SlowGuess/NeurIPS_2023_OCR huggingface dataset:
The GNN architecture used in all experiments consists of: [2 GIN layers] - [1 pooling layer with pooling ratio 0.1] - [1 GIN layer] - [global_sum_pool] - [dense readout]. Each GIN layer is configured with an MLP with 2 hidden layers of 64 units and ELU activation functions. The readout is a 3-layer MLP with units [64, 64, 32], ELU activations, and dropout 0.5. The GNN is trained with Adam optimizer with an initial learning rate of 1e-4 using batches with size 32. For SAGPool or ASAP we used only one GIN layer before pooling. For PanPool we used 2 PanConv layers with filter size 2 instead of the first 2 GIN layers. The auxiliary losses in DiffPool, MinCutPool, and DMoN are added to the cross-entropy loss with weights [0.1,0.1], [0.5, 1.0], [0.3, 0.3, 0.3], respectively. For  $k$ -MIS we used  $k = 5$  and we aggregated the features with the sum. For Graclus, we aggregated the node features with the sum.
C.3 EXPWL1 with different pooling ratios
In Table 3 we report the classification results obtained with different pooling ratios. Since in  $k$ -MIS we cannot specify the pooling ratio directly, we report the results obtained for  $k = 3,5,6$  that gives approximately a pooling ratio of 0.19, 0.09, and 0.07 respectively. As we discussed in the experimental section, Graclus, Cmp-Gracus, and ECPool roughly reduce the graph size by half. Therefore, to obtain the desired pooling ratio we apply them recursively. This creates a conspicuous computational overhead, especially in the case of ECPool which triggers an out-of-memory error for lower pooling ratios. Importantly, the results do not change significantly for the expressive pooling methods, while we notice a drastic improvement in the performance of the non-expressive ones for higher pooling ratios. Such sensitivity to different pooling ratios further highlights the practical difference between expressive and non-expressive operators.
Pooling	Pool Ratio 0.05	Pool Ratio 0.1	Pool Ratio 0.2
DiffPool	95.2±2.1	97.0±2.4	97.4±2.2
DMoN	98.5±1.1	99.0±0.7	98.1±1.7
MinCut	98.3±0.7	98.8±0.4	98.4±0.6
ECPool	OOM	OOM	100.0±0.0
Graclus	100.0±0.0	99.9±0.1	99.9±0.1
k-MIS	99.8±0.2	99.9±0.1	100.0±0.0
Top-k	69.7±7.1	67.9±13.9	89.4±10.5
PanPool	61.3±5.5	63.2±7.7	75.4±12.8
ASAPool	84.3±2.5	83.5±2.5	87.3±7.2
SAGPool	61.6±10.6	79.5±9.6	82.4±11.1
Rand-dense	91.3±1.9	91.7±1.3	91.6±0.8
Cmp-Graclus	91.7±1.1	91.9±1.2	92.1±1.4
Rand-sparse	59.6±3.3	62.8±1.8	67.1±2.3
C.4 Statistics of the datasets
Table 4 reports the information about the datasets used in the experimental evaluation. Since the COLLAB and REDDIT-BINARY datasets lack vertex features, we assigned a constant feature value of 1 to all vertices.
Table 3: Classification results on EXPWL1 using different pooling ratios.
Dataset	#Samples	#Classes	Avg. #vertices	Avg. #edges	Vertex attr.	Vertex labels
EXPWL1	3,000	2	76.96	186.46	-	yes
NCI1	4,110	2	29.87	64.60	-	yes
Proteins	1,113	2	39.06	72.82	1	yes
COLORS-3	10,500	11	61.31	91.03	4	no
Mutagenicity	4,337	2	30.32	61.54	-	yes
COLLAB	5,000	3	74.49	4,914.43	-	no
REDMIT-B	2,000	2	429.63	995.51	-	no
B-hard	1,800	3	148.32	572.32	-	yes
MUTAG	188	2	17.93	19.79	-	yes
PTC_MR	344	2	14.29	14.69	-	yes
IMDB-B	1,000	2	19.77	96.53	-	no
IMDB-MULTI	1,500	3	13.00	65.94	-	no
ENZYMES	600	6	32.63	62.14	18	yes
REDMIT-5K	4,999	5	508.52	594.87	-	no
C.5 Detailed performance on the benchmark datasets
Table 4: Details of the graph classification datasets.
DiffPool	DMoN	MinCut	ECPool	Graclus	k-MIS	Top-k	PanPool	ASAPool	SAGPool
0.96s	0.88s	1.02s	18.85s	6.44s	0.75s	0.87s	8.89s	1.29s	0.76s
71.4±3.7	72.3±3.4	72.8±3.7	72.4±3.5	71.6±3.8	72.2±3.3	66.1±5.2	61.6±5.2	63.9±4.7	62.6±4.9
Table 5: Average run-time in seconds per epoch (first row) and average classification accuracy (second row) achieved by the different pooling methods on the benchmark datasets.
The test accuracy of the GNNs configured with the different pooling operators on the graph classification benchmarks is reported in Table 6, while Table 7 reports the run-time of each model expressed in seconds per epoch. The overall average accuracy and average run-time computed across all datasets are presented in Table 5. For each dataset, we use the same GNN configured as described in C.2, as we are interested in validating the architecture used to classify EXPWL1. Clearly, by using less aggressive pooling, by fine-tuning the GNN models, and by increasing their capacity it is possible to improve the results on several datasets. Such results are reported in the original papers introducing the different pooling operators.
Pooling	NCI1	PROTEINS	COLORS-3	Mutagenity	COLLAB	REDMIT-B	B-hard
DiffPool	77.8±3.9	72.8±3.3	87.6±1.0	80.0±1.9	76.6±2.5	89.9±2.8	70.2±1.5
DMoN	78.5±1.4	73.1±4.6	88.4±1.4	81.3±0.3	80.9±0.7	91.3±1.4	71.1±1.0
MinCut	80.1±2.6	76.0±3.6	88.7±1.6	81.2±1.9	79.2±1.5	91.9±1.8	71.2±1.1
ECPool	79.8±3.3	69.5±5.9	81.4±3.3	82.0±1.6	80.9±1.4	90.7±1.7	74.5±1.6
Graclus	81.2±3.4	73.0±5.9	77.6±1.2	81.9±1.6	80.4±1.5	92.9±1.7	72.3±1.3
k-MIS	77.6±3.0	75.9±2.9	82.9±1.7	82.6±1.2	73.7±1.4	90.6±1.4	71.7±0.9
Top-k	72.6±3.1	73.2±2.7	57.4±2.5	74.4±4.7	77.9±2.1	87.4±3.5	68.1±7.7
PanPool	66.1±2.3	75.2±6.2	40.7±11.5	67.2±2.0	78.2±1.5	83.6±1.9	44.2±8.5
ASAPool	73.1±2.5	75.5±3.2	43.0±4.7	76.5±2.8	78.4±1.6	88.0±5.6	67.5±6.1
SAGPool	79.1±3.0	75.2±2.7	43.1±11.1	77.9±2.8	78.1±1.8	84.5±4.4	54.0±6.6
Rand-dense	78.2±2.0	75.3±1.3	83.3±0.9	81.4±1.8	69.3±1.6	89.3±2.1	71.0±2.2
Cmp-Graclus	77.8±1.8	73.6±4.7	84.7±0.9	80.7±1.8	OOM	OOM	OOM
Rand-sparse	69.1±3.3	74.6±4.2	35.5±1.1	69.8±1.0	68.8±1.6	84.5±1.9	50.1±4.0
Pooling	MUTAG	PTC_MR	IMDB-B	IMDB-MULTI	ENZYMES	REDDIT-5K
DiffPool	86.8±9.7	54.7±6.1	71.3±3.1	45.2±3.4	62.3±7.3	53.7±1.8
DMoN	86.3±7.1	55.5±7.3	71.9±3.3	47.0±5.5	61.0±5.0	56.6±2.3
MinCut	83.1±9.6	57.9±7.7	71.9±5.7	46.6±4.0	62.3±3.8	56.2±2.8
ECPool	90.0±7.2	54.7±8.4	71.3±3.4	49.2±2.9	59.6±3.7	53.6±2.2
Graclus	85.2±8.0	55.2±6.4	72.3±5.8	46.2±4.4	61.0±6.6	52.3±1.4
k-MIS	85.7±6.2	59.7±5.7	73.1±4.2	46.8±4.6	63.5±7.1	56.4±2.3
Top-k	78.4±11.8	58.2±8.9	70.9±3.3	44.8±2.9	45.5±10.5	50.4±3.7
PanPool	83.1±13.2	53.5±7.7	73.9±3.5	48.3±3.7	40.5±5.0	46.5±2.4
ASAPool	74.2±6.8	50.5±12.1	71.4±2.8	46.1±4.2	44.8±7.6	48.8±1.6
SAGPool	73.7±6.6	58.8±8.0	71.0±4.0	44.0±3.4	41.6±5.2	49.9±2.9
Rand-dense	88.9±4.3	56.1±9.7	70.5±3.4	45.2±5.6	62.1±5.0	54.5±2.1
Cmp-Graclus	83.2±9.1	55.9±4.6	OOM	OOM	63.5±5.0	OOM
Rand-sparse	68.9±17.3	56.4±5.9	71.6±3.6	45.8±3.7	62.1±5.0	50.6±2.4
Table 6: Graph classification test accuracy on benchmark datasets.
Pooling	NCI1	PROTEINS	COLORS-3	Mutagenity	COLLAB	REDBIT-B	B-hard
DiffPool	0.83s	0.23s	1.67s	0.90s	1.68s	1.74s	0.29s
DMoN	1.01s	0.28s	1.94s	1.06s	1.83s	1.04s	0.33s
MinCut	0.95s	0.28s	1.82s	1.10s	1.82s	1.78s	0.35s
ECPool	4.39s	1.97s	10.30s	4.22s	44.11s	3.17s	6.90s
Graclus	0.95s	0.27s	2.47s	0.98s	3.01s	0.75s	0.31s
k-MISPool	0.88s	0.25s	2.48s	0.95s	1.38s	0.48s	0.43s
Top-k	1.04s	0.29s	2.78s	1.04s	2.79s	0.47s	0.30s
PanPool	2.81s	0.81s	7.16s	5.48s	7.67s	46.15s	6.27s
ASAPool	1.83s	0.52s	4.48s	1.80s	3.97s	0.79s	0.52s
SAGPool	1.09s	0.30s	2.52s	1.07s	2.81s	0.43s	0.28s
Rand-dense	0.54s	0.14s	1.44s	0.55s	0.88s	1.44s	0.26s
Cmp-Graclus	7.94s	3.27s	34.05s	1.94s	-	-	-
Rand-sparse	0.64s	0.18s	1.72s	0.68s	1.00s	0.47s	0.31s
Pooling	MUTAG	PTC_MR	IMDB-B	IMDB-MULTI	ENZYMES	REDMIT-5K
DiffPool	0.04s	0.07s	0.14s	0.28s	0.12s	4.52s
DMoN	0.05s	0.09s	0.17s	0.37s	0.15s	3.21s
MinCut	0.04s	0.08s	0.16s	0.35s	0.14s	4.45s
ECPool	0.08s	0.12s	1.66s	1.01s	0.54s	37.13s
Graclus	0.09s	0.14s	0.24s	0.33s	0.14s	74.02s
k-MIS	0.06s	0.10s	0.28s	0.21s	0.12s	2.02s
Top-k	0.04s	0.08s	0.24s	0.32s	0.13s	1.75s
PanPool	0.26s	0.49s	1.30s	1.92s	0.98s	34.3s
ASAPool	0.14s	0.10s	0.43s	0.40s	0.16s	1.89s
SAGPool	0.04s	0.05s	0.23s	0.27s	0.08s	1.09s
Rand-dense	0.04s	0.06s	0.13s	0.19s	0.12s	3.48s
Cmp-Graclus	0.17s	0.47s	-	-	1.38s	-
Rand-sparse	0.04s	0.08s	0.15s	0.23	0.14s	1.67s
Table 7: Graph classification test run-time in s/epoch.