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Omni-Granular Ego-Semantic Propagation for Self-Supervised Graph Representation Learning

Unsupervised/self-supervised graph representation learning is critical for downstream node- and graph-level classification tasks. Global structure of graphs helps discriminating representations and existing methods mainly utilize the global structure by imposing additional supervisions. However, their global semantics are usually invariant for all nodes/graphs and they fail to explicitly embed the global semantics to enrich the representations. In this paper, we propose Omni-Granular Ego-Semantic Propagation for Self-Supervised Graph Representation Learning (OEPG). Specifically, we introduce instance-adaptive global-aware ego-semantic descriptors, leveraging the first- and second-order feature differences between each node/graph and hierarchical global clusters of the entire graph dataset. The descriptors can be explicitly integrated into local graph convolution as new neighbor nodes. Besides, we design an omni-granular normalization on the whole scales and hierarchies of the ego-semantic to assign attentional weight to each descriptor from an omni-granular perspective. Specialized pretext tasks and cross-iteration momentum update are further developed for local-global mutual adaptation. In downstream tasks, OEPG consistently achieves the best performance with a 2%~6% accuracy gain on multiple datasets cross scales and domains. Notably, OEPG also generalizes to quantity- and topology-imbalance scenarios.


Introduction
In the past few years, the paradigm of graph learning has been shifted from structural pattern discovery (Leskovec et al., 2005;Borgatti & Everett, 2000;Milo et al., 2004;Newman, 2006;Watts & Strogatz, 1998) to graph representation learning (Hamilton et al., 2017;Veličković et al., 2017). In many scientific domains and industrial scenarios, labeled graph-structured data is limited and hard to obtain. Thus learning the representations in an unsupervised or self-supervised manner becomes increasingly important. Traditional unsupervised graph representation learning approaches, such as DeepWalk (Perozzi et al., 2014), node2vec (Grover & Leskovec, 2016) and LINE (Tang et al., 2015), utilize a framework originated in the skip-gram model (Mikolov et al., 2013) which aim to learn an encoding function for transforming nodes to low-dimensional embeddings that preserve vital attributive and structural features. Recent works conduct graph representation learning in different ways including learning transferable prior knowledge with meta-learning (Lu et al., 2021;Thakoor et al., 2021) and predicting the informative substructures of graph (Hu et al., 2019;Jiao et al., 2020;Zhang et al., 2020;. Inspired by powerful contrastive learning paradigm (Oord et al., 2018;Chen et al., 2020;He et al., 2020), graph contrastive learning (GCL) methods  achieve a significant success. Recent works mainly focus on designing topology (Hassani & Khasahmadi, 2020;Qiu et al., 2020) or node/edge augmentations (You et al., 2020a;Zhu et al., 2021b) and defining the contrastive pairs at different levels (e.g., node-level and graph-level) of the graph (Peng et al., 2020;Sun et al., 2019). They perform local-global or global-global contrasting (Zhu et al., 2021a) to discriminate the representations, proving the effectiveness of global semantics. However, their use of the whole adjacency matrix is impractical when scaling to large graphs. Some recent global-level self-supervised methods (You et al., 2020b) focus on utilizing the global semantics to strengthen the graph representation learning, using the global structure information to guide the local pre-training (Jin et al., 2020) or impose the additional supervisions to conduct informative learning . Nevertheless, their utilization of global semantics are implicit since they can not explic-= ( )

Ego-Semantic Descriptor
( ) , Figure 1. Illustration of OEPG. (I) Obtain hierarchical global clusters. (II) Explicitly define instance-adaptive global-aware ego-semantic descriptors by measuring first-and second-order feature differences between target node (denoted in red) and hierarchical clusters. (III) Perform omni-granular normalization on ego-semantic descriptors and use them to form new subgraph for later feature propagation.
itly embed the global structure information into representations. Besides, their global semantics are invariant for all nodes/graphs, which fail to adequately describe the global diversity for better discriminativity and expressiveness.
In this paper, we propose Omni-Granular Ego-Semantic Propagation for self-supervised graph representation learning (OEPG) to address above problems. As in Figure.1-(I), we use k-means to obtain global clusters on hierarchical levels after node-or graph-level pre-training of the graph dataset. We characterize the instance-adaptive global-aware feature for each node/graph by the ego-semantic descriptor, which leverages the first-and second-order feature differences between itself and all global clusters in a 1-vs-N manner as in Figure.1-(II). We do not straightly use global clusters to represent the global semantic since it should be instance-adaptive from the ego perspective of different nodes/graphs. We explicitly incorporate the ego-semantic descriptor into local graph convolutional process to model the local-global feature propagation. Contrary to implicit methods such as predicting global structure related pairwise distances (Jin et al., 2020) or imposing additional global structural supervision , our ego-semantic descriptors can be explicitly utilized to make feature propagation between local context and global semantic.
We devise an omni-granular normalization over the hierarchies and scales of the ego-semantic to make crossgranularity attention between the coarse-grained and finegrained descriptors. It assigns attentional weight to each descriptor according to the contribution from an omni-granular perspective. After the normalization, the weighted egosemantic descriptors and original graph (or subgraph centered around node) are used to form a new one as in . To adequately utilize the local and global information of the new graph/subgraph, we further propose specialized pretext tasks involving local neighbors and global ego-semantic descriptors by performing mutual contrasting and prediction. Considering the dynamics of feature space in training process, we design a cross-iteration momentum update mechanism for omni-granular ego-semantic descriptors in a moving-average way to achieve local-global mutual adaptation. It is noteworthy that the ego-semantic and omnigranular normalization of our OEPG are also applicable for downstream tasks since they are explicit feature or operations, and thus narrow the gap between pre-training and applications. Additionally, OEPG is robust to quantity-and topology-imbalance (Chen et al., 2021) scenarios without additional operations (e.g., re-weighting or over-sampling).
Our main contributions are summarized as the following: • We firstly explicitly characterize the instance-adaptive global-aware feature by ego-semantic descriptors, utilizing combined first-and second-order feature differences to discriminate and enrich the representations.
• We propose an omni-granular normalization over all the hierarchies and scales of ego-semantic for assigning attentional weight to each descriptor according to the contribution from an omni-granular perspective.
• Specialized tasks and a cross-iteration omni-granular momentum update are both proposed for achieving local-global mutual adaptation in our framework.
• Our OEPG substantially outperforms previous methods with a 2%∼6% accuracy gain in multiple downstream tasks on datasets cross scales and domains, and generalizes to quantity-and topology-imbalance scenarios.

Related Work
Self-Supervised Graph Representation Learning. Existing self-supervised graph representation learning methods can be mainly divided into two categories: contrastive (You et al., 2020a) and predictive (Hu et al., 2019;Wu et al., 2021). The contrastive methods mainly design different augmented views and build loss function (Veličković et al., 2018;Sun et al., 2019;Peng et al., 2020) based on the predefined contrastive pairs to guide the graph representation learning (Suresh et al., 2021;Xu et al., 2021a;Zhang et al., 2021a). The predictive methods focus on the information embedded in the graph, generally based on pretext tasks such as reconstruction (Kipf & Welling, 2016b) or other predictive tasks, which exploit the attributes and structures of graph data as additional supervisions (Hu et al., 2020b;Rong et al., 2020;Xu et al., 2021b;Thakoor et al., 2021).
Global-Level Self-Supervised Objectives. Recent works begin to focus on global-level self-supervised objectives, utilizing the global structure of the whole graph dataset. Some methods use the whole adjacency matrix to do the local-global contrastive learning (Zhu et al., 2021a), which is impractical in large graphs. You et al. (2020b) applies graph topology partition and graph completion to improve the self-supervision. Jin et al. (2020) takes a bird's-eye view of the position of the node in the graph and proposes to pre-train by the guidance of global structure information. Xu et al. (2021b) learns the global structure by an online expectation-maximization (EM) algorithm (Dempster et al., 1977). Nevertheless, their approaches to capture the global semantic are all implicit with additional supervisions, and thus they are inapplicable in the inference stage and lead to a gap between pre-training and applications.

OEPG: Omni-Granular Ego-Semantic Propagation Graph
In this section, we elucidate our Omni-Granular Ego-Semantic Propagation Graph (OEPG) in detail. OEPG is designed to explicitly characterize the instance-adaptive global-aware feature for node/graph representations. Specifically, we derive first-and second-order ego-semantic descriptors for each subgraph in a locally-smoothed feature space, and perform omni-granular normalization on the descriptors to make attentional cross-granularity interactions. Then we aggregate the original subgraph and weighted ego-semantic (combine first and second orders) to form a new subgraph. Finally, specialized pretext tasks and a cross-iteration momentum update mechanism are devised for local-global feature propagation and mutual adaptation.
A Unified Perspective. In our framework, we treat the node-and graph-level graph representation learning from an unified perspective. Specifically, considering a set of unlabeled graphs or subgraphs G = {G 1 , G 2 , · · · , G N }, where a subgraph G = (V, E) is centered around node V. We only illustrate the node-level pre-training for simplicity. We aim at learning low-dimensional vectors The embed-ding process of a l-th GNN model is formulated as: where N V is the neighbor set of V, F 1 and F 2 denote the message passing and update function (Gilmer et al., 2017) respectively. F R is a permutation-invariant readout function (Ying et al., 2018;Zhang et al., 2018).

Ego-Semantic Descriptor
Motivation. Graph Neural Networks (GNNs) derive proximity-preserved feature vectors in a neighborhood aggregation way (Gilmer et al., 2017). Such a locallysmoothed latent space benefits the information propagation (Kipf & Welling, 2016a;Garcia Duran & Niepert, 2017;Wang & Leskovec, 2020) between the nodes. However, the local proximity based feature representations fail to capture the global pattern or structure of the whole graph, which is crucial for learning discriminative representations, especially in a self-supervised setting. An alternative way is to utilize the whole adjacency matrix Hassani & Khasahmadi, 2020), but it is impractical for when applied in large graph. Jin et al. (2020) and Xu et al. (2021b) firstly use global structure related supervisions for representation learning. We note that their utilization of global information are implicit, and the global structure information can not be explicitly embedded in learned representations. Besides, their global semantics are invariant for all subgraph instances, limiting the discriminativity and expressiveness. We argue that an effective expression of global semantic should be instance-adaptive and dynamic, which means it changes accordingly with the change of target node/graph and thus helps discriminating the representations from both local and global perspectives. Hence, we explicitly characterize an instance-adaptive global-aware feature, which integrates the local-global semantic differences.
The First-Order Ego-Semantic Descriptor. Considering that existing global-level methods only use instanceinvariant global information and also lead to over-smoothing problem, we propose the first-order ego-semantic descriptor to embed the instance-adaptive 1-vs-N global diversity with differential feature and alleviate the over-smoothing problem, which firstly considers the differences between local subgraph feature and global semantic clusters. In specific, we first use typical graph contrastive learning to acquire a locally-smoothed feature space and use K-means to make hierarchical clustering on the whole graph dataset for obtaining non-parametric global clusters {{C s,h } S h s=1 } H h=1 , where H is the total hierarchies and S h denotes the number of scales in h-th hierarchy (S 1 > S 2 > · · · > S H ). After that, we not only want to associate the local context with the global semantic but also expect that each target node can explicitly embeds the self-adaptive global-aware semantic. To achieve this goal, we define the first-order ego-semantic descriptor. : describe diverse global semantics from the ego perspective, calculated by the channel-wise feature difference between the target node and full global clusters. Thus for target nodes of different classes, their first-order ego-semantic descriptors can be also different, discriminating the final representations. Note that we use l 2 normalization along the feature channels to eliminate the influence caused by large numerical differences of different D 1st s,h .
The Second-Order Ego-Semantic Descriptor. Regarding enhancing the discriminativity and alleviating the oversmoothing, the first-order ego-semantic descriptor only leverages the instance-wise global diversity. Further, we try to model the instance-adaptive 1-vs-N global diversity correlations from an omni-granular perspective, which aims to leverage more structural information compared with first-order ones. Thus we propose the second-order egosemantic descriptor.
where • is the inner product. X is further specified as: {{D 2nd s,h } S h s=1 } H h=1 provide target node with a set of global similarity-distribution based features, which model the dense 1-vs-N correlations between each first-order descriptor D 1st s,h and {{D 1st s,h } S h s=1 } H h=1 from the ego perspective.
Such second-order descriptors integrate more contextual information to enrich the representations. Both first-and second-order descriptors discriminate the node-and graphlevel representations, further proved in Appendix.B.3. We combine the both to generate the final ego-semantic descrip- To the best of our knowledge, we are the first to explicitly characterize the instance-adaptive global-aware semantic features for GNNs. Notably, they can be easily utilized in existing GNN architectures. A typical local graph convolution in Eq.1 can be rewritten as (E is omitted for simplicity): where {V local i,l−1 } Nn i=1 denotes the set of total N n neighbor nodes at the l-th layer in subgraph, F denotes the aggregation and update functions. We note that previous GNN methods can only use the local context to aggregate feature and propagate the information for target node. In contrast, our final ego-semantic descriptors can be explicitly integrated into the local graph convolutional process as follow: Thus the target node explicitly embeds both local context and node-adaptive global semantics, improving the discriminativity and expressiveness from different perspectives.

Omni-Granular Normalization
The ego-semantic descriptors of each hierarchy jointly compose the global semantic space and each descriptor represents a semantic patch. The semantic patch changes from the fine granularity to the coarse one with the hierarchy changing from the bottom to the top. The ego-semantics of multiple granularities are critical for many downstream tasks such as community detection and molecular property prediction. Besides, different target node may have the different priorities of the ego-semantic granularities. Thus we perform an omni-granular normalization on the whole scales and hierarchies of the ego-semantic to conduct cross-granularity attention between fine-grained and coarse-grained descriptors, and weight them according to their contributions. We normalize first-and second-order descriptors separately since they describe different characteristics of the ego-semantic. For the first-order ego-semantic descriptors, we have: where D 1st s,h denotes the initial first-order ego-semantic descriptor. For each target node, we flatten the hierarchies and scales of the ego-semantic descriptors and calculate the attention weight a s,h (ranges between 0 and 1) with the omni-granular normalization technique. α is a trainable parameter (positive constant) that controls the decay of the response with the magnitude of the feature difference. We weight the each initial first-order ego-semantic descriptor: and we weight the each second-order ego-semantic descriptor in a similar way (β is another trainable parameter as α): Proposition 3.3. The value of the trainable parameters α ∈ (0, ∞) and β ∈ (0, ∞) could reflect the global semantical diversity from the ego perspective.
Proof. If α → ∞, the attention weight for the ego-semantic descriptor D 1st s,h will be a 1st s,h → 1 and it will be 0 for other ego-semantic descriptors, which means the target node is only related to the closest cluster C s,h because of the global diversity of the data distribution. Conversely, If α → 0, a 1st s,h → 1 S1+S2+···+S H , which means all clusters have the similar contribution to the target node because of the global proximity of the data distribution. It is the same for β.
We concatenate each weighted first-and second-order egosemantic descriptor along the channel for better description: where a linear W : R (d+S1+S2+···+S H ) → R d and nonlinear function follow the concatenation to fuse descriptors. Through above processes, each target captures the sufficient characteristics about diverse global information, which are beneficial for downstream classification tasks.
Finally, we group the local neighbors and projected egosemantic descriptors D s,h to construct a new subgraph for the target node. The adjacency matrix A ori of original subgraph is extended to the new adjacency matrix A: where local are the indexes of target and neighbor nodes, and global are the indexes of ego-semantic descriptors. The edges between the target and all ego-semantic descriptors are 1 and the edges are set to 0 between local neighbors and the descriptors. For the graph-level representation learning, we treat the readout feature as the V target and the A is: The new subgraph contains both local and global information for later self-supervised graph representation learning. The graph convolutional process would propagate the globalaware omni-granular ego-semantic information to the local context for discriminating the feature representations.

Model Optimization and Downstream Application
Specialized Local-Global Pretext Tasks. For better simultaneously leveraging both local and global information for embedding new subgraph, specialized pretext tasks are devised for pre-training. They can be grouped into two categories, predictive and contrastive pretext tasks. Node-based and edge-based contrastive modes have been fully explored to maximize the agreement of two graph augmentations in graph contrastive learning (Zhu et al., 2021a). We extend them to make it controllable for our OEPG. In specific, we simultaneously drop random portion of the local neighbors (edges) and diverse global-aware ego-semantic descriptors (edges) to produce augmented subgraphs for local-global contrastive learning. Given the local-global masking T i , T j , the loss is: where sim(·) is the similarity function and G − is negative subgraph. It enables the model robust to both local and global semantic noises and effectively combines the local proximity and global diversity for discriminating the feature.
Context prediction task has been well applied in selfsupervised graph representation learning (Hu et al., 2019). It enables the model to capture the local context by predicting surrounding graph structures. Inspired by this, we propose a cross-reconstruction task where we mask a substructure containing both local nodes and global ego-semantic descriptors in the subgraph, and use the rest subgraph G 1 to reconstruct the masked substructure G 2 . The loss is: where G 1 ∪G 2 = G, F aux is an auxiliary encoder to produce substructure embedding and σ(·) is the sigmoid function. It builds a strong correlation between local and global semantics by predicting the co-concurrences of them.

Cross-Iteration Omni-Granular Momentum Update.
In the pre-training procedure guided by specialized contrastive and predictive pretext tasks, Considering the shifts of global feature space in pre-training process, we introduce a cross-iteration momentum update mechanism of multigranular clusters {{C s,h } S h s=1 } H h=1 . Specifically, for each node, we assign it to a closest cluster of each hierarchy. Given the hierarchy index h ∈ [1, H], then the scale index s of each hierarchy is calculated by: Instead of updating node queue after each training iteration, we construct a node queue with a budget B for each cluster. If the number of nodes in a queue equals to the B, the corresponding cluster will be update as follow: where momentum coefficient m is set 0.999 in our experiments. Such a cross-iteration momentum update effectively facilitates the training process. And the updated clusters would in turn have a positive effect on the node update, which achieves a local-global mutual adaptation. We summarize the entire procedure of our OEPG in Algorithm.1.
After the self-supervised representation learning, the derived GNN model explicitly propagates the information between the local and global contexts, discriminating the representations. Benefiting from the explicit definition of the egosemantic descriptors, such instance-adaptive global-aware features are also applicable (in Appendix.B) in downstream tasks under self-supervised, semi-supervised and transfer learning settings, substantially outperforming SOTA methods. Notably, OEPG also generalizes to the quantity-and topology-imbalance scenarios, never explored by previous self-supervised graph representation learning methods. Evaluation Protocols. To adequately evaluate our algorithm, we perform downstream evaluations in three settings, following the same protocols as in State-of-the-Art. (1) In self-supervised learning , we pre-train the model using the whole graph dataset to learn a graph encoder and compute embedding for each node or graph, then the embeddings are feed into a downstream SVM (You et al., 2021) or linear classifier (Veličković et al., 2018) for evaluation.

Experiments and Analysis
(2) In transfer learning, we first pre-train the model on a larger graph dataset (You et al., 2021;Xu et al., 2021b). Then we finetune and evaluate the model on smaller datasets of the same category using the given training/validation/test split. (3) In semi-supervised learning (You et al., 2020a), for datasets without explicit train/validation/test split, we conduct pre-training process with all graph data at first. Then we finetune and evaluate the model with K folds (You et al., 2021). For datasets with the explicit split, we only pre-train the model with the training split, finetune on the partial training split and evaluate on the validation/test splits.
GNN Architectures. We use the same GNN architectures with default training hyper-parameters as in the SOTA methods under three experiment settings. In specific, (1) in selfsupervised representation learning, GIN (Xu et al., 2018) is used with 3 layers and 32 hidden dimensions, (2) in semisupervised learning, a 2-layer or 3-layer (for large graphs) GCN encoder network with 128 or 256 hidden dimensions, and (3) in transfer learning and on large-scale OGB datasets, GIN is used with 5 layers and 300 hidden dimensions.
Self-Supervised Learning. We follow the previous methods (You et al., 2021) to perform pre-training and evaluation, and report the results in Table.1. Our OEPG substantially outperforms all previous methods with a 3%∼4% accuracy gain compared to SOTA. It demonstrates that OEPG exploits the local proximity and global diversity, improving the expressiveness and discriminativity of repersentations.
Transfer Learning. For fair comparison, we use the same subset of ZINC15 (Sterling & Irwin, 2015) for pre-training as , and perform evaluation on smaller datasets with the results in Table.2. Among all methods, our OEPG achieves the best performance on all datasets, and a 2%∼6% performance gain is obtained in terms of ROC-AUC compared to SOTA method.
Semi-Supervised Learning. We report our comparison results in Table.3 with the same evaluation setting as (Bielak et al., 2021). Our OEPG surpasses the SOTA by a large margin, with a 2%∼4% accuracy improvement. The proposed model could strengthen the label propagation with the effective local-global semantic feature propagation.
Generalizing to Quantity-and Topology-Imbalance Scenarios. Quantity-imbalance denotes the unequal quantity of labeled examples while topology-imbalance (Chen et al., 2021) means the asymmetric topological properties of the labeled nodes. They are both practical issues and hard to address. Notably, we find that our OEPG also generalizes well in tasks where quantity-and topology-imbalance problems co-occur as reported in Table.4, outperforming previous   Table 5. Semi-supervised learning on large-scale OGB datasets on (accuracy in % on ogbg-ppa, F1 score in % on ogbg-code, ROC-AUC in % on ogbg-molhiv). L.R. denotes the label ratio. methods by 2%∼7% including supervised model. The main reason is OEPG enables each node to adequately embed the global semantic features to enrich and discriminate itself despite sparse samples as illustrated in Appendix.B.3.
Scaling to Large-Scale Graph Datasets. We evaluate our OEPG on large-scale OGB datasets, with results in Table.5. Note that our OEPG significantly outperforms the previous SOTA methods with 2%∼5% performance gains at 1% and 10% label ratios, demonstrates the scalability of OEPG.

Model Analysis
Effectiveness of Ego-Semantic Descriptors. We investigate the effect of our proposed ego-semantic descriptors on downstream tasks and we reproduce GraphCL as our baseline. We report the result on the MUV dataset in Figure.2. The performance is obviously promoted by 2%-4% when adding the ego-semantic descriptors, which demonstrates the sufficient utilization of global information in OEPG. Besides, we also conclude that second-order ego-semantic is more effective than first-order one, and the concatenation of them performs the best with the diverse global perspective.
The Impact of Omni-Granular Normalization. In Subsection.3.2, we have theoretically proved the value of parameters α, β can reflect the global diversity of graph dataset. We also provide the experimental proof in Figure.3 of Appendix.B.3. We gradually increase the diversity of graph datasets, and find the values of α, β become correspondingly larger. The phenomenon implies the omni-granular normalization effectively reflects and leverages the global diversity. We also find the second-order ego-descriptors (related with β) are more sensitive to the global diversity.
Ba se lin e +E go -S em an tic +O m ni -G ra nu la r No rm +P re te xt Ta sk s +M om en tu m Up da te Ablation Study of OEPG. Illustrated in Section.3, OEPG consists of four cascade modules. The ablation study is conducted by gradually adding each module in order, and the results are in Figure.2. We conclude that each module gives OEPG a accuracy gain, demonstrates the efficacy of the designs in our framework. The x-axis in is order invariant since the modules are conducted in sequence. Note that the ego-semantic descriptors provide the most performance gain among all modules, further enhanced when empowered by omni-granular normalization. And specified pretext tasks and cross-iteration momentum update are also vital for the integrity and effectiveness of our framework. The results on more datasets are illustrated in Appendix.B.3. More analysis about the sensitivity of the hyperparameters are in Appendix.B.3.

Conclusions
In this paper, we propose a unified framework to explicitly characterize the global-aware feature and enable the local-global semantic propagation, namely Omni-Granular Ego-Semantic Propagation for self-supervised Graph representation learning (OEPG). We first propose the fused first-and second-order ego-semantic descriptors, weighted by omni-granular normalization, to explicitly represent the instance-adaptive global semantic information. New localglobal pretext tasks and a cross-iteration omni-granular momentum update are further developed to improve the localglobal mutual adaptation. Extensive experiments on datasets cross scales and domains are conducted and our OEPG substantially outperforms all the SOTA methods, demonstrating the effectiveness. Notably, OEPG also achieves the best performance in the quantity-and topology-imbalance scenarios.
For the future work, we will further improving the scalability and transferability of our framework.

A. Dataset Descriptions and Statistics
In this section, we provide the detailed descriptions and statistics about all the datasets used in Section.4.
TUDataset (Morris et al., 2020) contains the datasets of different natures with graph data for small molecules & proteins, computer vision and various relation networks.  et al., 2015) are two networks extracted from Amazon's copurchase data. Nodes are products and edges denote that these products were often bought together.
Coauthor CS, Coauthor Physics are two networks extracted from the Microsoft Academic Graph (Sinha et al., 2015). Nodes are authors and edges denote a collaboration of two authors. Citeseer, Cora, Pubmed (Sen et al., 2008) are widely-used citation networks where documents (nodes) are connected through citations (edges). MoleculeNet (Wu et al., 2018) has a full collection currently includes over 700 000 compounds tested on a range of different properties. These properties can be subdivided into four categories: quantum mechanics, physical chemistry, biophysics and physiology.
Open Graph Benchmark (OGB) (Hu et al., 2020a) contains a diverse set of large-scale and realistic benchmark datasets to validate the scalablity and robustness of graph machine learning algorithms.  (You et al., 2020a) for several epochs and then train the whole model with the specified pretext tasks, optimized by the contrastive loss L contrastive in Eq.14 and predictive loss L predictive in Eq.15. The total epochs are same or comparable with previous methods. At the start of every two epochs, we would perform K-means clustering for the initialization of hierarchical clusters. The parameter budget B is set to 4 and we adopt the 4-hierarchies (H = 4) ego-semantic descriptors for each subgraph/graph, with the specified cluster numbers [S 1 , S 2 , S 3 , S 4 ] = [16,12,8,4].
Fine-tuning Details. For fine-tuning on a certain downstream task, we follow the default setting in the previous methods You et al., 2021;Bielak et al., 2021). There are some differences between self-supervised/semi-supervised learning and transfer learning when applying our OEPG. In self-supervised/semi-supervised learning tasks, the classes between the pre-training and fine-tuning are the same, so the hierarchical clusters can be straightly used in computing the ego-semantic descriptors for downstream classification tasks. In contrary, the hierarchical clusters will be recomputed for obtaining the ego-semantic descriptors in transfer learning downstream tasks since the classes have changed from the pre-training to the fine-tuning. More Comparison Results. Due to the page limit, we have one experiment comparison results left as in Table.11. As in the main text, our OEPG also achieves the best performance in all methods with a large accuracy promotion.

B.3. More Analysis of OEPG
The Discriminativity of Ego-Semantic Descriptors. In this paper, we mainly propose the ego-semantic descriptors in the graph feature propagation process to discriminating the final representations. Next, we will illustrate how it works in specific and we take graph-level pre-training for example. Given the full graph dataset: {G 1 , G 2 , · · · , G g C1 , G g+1 , · · · , G 2·g

C2
, · · · , G (n−1)·g+1 · · · , G n·g Cn }, where we assume each graph cluster C n contains a group of same number of graphs and only consider 1-hierarchy clusters for simplicity. When we judge whether a pair of graphs G i , G j belong to the same class or not, we typically compute the similarity between the graph embeddings f Gi , f Gj as follow: where F denotes all the linear and non-linear operations and sim(·, ·) is the similarity function. Equipped with ego-semantic descriptors, the process can be reformulated as follow: 1st−order ego−semantic )), since clusters {C k } n k=1 are all non-parametric, and thus ego-semantic descriptors only vary with G i , G j , which are graphadaptive. When G i , G j are of different classes, all their ego-semantic descriptors can be different, enlarging the distances between f Gi and f Gj . Conversely, if G i , G j are of the same class, all their ego-semantic descriptors can be similar, minimizing the distances between f Gi and f Gj . The proposed ego-semantic descriptors not only discriminate the representations but also enhance the expressiveness by explicitly embedding the instance-adaptive global semantic features. Experimental Analysis Involving Omni-Granular Normalization. We make the controllable experiments to validate the effectiveness of omni-granular normalization. We use the 10-class WikiCS dataset for illustration. Specifically, we pre-train the OEPG on the graph dataset with the different number of total classes (from 2 to 10 classes). And we print the learned value of α, β and report them in Figure.3. We can find that with the increasing classes (global diversity), the values also become larger and the value of β (related with second-order ego-semantic descriptors) is more sensitive to the classes increasing.
Omni-Granular Ego-Semantic Propagation for Self-Supervised Graph Representation Learning More Ablation Results. Here we provide more ablation results in Table.12. The results including Fig.2 consistently illustrate that the Ego-Semantic is most relevant contribution to accuracy gains, because it is the Ego-Semantic which explicitly introduces the additional global information to the representations and other modules are based on it to make optimizations.  Sensitivity of Hierarchy h in Ego-Semantic. In this part, we discuss the selection of parameter h in our OEPG and report the results on MUV dataset for example. From the Figure.4, we can see that the performance is gradually promoted with the higher h, which implies that our omni-granular ego-semantic does enable the model to capture the global diversity and discriminate the representations. And the performance has a slight degradation when h is higher than 4. We think it may be caused by the large redundancy of the ego-semantics.