SlowGuess/NeurIPS_2023_OCR huggingface dataset:
k-OSAN [26] proposes a data-driven subgraph sampling strategy to find informative subgraphs by another MPNN. This strategy reduces to random sampling when the graph requires higher expressivity (e.g., regular graphs) and has no features because the MPNN will generate the same embedding for all nodes and hence cannot identify subgraphs that most benefit the prediction like MAG-GNN can. MAG-GNN does not solely depend on the data and finds more expressive subgraphs even without node features. Moreover, sampled subgraphs in previous methods are essentially independent. In contrast, MAG-GNN also models their correlation using RL. This allows MAG-GNN to obtain better expressivity with fewer samples and better consistency (More discussions in Appendix C.3).
4.1 Training MAG-GNN
With the state and action space, reward, and Q-network defined, we can use any DQN techniques to train the RL agent. However, to evaluate the framework's capability, we select the basic Experience Replay method [19] to train the Q-network. MAG-GNN essentially consists of two systems, an RL agent and a prediction MPNN. Making them coordinate is more critical to the method. The most natural way to train our system is first to train  $g_{p}$ , as introduced in Section 3 with random node tuples. We then use  $g_{p}$  as part of  $f_{p}$ , the marked-graphs encoder, and treat  $f_{p}$  as the fixed environment to train our Q-network. The advantage of the paradigm is that the environment is stable. Consequently, all experiences stored in the memory have the correct reward value for the action. This encourages stability and fast convergence during RL training. We term this paradigm ORD for ordered training.
However, not all  $g_{p}$  trained from random node tuples are good. When we train  $g_{p}$  for the ZINC dataset and evaluate all node-marked graphs to as in Section 3. The average standard deviation among all scores of node-marked graphs is  $\sim 0.003$ , and the average difference between the worst and best score is  $\sim 0.007$ . Hence, the maximal improvement is minimal if we use this MPNN as the environment. Intuitively, when the graph has informative initial features,  $X$ , like those in the ZINC dataset, the MPNN quickly recognizes patterns from these features while gradually learning to ignore node marking features  $X(\boldsymbol{v}_{i})$ , as not all node markings carry helpful information. In such cases, we need to adjust  $g_{p}$  while the RL agents become more powerful in finding discriminative subgraphs.
One way is to train the RL agent and  $g_{p}$  simultaneously. Concretely, we sample a state  $s$ , run the RL agent  $t$  steps to update it to state  $s^t$ , and train  $g_{p}$  on the marked graphs of  $s^t$ . Then, in the same step, the RL agent samples a different state and treats the current  $g_{p}$  as the environment to generate experience. Lastly, the RL agent is optimized on the sampled previous experiences. Because we adjust  $g_{p}$  to capture node tuple information better, the score standard deviation of the node-marked ZINC graphs is kept at  $\sim 0.036$ . We term this paradigm SIMUL. Compared to ORD, SIMUL makes
the RL agent more difficult to train when  $g_{p}$  evolves rapidly. Nevertheless, we observe that as  $g_{p}$  gradually becomes stable, the RL agent can still learn effective policies.
One of the critical goals of MAG-GNN is to identify complex structural information that MPNN cannot. Hence, instead of training the agent on real-world datasets from scratch, we can transfer the knowledge from synthetic expressivity data to real-world data. As mentioned above, training MAG-GNN on graphs with features is difficult. Alternatively, we first use the ORD paradigm to train the agent on expressivity data without node features such as SR25. On these datasets,  $g_{p}$  relies on the node markings to make correct predictions. We then fix the RL agent and only use the output state from the agent to train a new  $g_{p}$  for the actual tasks, such as ZINC graph regression. Using this paradigm, we only need to train one MAG-GNN with good expressivity and adapt it to different tasks without worrying about overfitting and the aforementioned stability issue, we name this paradigm PRE. We experimented with different schemes in Section 6.
More expressive GNNs. A substantial amount of work strive to improve the expressivity of GNNs. They can be classified into the following categories: (1) Just like MPNN simulates the 1-WL test, Higher-order GNNs design GNNs to simulate higher-order WL tests. They include k-GNN [18], RingGNN [5], and PPGN [20]. These methods perform message passing on node tuples and have complexity that grows exponentially with  $k$  and hence does not scale well to large graphs. (2) Realizing the symmetry-breaking power of subgraphs, Subgraph GNNs, including NGNN [35], GNN-AK [37], KPGNN [8], and ID-GNN [33], use MPNNs to encode subgraphs instead of subtrees around graph nodes. Later works, like  $\mathrm{I}^2$ -GNN [15], further use 2-order(edge) subgraph information to improve the expressivity of subgraph GNNs beyond 3-WL. Recent works, such as OSAN [26] and SUN [10], unify subgraph GNNs into the WL hierarchy, showing improvement in expressivity for subgraph GNNs also requires exponentially more subgraphs. (3) Substructure counting methods, including GSN [3] and MotifNet [22], employ substructure counting in GNNs. They count predefined substructures undetectable by the 1-WL test as features for MPNN to break its expressivity limit. However, the design of the relevant substructures usually requires human expertise. (4) Many previous works also realize the complexity issue of more expressive GNNs and strive to reduce it. SetGNN [38] proposes to reduce node tuple to set and thus reduce the number of nodes in message passing. GDGNN [16] designs geodesic pooling functions to have strong expressivity without running MPNN multiple times. (5) Non-equivariant GNNs. Abboud et al. [1] proves the universality of MPNN with randomly initialized node features, but due to the large search space, such expressivity is hardly achieved. DropGNN [24] randomly drops out nodes from the graph to break symmetries in graphs. PF-GNN [7] implements a neural version of canonical labeling and uses particle filtering to sample branches in the labeling process. OSAN [26] proposes to use input features to select important subgraphs. Agent-based GNN [21] initializes multiple agents on a graph without the message-passing paradigm to iteratively update node embeddings. MAG-GNN also falls into this category.
Reinforcement Learning and GNN. There has been a considerable effort to combine RL and GNN. Most work is on application. GraphNAS [11] uses GNN to encode neural architecture and use reinforcement learning to search for the best network. Wang et al. [30] uses GNN to model circuits and RL to adjust the transistor parameters. On graph learning, most works use RL to optimize particular parameters in GNN. For example, SUGAR [27] uses Q-learning to learn the best top-k subgraphs for aggregations; Policy-GNN [17] learns to pick the best number of hops to aggregate node features. GPA [13] uses Deep Q-learning to locate the valuable nodes in active search. These works leverage the node feature to improve the empirical performance but fail to identify graphs with symmetries, while MAG-GNN has good expressivity without node features. To the best of our knowledge, our work is the first to apply reinforcement learning to the graph expressivity problem.
In the experiment  $^{1}$ , we answer the following questions: Q1: Does MAG-GNN have good expressivity, and is the RL agent output more expressive than random ones? Q2: MAG-GNN is graph-level;
<sup>1</sup>The code can be found at https://github.com/LechengKong/MAG-GNN
Table 1: Synthetic results. (↑)
EXP	CSL	SR25
GIN [32]	50.0	10.0	6.7
RNI [1]	99.7	16.0	6.7
NGNN [35]	100	100	6.7
GNNAK+ [37]	100	100	6.7
SSWL+ [36]	100	100	6.7
RNM	100	100	93.8
I²GNN [15]	100	100	100
MAG-GNN	100	100	100
Table 2: Cycle counting results. (↓)
Method	3-CYCLES	4-CYCLES	5-CYCLES	6-CYCLES
GIN [32]	0.3515	0.2742	0.2088	0.1555
RNM	0.0041	0.0129	0.0351	0.0327
ID-GNN [33]	0.0006	0.0022	0.0490	0.0495
NGNN [35]	0.0003	0.0013	0.0402	0.0439
GNNAK+ [37]	0.0004	0.0041	0.0133	0.0238
I²GNN [15]	0.0003	0.0016	0.0028	0.0082
MAG-GNN	0.0029	0.0037	0.0097	0.0286
can the expressivity generalize to node-level tasks? Q3: How is the RL method's performance on real-world datasets? Q4: Does MAG-GNN have the claimed computational advantages?
For the experiment, we update all node tuples simultaneously as it allows more parallelizable computation. More experiment and dataset details can be found in Appendix F.
6.1 Discriminative power
Dataset. To answer Q1, we use synthetic datasets (Accuracy) to test the expressivity of models. (1) EXP contains 600 pairs of non-isomorphic graphs that cannot be distinguished by 1-WL/2-WL bounded GNN. The task is to differentiate all pairs. (2) SR25 contains 15 strongly regular graphs with the same configuration and cannot be distinguished by 3-WL bounded GNN. (3) CSL contains 150 circular skip link graphs in 10 isomorphism classes. (4) BREC contains 400 pairs of synthetic graphs to test the fine-grained expressivity of GNNs (Appendix E). We use the ORD training scheme.
Results. We compare to MPNN [32], Random Node Marked (RNM) GNN with the same hyperparameter search space as MAG-GNN, subgraph GNNs[35, 15, 36, 37], and Nonequivariant GNNs [1] as baselines. Table 1 shows that MAG-GNN achieved a perfect score on all datasets, which verifies our observation and theory. Note that subgraph GNNs like NGNN take at least  $|V|$  MPNN runs, while MAG-GNN takes constant times of MPNN runs. However, MAG-GNN successfully distinguished all strongly regular graphs in SR25, which NGNN failed. RNI, despite being universal, is challenging to train and can only make random guesses on SR25. Compared to the RNM approach, MAG-GNN finds more representative subgraphs for the SR25 dataset and performs better. Figure 4 shows a graph in the EXP dataset. We observe that MAG-GNN moves the initial node marking on the same component to different components, allowing better structure learning. Another example of strongly regular graphs is in Appendix E. In Figure
5, we also plot the performance of MAG-GNN against the number of MAG-GNN search steps when we only use one node tuple of size two. We see that node tuples from MAG-GNN are significantly more expressive than random node tuples (step 0). On EXP and CSL datasets, MAG-GNN can achieve perfect scores in one or two steps, whereas in SR25, it takes six steps but with a consistent performance increase over steps. We plot the reward curve during training in Appendix E.
Figure 4: Initial Random Node Marking (Left). MAG-GNN generated markings. (Right)
Figure 5: Performance versus the number of RL steps.
6.2 Node-level tasks.
Datasets. To answer Q2, we adopt the synthetic cycle counting dataset, CYCLE (MAE), in Zhao et al. [37]. The task is to count the number of  $(3,4,5,6)$ -cycles on each node. MPNN cannot count these structures. Node-rooted GNN can count  $(3,4)$ -cycles, while only models with expressivity beyond 3-WL can count all cycles. We use the ORD training scheme.
Table 5: QM9 experimental results on all targets. (↓)
Target	RNM	1-2-3-GNN [23]	PPGN [20]	NGNN [35]	I²-GNN [15]	MAG-GNN
Comp.	O(kT|V²|)	O(T|V4|)	O(T|V3|)	O(T|V3|)	O(T|V4|)	O(mtT|V2|)
μ	0.426	0.476	0.231	0.428	0.428	0.353
α	0.306	0.27	0.382	0.230	0.230	0.226
εHOMO	0.00258	0.00337	0.00276	0.00265	0.00261	0.00257
εLUMO	0.00269	0.00351	0.00287	0.00297	0.00267	0.00252
Δε	0.0047	0.0048	0.00406	0.0038	0.0038	0.0035
〈R²〉	20.9	22.9	16.07	20.5	18.64	15.44
ZPVE	0.0002	0.00019	0.0064	0.0002	0.00014	0.0002
U0	0.281	0.0427	0.234	0.295	0.211	0.111
U	0.193	0.111	0.234	0.361	0.206	0.105
H	0.384	0.0419	0.229	0.305	0.269	0.089
G	0.250	0.0469	0.238	0.489	0.261	0.116
Cv	0.177	0.0944	0.184	0.174	0.073	0.093
Results. Following Huang et al. [15], we say a model has the required counting power if the error is below 0.01 and report the result in Table 2. We compare to MPNN baseline [32], RNM GNN, node-level subgraph GNNs [33, 37, 35], and edge-level subgraph GNNs [15]. We can see that MAG-GNN successfully counts (3,4,5)-cycles, which indicates that node-marking also helps non-marked nodes to count cycles. We also note that MAG-GNN does not count 6-cycles well, although we use  $>2$ -order node tuples. We suspect this is because MAG-GNN takes the average score improvement of nodes as the reward, which might not be the best reward for a node-level task. Even so, we can still observe MAG-GNN's performance improvement over NGNN, which shows that MAG-GNN with larger node tuples indeed increases expressivity. We leave the design of a more proper node-level reward to future works.
6.3 Real-world datasets
Datasets. To answer Q3, we adopt the following real-world molecular property prediction datasets: (1) QM9 (MAE) contains 130k molecules for twelve graph regression targets. (2) ZINC and ZINC-FULL (MAE) include 12k and 250k chemical compounds for graph regression. (3) OGBG-MOLHIV (AUROC) contains 41k molecules, and the goal is to classify whether a molecule inhibits HIV. We use the SIMUL training scheme for a fair comparison to other methods.
Results. On QM9 targets, MAG-GNN significantly outperforms NGNN on all targets with an average of  $33\%$  MAE reduction. It also outperforms  $\mathrm{I}^2$ -GNN, with partially  $>3$ -WL expressivity, on most of the targets (16% average MAE reduction), showing that with much fewer MPNN runs, MAG-GNN can still achieve better performance. This is because despite
using fewer subgraphs, we use node tuples with a size greater than two, which grants MAG-GNN the power to distinguish graphs that require higher expressivity. MAG-GNN also performs comparably to 1-2-3-GNN, which simulates 3-WL. We observe that 1-2-3-GNN performs well on the last five targets (which are global properties hard for subgraph GNNs) while outcompeted by MAG-GNN on the rest of the targets. We suspect that 1-2-3-GNN has a constant high expressivity, which might easily lead to overfitting during training. At the same time, MAG-GNN can automatically adjust the expressivity by subgraph selection, reducing the risk of overfitting.
The results on other molecular datasets are shown in Table 6. We see that MAG-GNN outperforms base GNN by a large margin showing its better expressivity. Another important comparison is between MAG-GNN and other non-equivariant GNNs, including PF-GNN, k-OSAN, and RNM. We see that MAG-GNN achieves significantly better results on ZINC, where only models with high expressivity get good performance. This verifies that using an RL agent to capture the inter-subgraph relation is essential in finding expressive subgraphs. MAG-GNN does not perform as well on the
Table 3: Transfer learning.
ZINC↓	ZINC-FULL↓	MOLHIV↑
No PRE	0.106	0.030	77.12
3-CYCLES	0.104	0.030	76.98
6-CYCLES	0.096	0.023	78.30
SR25	0.103	0.028	77.26
Table 4: Inference time.
Time (ms)	ZINC-FULL	CYCLE	QM
MPNN	100.1	58.4	222.9
NGNN	402.9	211.7	776.8
I²GNN	1864.1	1170.4	3524.0
PPGN	2097.3	1196.8	4108.7
MAG-GNN	385.8	155.1	704.9
Table 6: Molecular datasets results.  $(\downarrow)$
# Params	ZINC (↓)	ZINC-FULL (↓)	OGBG-MOLHIV (↑)
GIN	-	0.163±0.004	0.088±0.002	77.07±1.49
PNA	-	0.188±0.004	-	79.05±1.32
k-OSAN	-	0.155±0.004	-	-
PF-GNN	-	0.122±0.010	-	80.15±0.68
RNM	453k	0.128±0.027	0.062±0.004	76.79±0.94
GSN	-	0.115±0.012	-	78.80±0.82
CIN	~100k	0.079±0.006	0.022±0.002	80.94±0.57
NGNN	~500k	0.111±0.003	0.029±0.001	78.34±1.86
GNNAK+	~500k	0.080±0.001	-	79.61±1.19
SUN	526k	0.083±0.003	-	80.03±0.55
KPGNN	489k	0.093±0.007	-	-
I²GNN	-	0.083±0.001	0.023±0.001	78.68±0.93
SSWL+	387k	0.070±0.005	0.022±0.002	79.58±0.35
MAG-GNN	496k	0.106±0.014	0.030±0.002	77.12±1.13
MAG-GNN-PRE	496k	0.096±0.009	0.023±0.002	78.30±1.08
OGBG-MOLHIV dataset. We observe that 1-WL bounded methods PNA also achieves good results on the datasets, meaning that the critical factor determining the performance on this dataset is likely the implementation of the base GNN but not the expressivity. MAG-GNN is highly adaptable to any base GNN, potentially improving MAG-GNN's performance further. We leave this to future work.
We use the PRE training scheme to conduct transfer learning tasks on ZINC, ZINC-FULL, and OGBG-MOLHIV datasets. We pre-train the expressivity datasets shown on the left column of Table 3 and train the attached MPNN head using the datasets on the top row. We see that pretraining consistently brings performance improvement to all datasets. Models pre-trained on CYCLE are generally better than the one pretrained on SR25, possibly due to the abundance of cycles in molecular graphs.
6.4 Runtime comparison
We conducted runtime analysis on previously mentioned datasets. Since it is difficult to match the number of parameters for all models strictly, we fixed the number of GNN layers to five and the embedding dimension to 100. We set a 1 GB memory budget for all models and measured their inference time on the test datasets. We use  $m = 2$  and  $T = 2$  for MAG-GNN. The results in Table 4 show that MAG-GNN is more efficient than all subgraph GNNs and is significantly faster than the edge-rooted subgraph GNN,  $I^2$ GNN. NGNN achieves comparable efficiency to MAG-GNN because it takes a fixed-hop subgraph around nodes, reducing the subgraph size. However, MAG-GNN outperforms NGNN on most targets with its better expressivity.
Limitations. Despite the training schemes in Section 4.1, MAG-GNN is harder to train. Also, MAG-GNN's design for node-level tasks might not be proper. This motivates research on extending MAG-GNN to node-level or even edge-level tasks. We discuss this further in Appendix C.4.
In this work, we closely examine one popular GNN paradigm, subgraph GNN, and discover that a small subset of subgraphs is sufficient for obtaining high expressivity. We then design MAG-GNN, which uses RL to locate such a subset, and propose different schemes to train the RL agent effectively. Experimental results show that MAG-GNN achieved very competitive performance to subgraph GNNs with significantly less inference time. This opens up new pathways to design efficient GNNs.
Acknowledgement. Lecheng Kong, Jiarui Feng, Hao Liu, and Yixin Chen are supported by NSF grant CBE-2225809. Muhan Zhang is partially supported by the National Natural Science Foundation of China (62276003) and Alibaba Innovative Research Program.
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[35] Muhan Zhang and Pan Li. Nested graph neural networks. In A. Beygelzimer, Y. Dauphin, P. Liang, and J. Wortman Vaughan, editors, Advances in Neural Information Processing Systems, 2021.
[36] Bohang Zhang, Guhao Feng, Yiheng Du, Di He, and Liwei Wang. A complete expressiveness hierarchy for subgraph gnns via subgraph weisfeiler-lehman tests, 2023.
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A Proof of theorems
To prove theorem 1, we leverage the following lemma from [35], and we restate it here.
Lemma 1. For two graphs  $G_{1}$  and  $G_{2}$  that are uniformly independently sampled from all  $n$ -node  $r$ -regular graphs, where  $3 \leq r < \sqrt{2\log n}$ , we pick any two nodes, each from one graph, denoted by  $v_{1}$  and  $v_{2}$  respectively, and do  $\lceil (\frac{1}{2} + \epsilon)\frac{\log n}{\log(r - 1 - \epsilon)} \rceil$ -height rooted subgraph extraction around  $v_{1}$  and  $v_{2}$ . With at most  $\epsilon \left\lceil \frac{\log n}{\log(r - 1 - \epsilon)} \right\rceil$  many layers, a proper message passing GNN will generate different representations for the extracted two subgraphs with probability at least  $1 - o(n^{-1})$ .
For completeness, we recall Theorem 1:
Proof. Since the pooling function  $R^{(G)}$  is injective and the GNN  $g$  has 1-WL expressiveness, which is the maximal expressiveness for message-passing GNNs,  $f_{r}$  satisfies the proper GNN requirement in Lemma 1. We also let  $g$  have  $\epsilon \left\lceil \frac{\log n}{\log(r - 1 - \epsilon)} \right\rceil + d$  layers, where  $d = \left\lceil \left(\frac{1}{2} + \epsilon\right) \frac{\log n}{\log(r - 1 - \epsilon)} \right\rceil$ . Let  $\pmb{v} = (v_{1})$  on  $G_{1}$ , since  $v_{1}$  is marked, we can have the first  $d$  layers of  $g$  perform breadth-first-search on the graph rooted from  $v_{1}$ , and afterward, each node's representation embeds whether it's within  $d$  distance from  $v_{1}$ . We then let the last  $\epsilon \left\lceil \frac{\log n}{\log(r - 1 - \epsilon)} \right\rceil$  layers of  $g$  only to perform message passing on the nodes that are within  $d$  distance from  $v_{1}$ . This can be easily done by implementing an injective COMBINE function. We do the same operation for  $\pmb{v}' = (v_{2})$  on  $G_{2}$ . According to lemma 1, since  $f_{r}$  is proper, the probability that  $f_{r}(G_{1}, \pmb{v}) \neq f_{r}(G_{1}, \pmb{v})$  is at least  $1 - o(n^{-1})$ . Then, by union bound, the probability that  $f_{r}(G_{1}, \pmb{v}) \notin \{f_{r}(G_{2}, \pmb{v}') | \forall \pmb{v}' \in V(G_{2})\}$  is at least  $1 - o(1)$
We first introduce two non-isomorphic graphs in Figure 6.
Figure 6: Two CSL graphs.
They are CSL graphs both with 8 nodes, one with a skip factor of 2 and the other with a skip factor of 3. They have the following properties,
- MPNN cannot differentiate them since they are regular graphs.
- They are vertex-transitive graphs. Meaning that all node-rooted subgraphs from one graph are the same.
- The rooted subgraphs from the two graphs are different and can be differentiated by an MPNN.
The properties imply that we cannot use an MPNN to differentiate the two graphs, but we can apply MPNN once on one subgraph from each graph to differentiate them since all subgraphs for one graph are the same.
We name the two CSL graphs A and B and use them to build a super graph. Let  $d = 2n$  be a super graph's diameter where  $n \in \mathbb{Z}^+$ . We randomly permute  $n$  type A graphs and  $n$  type B graphs to form a graph sequence  $S$ . An example is  $S_e = (A, B, B, A, A, B)$ .
We then define  $S^{\prime}$  as the inverse of  $S$ , that is
$$
[ S ^ {\prime} ] _ {i} = \left\{ \begin{array}{l l} A & \text {i f} [ S ] _ {i} = B \\ B & \text {i f} [ S ] _ {i} = A \end{array} \right. \tag {11}
$$
The inverse of  $S_{e}, S_{e}^{\prime} = (B, A, A, B, B, A)$ .
We can then use concatenation to form two new sequences.
$$
S ^ {+} = S \oplus S \quad S ^ {-} = S \oplus S ^ {\prime} \tag {12}
$$
For example,  $S_{e}^{+} = (A,B,B,A,A,B,A,B,A,B,A,A,B)$  and  $S_{e}^{-} = (A,B,B,A,A,B,B,A,A,B,A,B,A)$ . We then use full connections to connect the graphs in a sequence and create a super graph  $G(S)$ , specifically,
$$
\begin{array}{l} G (S) = \{V (G), E (G) \}, \quad V (G) = \cup_ {i} V ([ S ] _ {i}), \quad E (G) = \cup_ {i} (E ([ S ] _ {i}) \cup W (i))) \\ W (i) = V ([ S ] _ {i}) \times V ([ S ] _ {i + 1 \bmod 4 n}) \\ \end{array}
$$
The nodes in graph  $S[i]$  connect to every node in graph  $S[j]$  if the two graphs are adjacent in the sequence or are the head and tail of the sequence. Figure 7 shows the example super graphs  $G(S_{e}^{+})$  and  $G(S_{e}^{-})$
Figure 7: Two example super graphs.
The properties of  $G(S)$  is,
- The diameter of the super graph is  $2n$ .
-  $G(S)$  has an equal number of graphs A and B.
- Because adjacent graphs in the super graph are fully connected, adding node marking to graph  $[S]_i$  in the super graph does not help us identify any other graphs  $[S]_j$  without node marking. To identify  $[S]_j$ , we must add node marking to  $[S]_j$ .
- For positive super graphs,  $[S]_i = [S]_{i+2n}$ , that is, for any graph in the super graph, the graph on its direct opposite shares its type. Whereas for negative super graphs,  $[S]_i \neq [S]_{i+2n}$ .
Let the task be to discriminate between positive super graphs and negative super graphs. We then prove that MAG-GNN is superior in this case to show strictness.