&D ↑	ENZYMES ↑	Average
GCN	92.46 ±6.55	82.55 ±0.99	77.82 ±4.52	80.56 ±2.40	74.17 ±5.59	81.51
GFuN	93.01 ±7.96	82.80 ±1.30	80.60 ±3.83	82.43 ±2.60	73.00 ±5.31	82.37
Model	IMDB-B ↑	IMDB-M ↑	RE-B ↑	RE-M5K ↑	RE-M12K ↑	Average
GCN	76.00 ±3.44	50.33 ±1.89	91.15 ±1.63	56.47 ±1.54	48.71 ±0.88	64.53
GFuN	68.90 ±3.42	51.27 ±3.22	92.25 ±1.12	57.53 ±1.31	51.54 ±1.19	64.30
Figure 7: Node Classification accuracy for MeGraph model on TreeCycle (left) and TreeGrid (right) datasets, varying the height  $h$  and the number of Mee layers  $n$ . A clear gap can be observed between heights 1, 2 and  $\geq 3$ . The concrete number of accuracy can be found in Table 14.
Figure 8: Node Classification accuracy for MeGraph model on BASHape (left) and BACommunity (right) datasets, varying the height  $h$  and the number of Mee layers  $n$ . A clear gap can be observed between heights 1 and  $\geq 2$ . The concrete number of accuracy can be found in Table 15.
F.7 Graph Theory Dataset
We provide a list of tables (from Table 19 to 30) showing the individual results of Table 1 for each possible graph generation method. Each table contains a list of variants of models and 5 tasks. Some graph generation methods and task combinations are trivial so we filter them out.
Table 14: Node Classification accuracy for MeGraph model on TreeCycle (above) and TreeGrid (below).
layer\height	1	2	3	4	5	6
1	61.48 ±6.04	76.59 ±4.41	91.48 ±2.70	98.52 ±1.69	97.95 ±2.32	98.52 ±1.35
2	67.27 ±6.91	81.59 ±4.03	97.39 ±1.25	98.98 ±0.94	98.75 ±1.29	98.86 ±1.14
3	74.43 ±3.60	90.80 ±2.61	98.64 ±1.11	99.09 ±1.11	98.75 ±1.56	99.09 ±0.85
4	79.55 ±4.34	93.41 ±2.82	99.20 ±0.73	99.20 ±0.89	99.66 ±0.73	99.20 ±1.69
5	82.73 ±4.06	93.41 ±1.89	99.43 ±1.05	99.20 ±1.35	99.32 ±1.16	99.32 ±0.56
6	83.18 ±3.51	94.09 ±2.02	99.43 ±0.76	99.09 ±0.85	99.20 ±1.69	99.20 ±0.89
7	84.43 ±3.74	94.43 ±2.24	99.89 ±0.34	99.20 ±1.02	99.20 ±0.89	99.66 ±0.73
8	84.20 ±3.82	94.20 ±2.00	98.98 ±1.19	99.32 ±0.75	99.66 ±0.52	99.20 ±0.73
9	84.43 ±3.87	94.20 ±2.06	99.77 ±0.45	99.20 ±1.02	98.98 ±1.07	99.32 ±0.75
10	84.77 ±3.98	94.43 ±2.18	99.32 ±0.75	98.86 ±1.14	99.09 ±1.67	99.66 ±0.52
layer\height	1	2	3	4	5	6
1	79.11 ±3.07	91.13 ±2.01	96.85 ±1.11	97.18 ±1.31	97.10 ±1.45	97.42 ±1.34
2	89.68 ±1.76	93.55 ±1.53	98.31 ±0.76	97.82 ±1.14	97.42 ±1.24	97.98 ±0.74
3	90.81 ±1.36	96.13 ±1.48	97.66 ±1.22	98.23 ±0.94	98.87 ±0.82	98.39 ±0.62
4	91.53 ±1.04	96.69 ±1.05	98.06 ±1.03	98.55 ±1.01	98.63 ±1.08	97.98 ±1.15
5	93.95 ±1.58	96.13 ±1.76	98.47 ±1.17	98.47 ±0.92	98.31 ±0.84	97.90 ±0.65
6	94.35 ±1.25	96.69 ±1.46	98.06 ±1.03	98.31 ±1.05	98.15 ±1.20	98.39 ±1.20
7	94.76 ±1.10	97.02 ±1.44	98.47 ±0.84	98.47 ±1.05	98.71 ±0.74	98.87 ±0.90
8	95.08 ±0.76	97.02 ±1.20	98.55 ±1.24	98.87 ±0.82	98.47 ±0.92	98.71 ±1.15
9	94.68 ±1.09	96.94 ±1.19	98.47 ±0.43	98.15 ±1.20	98.15 ±0.89	98.39 ±1.08
10	94.84 ±1.21	96.77 ±1.20	98.47 ±0.92	97.98 ±1.50	98.15 ±1.02	98.23 ±1.19
Table 15: Node Classification accuracy for MeGraph model on BASHape (above) and BACommunity (below).
layer\height	1	2	3	4	5	6
1	98.71 ±1.00	99.14 ±1.14	99.86 ±0.43	99.43 ±0.70	99.43 ±0.95	99.57 ±0.91
2	98.71 ±1.00	99.29 ±0.96	99.57 ±0.91	99.71 ±0.57	99.57 ±0.91	99.57 ±0.91
3	99.00 ±0.91	99.43 ±0.95	99.86 ±0.43	99.86 ±0.43	99.86 ±0.43	99.86 ±0.43
4	99.00 ±0.91	99.71 ±0.57	99.86 ±0.43	99.86 ±0.43	99.43 ±0.95	99.71 ±0.57
5	99.00 ±0.91	99.86 ±0.43	99.86 ±0.43	99.86 ±0.43	99.86 ±0.43	99.43 ±0.95
6	99.00 ±0.91	99.86 ±0.43	99.86 ±0.43	99.86 ±0.43	99.86 ±0.43	99.86 ±0.43
7	99.00 ±0.91	99.86 ±0.43	99.86 ±0.43	99.86 ±0.43	99.86 ±0.43	99.86 ±0.43
8	99.00 ±0.91	99.86 ±0.43	99.86 ±0.43	99.86 ±0.43	99.71 ±0.57	99.57 ±0.65
9	99.00 ±0.91	99.86 ±0.43	99.57 ±0.91	99.57 ±0.91	99.57 ±0.91	99.57 ±0.91
10	99.00 ±0.91	99.71 ±0.57	99.57 ±0.91	99.71 ±0.57	99.86 ±0.43	99.43 ±0.95
layer\height	1	2	3	4	5	6
1	94.93 ±1.30	97.00 ±1.80	96.93 ±1.60	97.00 ±1.88	97.21 ±1.70	96.86 ±1.67
2	97.93 ±0.87	98.36 ±0.72	98.79 ±0.46	98.79 ±0.46	98.57 ±0.55	98.50 ±1.03
3	98.07 ±0.91	98.64 ±0.87	98.86 ±0.91	98.86 ±0.80	98.50 ±0.98	98.93 ±0.80
4	98.21 ±0.97	98.86 ±0.65	98.86 ±0.80	98.79 ±0.64	99.00 ±0.73	99.07 ±0.64
5	98.50 ±0.87	98.86 ±0.91	99.07 ±0.64	99.21 ±0.67	99.14 ±0.70	99.00 ±0.73
6	98.71 ±0.83	98.64 ±0.87	99.07 ±0.64	99.14 ±0.70	99.07 ±0.85	99.14 ±0.70
7	98.29 ±0.91	98.86 ±0.65	99.07 ±0.56	98.79 ±0.56	98.79 ±0.72	98.86 ±0.57
8	98.43 ±0.77	99.00 ±0.47	99.14 ±0.53	98.93 ±0.58	99.14 ±0.29	99.14 ±0.43
9	98.79 ±0.79	99.07 ±0.56	99.21 ±0.50	99.00 ±0.73	99.29 ±0.45	99.36 ±0.50
10	98.86 ±0.73	98.93 ±0.80	99.21 ±0.87	99.14 ±0.70	99.00 ±0.73	99.29 ±0.64
Table 16: Comparison results of MeGraph with ATT and GATE on Graph Theory Benchmark.
Category	Model	SPsssd	MCC	Diameter	SPss	ECC
MeGraph w. ATT	h=1	2.990 ±3.411	3.346 ±3.228	44.41 ±36.33	16.39 ±13.57	29.04 ±27.59
h=5	0.594 ±0.903	1.706 ±1.409	3.256 ±2.956	1.018 ±1.071	14.80 ±17.09
h=5,ηv=0.3,τc=4	0.749 ±1.131	1.128 ±0.794	4.430 ±4.329	0.640 ±0.833	5.649 ±4.496
MeGraph w. GATE	h=1	4.144 ±4.181	0.908 ±0.934	6.343 ±7.152	13.94 ±12.78	19.73 ±19.47
h=5	0.809 ±0.993	0.660 ±0.601	2.506 ±2.639	0.669 ±0.546	7.508 ±7.558
h=5,ηv=0.3,τc=4	0.602 ±0.622	0.599 ±0.520	0.544 ±0.490	0.342 ±0.193	0.859 ±0.712
Table 17: Comparison results of MeGraph with ATT and GATE on GNN Benchmark.
ZINC	AQSOL	CIFAR10	MNIST	PATTERN	CLUSTER
MeGraph w. ATT (h=1)	0.4258 ±0.0054	1.1421 ±0.0270	69.890 ±0.209	97.570 ±0.168	78.232 ±0.827	59.497 ±0.207
MeGraph w. ATT (h=5)	0.3637 ±0.0116	1.0767 ±0.0105	69.925 ±0.631	97.860 ±0.098	83.798 ±0.885	68.930 ±0.68.76
MeGraph w. GATE (h=1)	0.3336 ±0.0036	1.0766 ±1.0556	64.200 ±0.586	96.812 ±0.205	85.391 ±0.029	59.321 ±0.290
MeGraph w. GATE (h=5)	0.2897 ±0.0291	1.0240 ±0.0098	64.935 ±0.829	97.290 ±0.140	86.611 ±0.041	67.122 ±3.323
Table 18: Comparison results of MeGraph with ATT and GATE on OGB-G.
Model	molhiv ↑	molbace ↑	molbbbp ↑	molclintox ↑	molsider ↑
MeGraph w. ATT (h=1)	77.33 ±0.78	74.83 ±4.87	64.74 ±1.14	86.34 ±1.04	58.12 ±0.53
MeGraph w. ATT (h=5)	77.15 ±1.37	76.13 ±3.85	68.68 ±2.07	87.17 ±0.76	58.03 ±1.58
MeGraph w. GATE (h=1)	76.35 ±0.70	76.36 ±1.55	65.97 ±1.98	87.12 ±0.74	58.11 ±1.29
MeGraph w. GATE (h=5)	78.14 ±0.91	78.90 ±1.29	66.53 ±0.74	89.02 ±2.56	59.58 ±1.88
Model	moltox21 ↑	moltoxcast ↑	molesol ↓	molfreesolv ↓	mollipo ↓
MeGraph w. ATT (h=1)	75.88 ±0.64	64.65 ±0.59	1.091 ±0.030	2.318 ±0.089	0.790 ±0.012
MeGraph w. ATT (h=5)	76.71 ±0.98	66.98 ±0.70	1.007 ±0.617	2.065 ±0.151	0.736 ±0.023
MeGraph w. GATE (h=1)	75.30 ±0.43	64.34 ±0.62	1.064 ±0.015	2.191 ±0.068	0.766 ±0.008
MeGraph w. GATE (h=5)	76.91 ±0.25	66.78 ±0.13	1.003 ±0.086	2.048 ±0.199	0.700 ±0.014
Table 19: Graph Theory Benchmark results on Grid graphs, all results are obtained using our codebase.
Category	Model	SPsssd	MCC	Diameter	SPss	ECC
Baselines (h=1)	n=1	6.60 ±0.541	1.50 ±0.050	22.49 ±1.36	26.74 ±0.347	20.99 ±0.232
n=5	4.18 ±0.737	1.29 ±0.124	5.04 ±1.26	15.54 ±0.155	20.32 ±0.326
n=10	3.70 ±0.422	1.33 ±0.100	0.737 ±0.116	7.24 ±0.243	20.32 ±0.422
MeGraph(h=5)	n=1	1.19 ±0.486	1.24 ±0.154	6.78 ±1.95	5.34 ±0.265	18.00 ±0.910
EdgePool(τc=2)	n=5	0.738 ±0.322	1.11 ±0.043	0.616 ±0.310	0.617 ±0.099	13.3 ±3.31
MeGraph S-EdgePool Variants (h=5, n=5)	τc=3	0.361 ±0.182	1.24 ±0.113	0.382 ±0.120	0.442 ±0.130	0.918 ±0.220
ηv=0.3	4.77 ±2.50	1.33 ±0.161	0.349 ±0.074	5.40 ±0.954	3.59 ±0.354
ηv=0.3, τc=4	0.745 ±0.316	1.35 ±0.168	0.385 ±0.180	0.552 ±0.113	0.622 ±0.100
ηv=0.5, τc=4	1.61 ±0.394	1.28 ±0.138	0.458 ±0.220	1.71 ±0.535	1.48 ±0.283
ηv=0.3, τc=4 (X-Pool)	1.03 ±0.365	1.50 ±0.142	0.626 ±0.216	1.70 ±0.185	3.44 ±0.991
ηv=0.3, τc=4 (w/o pw)	0.616 ±0.194	1.66 ±0.083	0.361 ±0.147	0.678 ±0.139	1.70 ±0.485
Graph-UNets	h=5,n=9,ηv=0.3,τc=4	0.969 ±0.643	2.18 ±0.381	0.111 ±0.091	0.773 ±0.086	0.548 ±0.131
Table 20: Graph Theory Benchmark results on Tree graphs. All results are obtained using our codebase.
Category	Model	\(SP_{sssd}\)	MCC	Diameter	\(SP_{ss}\)	ECC
Baselines (h=1)	n=1	5.21 ±0.209	1.28 ±0.050	3.77 ±1.22	17.16 ±0.168	24.63 ±0.427
n=5	3.34 ±0.375	0.405 ±0.089	0.504 ±0.109	7.66 ±0.325	18.11 ±1.85
n=10	3.16 ±0.252	0.338 ±0.046	0.100 ±0.059	2.28 ±0.209	14.93 ±0.800
MeGraph(h=5)	n=1	1.62 ±0.314	0.846 ±0.071	0.725 ±0.249	6.99 ±0.610	12.27 ±0.843
EdgePool(τc=2)	n=5	0.83 ±0.667	0.490 ±0.118	0.084 ±0.030	1.27 ±0.442	2.87 ±0.420
MeGraph S-EdgePool Variants (h=5, n=5)	τc=3	0.599 ±0.200	0.483 ±0.081	0.075 ±0.012	0.497 ±0.121	0.429 ±0.105
ηv=0.3	0.868 ±0.230	0.413 ±0.054	0.142 ±0.047	0.789 ±0.092	0.534 ±0.074
ηv=0.3, τc=4	0.615 ±0.209	0.418 ±0.024	0.081 ±0.017	0.440 ±0.106	0.436 ±0.097
ηv=0.5, τc=4	1.06 ±0.327	0.424 ±0.042	0.214 ±0.018	1.20 ±0.128	2.03 ±0.507
ηv=0.3, τc=4 (X-Pool)	0.666 ±0.118	0.596 ±0.067	0.182 ±0.057	1.22 ±0.281	1.11 ±0.122
ηv=0.3, τc=4 (w/o pw)	0.771 ±0.141	0.455 ±0.056	0.124 ±0.032	0.700 ±0.190	1.07 ±0.260
Graph-UNets	h=5,n=9,ηv=0.3,τc=4	0.873 ±0.247	0.667 ±0.043	0.804 ±0.284	0.606 ±0.123	1.00 ±0.221
Table 21: Graph Theory Benchmark results on Ladder graphs, all results are obtained using our codebase.
Category	Model	SPsssd	MCC	Diameter	SPss	ECC
Baselines (h=1)	n=1	5.06 ±0.330	1.73 ±0.249	1.17 ±0.149	13.20 ±0.126	20.10 ±0.583
n=5	0.692 ±0.204	0.734 ±0.106	1.39 ±0.078	5.02 ±0.876	19.81 ±0.669
n=10	0.257 ±0.078	0.691 ±0.119	1.55 ±0.069	1.60 ±0.194	20.40 ±0.995
MeGraph(h=5)	n=1	0.662 ±0.165	0.866 ±0.071	1.57 ±0.992	2.18 ±0.181	6.61 ±1.32
EdgePool(τc=2)	n=5	0.251 ±0.108	0.753 ±0.091	0.175 ±0.169	0.321 ±0.058	1.18 ±0.746
MeGraph S-EdgePool Variants (h=5, n=5)	τc=3	0.296 ±0.070	0.754 ±0.086	0.226 ±0.069	0.228 ±0.021	0.285 ±0.069
ηv=0.3	0.507 ±0.204	0.768 ±0.050	0.156 ±0.053	0.969 ±0.148	0.787 ±0.059
ηv=0.3, τc=4	0.297 ±0.113	0.712 ±0.059	0.095 ±0.046	0.180 ±0.026	0.225 ±0.043
ηv=0.5, τc=4	0.375 ±0.196	0.656 ±0.064	0.058 ±0.019	0.612 ±0.191	0.464 ±0.121
ηv=0.3, τc=4 (X-Pool)	0.442 ±0.108	0.742 ±0.047	0.158 ±0.074	0.710 ±0.076	0.765 ±0.089
ηv=0.3, τc=4 (w/o pw)	0.245 ±0.021	0.682 ±0.105	0.106 ±0.052	0.271 ±0.039	0.618 ±0.188
Graph-UNets	h=5,n=9,ηv=0.3,τc=4	0.339 ±0.11	0.797 ±0.138	0.013 ±0.005	0.287 ±0.023	0.230 ±0.054
Table 22: Graph Theory Benchmark results on Line graphs, all results are obtained using our codebase.
Category	Model	\(SP_{sssd}\)	MCC	Diameter	\(SP_{ss}\)	ECC
Baselines (h=1)	n=1	30.37 ±1.41	0.458 ±0.035	21.49 ±8.84	68.99 ±0.247	75.46 ±1.86
n=5	10.55 ±2.40	0.019 ±0.004	9.97 ±10.85	46.39 ±3.09	78.49 ±4.38
n=10	3.29 ±0.813	0.012 ±0.003	10.18 ±10.59	35.07 ±2.71	77.23 ±3.42
MeGraph(h=5)	n=1	1.45 ±0.598	0.056 ±0.014	7.62 ±4.43	10.13 ±2.33	45.19 ±8.64
EdgePool(\(τ_c\)=2)	n=5	0.536 ±0.149	0.016 ±0.007	0.611 ±0.238	1.06 ±0.341	14.12 ±13.82
MeGraph S-EdgePool Variants (h=5, n=5)	\(\tau_c\)=3	0.349 ±0.206	0.013 ±0.003	0.724 ±0.479	0.339 ±0.102	1.15 ±0.267
\(\eta_v\)=0.3	3.65 ±2.13	0.017 ±0.005	1.75 ±1.63	13.99 ±2.09	7.45 ±0.989
\(\eta_v\)=0.3, \(\tau_c\)=4	0.283 ±0.072	0.019 ±0.006	0.584 ±0.337	0.515 ±0.044	1.27 ±1.08
\(\eta_v\)=0.5, \(\tau_c\)=4	1.81 ±0.121	0.022 ±0.006	0.711 ±0.213	2.64 ±0.047	3.77 ±0.763
\(\eta_v\)=0.3, \(\tau_c\)=4 (X-Pool)	1.06 ±0.510	0.101 ±0.016	0.767 ±0.522	2.29 ±0.472	3.89 ±1.02
\(\eta_v\)=0.3, \(\tau_c\)=4 (w/o pw)	0.377 ±0.106	0.022 ±0.007	1.19 ±1.17	1.12 ±0.115	3.34 ±0.904
Graph-UNets	h=5,n=9,\(\eta_v\)=0.3,\(\tau_c\)=4	0.426 ±0.223	0.062 ±0.008	2.89 ±1.89	0.767 ±0.129	4.78 ±1.94
Table 23: Graph Theory Benchmark results on Caterpillar graphs, all results are obtained using our codebase.
Category	Model	\(SP_{sssd}\)	MCC	Diameter	\(SP_{ss}\)	ECC
Baselines (h=1)	n=1	24.24 ±1.57	1.25 ±0.082	28.62 ±2.55	19.08 ±0.208	35.32 ±0.462
n=5	8.32 ±2.10	0.561 ±0.070	4.59 ±0.346	9.62 ±0.357	37.01 ±1.48
n=10	6.40 ±0.652	0.630 ±0.127	5.06 ±0.499	4.06 ±0.297	37.87 ±3.22
MeGraph(h=5)	n=1	5.04 ±1.03	0.685 ±0.077	6.08 ±1.40	5.40 ±0.843	28.52 ±2.16
EdgePool(\(τ_c\)=2)	n=5	3.44 ±1.13	0.533 ±0.064	2.00 ±1.28	0.921 ±0.149	5.20 ±1.57
MeGraph S-EdgePool Variants (h=5, n=5)	\(\tau_c\)=3	2.47 ±0.529	0.607 ±0.081	0.591 ±0.172	0.574 ±0.073	1.21 ±0.148
\(\eta_v\)=0.3	3.61 ±1.36	0.582 ±0.052	0.578 ±0.231	1.69 ±0.572	1.95 ±0.322
\(\eta_v\)=0.3,\(\tau_c\)=4	1.59 ±0.444	0.535 ±0.091	0.317 ±0.104	0.474 ±0.170	1.32 ±0.272
\(\eta_v\)=0.5,\(\tau_c\)=4	2.00 ±0.648	0.514 ±0.040	1.10 ±0.288	0.986 ±0.130	2.11 ±0.766
\(\eta_v\)=0.3,\(\tau_c\)=4 (X-Pool)	1.39 ±0.478	0.602 ±0.110	0.736 ±0.230	1.78 ±0.254	3.36 ±0.873
\(\eta_v\)=0.3,\(\tau_c\)=4 (w/o pw)	1.82 ±0.627	0.628 ±0.093	0.604 ±0.067	0.797 ±0.299	2.25 ±0.230
Graph-UNets	h=5,n=9,\(\eta_v\)=0.3,\(\tau_c\)=4	1.57 ±0.670	0.679 ±0.098	3.18 ±0.583	0.976 ±0.270	3.83 ±1.06
Table 24: Graph Theory Benchmark results on Lobster graphs, all results are obtained using our codebase.
Category	Model	\(SP_{sssd}\)	MCC	Diameter	\(SP_{ss}\)	ECC
Baselines (h=1)	n=1	23.92 ±0.319	1.06 ±0.166	11.93 ±1.32	38.44 ±0.065	40.46 ±0.350
n=5	10.89 ±1.47	0.544 ±0.067	3.66 ±0.424	20.12 ±0.105	28.81 ±1.14
n=10	7.35 ±2.50	0.631 ±0.067	2.59 ±0.517	10.52 ±0.619	28.47 ±1.65
MeGraph(h=5)	n=1	6.00 ±1.82	0.785 ±0.062	4.35 ±1.51	13.75 ±0.675	30.49 ±2.18
EdgePool(\(τ_c\)=2)	n=5	1.93 ±0.861	0.543 ±0.073	1.07 ±0.114	2.05 ±0.393	11.39 ±5.43
MeGraph S-EdgePool Variants (h=5, n=5)	\(\tau_c\)=3	2.02 ±0.791	0.447 ±0.123	0.705 ±0.133	1.66 ±0.270	2.23 ±0.378
\(\eta_v\)=0.3	6.01 ±1.52	0.521 ±0.028	0.707 ±0.202	3.04 ±0.250	2.70 ±0.212
\(\eta_v\)=0.3,\(τ_c\)=4	1.90 ±0.449	0.489 ±0.069	0.671 ±0.165	1.30 ±0.106	2.62 ±0.849
\(\eta_v\)=0.5,\(τ_c\)=4	3.27 ±0.716	0.451 ±0.090	0.941 ±0.324	2.82 ±0.803	4.04 ±0.527
\(\eta_v\)=0.3,\(τ_c\)=4 (X-Pool)	2.67 ±0.486	0.494 ±0.109	1.01 ±0.194	2.79 ±0.343	4.16 ±0.886
\(\eta_v\)=0.3,\(τ_c\)=4 (w/o pw)	1.85 ±0.432	0.473 ±0.069	0.892 ±0.277	1.77 ±0.329	4.33 ±1.71
Graph-UNets	h=5,n=9,\(\eta_v\)=0.3,\(τ_c\)=4	4.85 ±1.48	0.782 ±0.026	3.74 ±0.361	2.96 ±0.443	4.25 ±0.544
Table 25: Graph Theory Benchmark results on Cycle graphs, all results are obtained using our codebase.
Category	Model	\(SP_{sssd}\)	MCC	Diameter	\(SP_{ss}\)	ECC
Baselines (h=1)	n=1	18.75 ±0.066	0.534 ±0.022	22.35 ±0.149	24.07 ±0.009	21.47 ±0.060
n=5	3.39 ±0.304	0.027 ±0.001	25.11 ±0.325	12.44 ±1.05	21.81 ±0.102
n=10	0.352 ±0.060	0.011 ±0.003	26.54 ±1.16	8.65 ±1.02	24.09 ±0.360
MeGraph(h=5) EdgePool(\(τ_c\)=2)	n=1	0.594 ±0.212	0.074 ±0.029	9.11 ±1.88	4.07 ±0.364	21.53 ±0.070
n=5	0.060 ±0.032	0.014 ±0.003	13.44 ±6.40	0.103 ±0.016	24.05 ±0.204
MeGraph S-EdgePool Variants (h=5, n=5)	\(\tau_c\)=3	0.066 ±0.036	0.015 ±0.006	0.241 ±0.049	0.090 ±0.037	0.342 ±0.186
\(\eta_v\)=0.3	2.45 ±0.873	0.015 ±0.001	0.709 ±0.226	8.36 ±0.261	0.488 ±0.267
\(\eta_v\)=0.3,\(τ_c\)=4	0.060 ±0.030	0.019 ±0.003	0.312 ±0.236	0.226 ±0.050	0.562 ±0.209
\(\eta_v\)=0.5,\(τ_c\)=4	0.451 ±0.203	0.014 ±0.004	0.252 ±0.124	1.05 ±0.524	4.30 ±1.90
\(\eta_v\)=0.3,\(τ_c\)=4 (X-Pool)	0.494 ±0.292	0.096 ±0.028	0.468 ±0.220	1.08 ±0.130	0.860 ±0.292
\(\eta_v\)=0.3,\(τ_c\)=4 (w/o pw)	0.159 ±0.209	0.017 ±0.008	1.23 ±0.928	0.461 ±0.118	8.26 ±3.70
Graph-UNets	h=5,n=9,\(\eta_v\)=0.3,\(τ_c\)=4	0.144 ±0.073	0.035 ±0.010	3.21 ±0.893	0.439 ±0.089	5.91 ±1.31