ABSTRACT

1
Introduction
Many real-world data can be naturally represented as graphs such as social networks, chemical molecules, transportation networks, and recommender systems (Battaglia et al.) [2018] [Wu et al.] (2019b) Zhou et al. (2018).  As a generalization of deep neural networks for graph-structured data, graph neural networks (GNNs) have achieved great success in capturing the abundant information residing in graphs and tackle various graph-related applications (Wu et al.J 2019b, Zhou et al.J 2018). However, the prevalence of large-scale graphs in real-world scenarios, often on the scale of millions of nodes and edges, poses significant computational challenges for training GNNs. More dramat- ically, the computational cost continues to increase when we need to retrain the models multiple times, e.g., under incremental learning settings, hyperparameter and neural architecture search. To address this challenge, a natural idea is to properly simplify, or reduce the graph so that we can not only speed up graph algorithms (including GNNs) but also facilitate storage, visualization and retrieval for associated graph data analysis tasks. There are two main strategies to simplify graphs: graph sparsification (Peleg & Schäffer) [1989] (Spielman & Teng) (2011) and graph coarsening (Loukas & Vandergheynst) (2018) Loukas, (2019) . Graph sparsification approximates a graph with a sparse graph by reducing the number of edges, while graph coarsening directly reduces the number of nodes by replacing the original node set with its subset. However, these methods have some shortcomings: (1) sparsification becomes much less promising in simplifying graphs when nodes are also associated with attributes as sparsification does not reduce the node attributes; (2) the goal of sparsification and coarsening is to preserve some graph 1 Figure 1: We study the graph condensation problem, which seeks to learn a small, synthetic graph, features and labels { A ′ , X ′ , Y ′ } from a large, original dataset { A , X , Y } , which can be used to train GNN models that generalize comparably to the original. Shown: An illustration of our proposed GC OND graph condensation approach's empirical performance, which exhibits 95.3% of original graph test performance with 99.9% data reduction. properties such as principle eigenvalues (Loukas & Vandergheynst) (2018) that could be not optimal for the downstream performance of GNNs. In this work, we ask if it is possible to significantly reduce the graph size while providing sufficient information to well train GNN models. Motivated by dataset distillation (Wang et al.) [2018] and dataset condensation (Zhao et al.) [2021] which generate a small set of images to train deep neural networks on the downstream task, we aim to condense a given graph through learning a synthetic graph structure and node attributes. Correspondingly, we propose the task of graph condensation 1 It aims to minimize the performance gap between GNN models trained on a synthetic, simplified graph and the original training graph. In this work, we focus on attributed graphs and the node classification task. We show that we are able to reduce the number of graph nodes to as low as 0.1% while training various GNN architectures to reach surprisingly good performance on the synthetic graph. For example, in Figure Π , we condense the graph of the Reddit dataset with 153,932 training nodes into only 154 synthetic nodes together with their connections.  In essence, we face two challenges for graph condensation:  (1) how to formulate the objective for graph condensation tractable for learning; and (2) how to parameterize the to-be-learned node features and graph structure.  To address the above challenges, we adapt the gradient matching scheme in (Zhao et al.) (2021) and match the gradients of GNN parameters w.r.t. the condensed graph and original graph. In this way, the GNN trained on condensed graph can mimic the training trajectory of that on real data. Further, we carefully design the strategy for parametrizations for the condensed graph. In particular, we introduce the strategy of parameterizing the condensed features as free parameters and model the synthetic graph structure as a function of features, which takes advantage of the implicit relationship between structure and node features, consumes less number of parameters and offers better performance. Our contributions can be summarized as follows: 1. We make the first attempt to condense a large-real graph into a small-synthetic graph, such that the GNN models trained on the large graph and small graph have comparable performance. We introduce a proposed framework for graph condensation (GC OND ) which parameterizes the con- densed graph structure as a function of condensed node features, and leverages a gradient match- ing loss as the condensation objective. 2. Through extensive experimentation, we show that GC OND is able to condense different graph datasets and achieve comparable performance to their larger counterparts. For instance, GC OND approximates the original test accuracy by 95.3% on Reddit, 99.8% on Flickr and 99.0% on Cite- seer, while reducing their graph size by more than 99.9%. Our approach consistently outperforms coarsening, coreset and dataset condensation baselines. 3. We show that the condensed graphs can generalize well to different GNN test models. Addition- ally, we observed reliable correlation of performances between condensed dataset training and whole-dataset training in the neural architecture search (NAS) experiments.
2
RELATED WORK
Dataset Distillation & Condensation. Dataset distillation (DD) (Wang et al.) [2018] [Bohdal et al.] (2020) Nguyen et al.] (2021) (Wang et al.) (2022) Cazenavette et al.) (2022) aims to distill knowledge of a large training dataset into a small synthetic dataset, such that a model trained on the synthetic set is able to obtain the comparable performance to that of a model trained on the original dataset. To improve the efficiency of DD, dataset condensation (DC) (Zhao et al.) [2021] [Zhao & Bilen] [2021] is proposed to learn the small synthetic dataset by matching the gradients of the network parameters w.r.t. large-real and small-synthetic training data. However, these methods are designed exclusively for image data and are not applicable to non-Euclidean graph-structured data where samples (nodes) are interdependent. In this work, we generalize the problem of dataset condensation to graph domain and we seek to jointly learn the synthetic node features as well as graph structure. Additionally, our work relates to coreset methods (Welling) 2009) Sener & Savaresel (2018) Rebuffi et al.1 (2017), which seek to find informative samples from the original datasets. However, they rely on the presence of representative samples, and tend to give suboptimal performance. Graph Sparsification & Coarsening. Graph sparsification and coarsening are two means of reduc- ing the size of a graph. Sparsification reduces the number of edges while approximating pairwise distances (Peleg & Schäffer) [1989), cuts (Karger) [1999) or eigenvalues (Spielman & Teng] [2011) while coarsening reduces the number of nodes with similar constraints  (Loukas & Vandergheynst) (2018) [Loukas] [2019] Deng et al.] [2020], typically by grouping original nodes into super-nodes, and defining their connections. Cai et al. (2021) proposes a GNN-based framework to learn these con- nections to improve coarsening quality. [Huang et al] (2021b) adopts coarsening as a preprocessing method to help scale up GNNs. Graph condensation also aims to reduce the number of nodes, but aims to learn synthetic nodes and connections in a supervised way, rather than unsupervised group- ing as in these prior works.  Graph pooling is also related to our work, but it targets at improving graph-level representation learning (see Appendix D). Graph Neural Networks. Graph neural networks (GNNs) are a modern way to capture the intuition that inferences for individual samples (nodes) can be enhanced by utilizing graph-based information from neighboring nodes (Kipf & Welling) [2017] [Hamilton et al.] [2017] [Klicpera et al.] [2019] [Velick- ovic et al.] 2018; Wu et al.] 2019bla, Liu et al.] 2020; 2021; You et al.] 2021; Zhou et al.] 2021; Zhao et al.) (2022). Due to their prevalence, various real-world applications have been tremendously fa- cilitated including recommender systems (Ying et al.) (2018a) Fan et al.) (2019), computer vision (Li et al.] (2019) and drug discovery (Duvenaud et al.) (2015). Graph Structure Learning. Our work is also related to graph structure learning, which explores methods to learn graphs from data. One line of work (Dong et al.) (2016) Egilmez et al.) (2017) learns graphs under certain structural constraints (e.g. sparsity) based on graph signal processing. Recent efforts aim to learn graphs by leveraging GNNs (Franceschi et al.) [2019] [in et al.] [2020] [Chen et al.] (2020). However, these methods are incapable of learning graphs with smaller size, and are thus not applicable for graph condensation. For more related works on graph structure learning and graph generation, we kindly refer the reader to recent surveys (Ding et al.) [2022] Zhu et al.] [2021b).
3
METHODOLOGY
In this section, we present our proposed graph condensation framework, GC OND . Consider that we have a graph dataset T = {A,X, Y}, where A ∈ R N×N is the adjacency matrix, N is the number of nodes, X ∈ R N × d is the d -dimensional node feature matrix and Y ∈ {0, . . . , C − 1} N denotes the node labels over C classes. Graph condensation aims to learn a small, synthetic graph dataset S = {A',X',Y'} with A' ∈ RN'×N',X' ∈ RN'×D , Y' ∈ {0,...,C − 1} N' and N' ≪ N , such that a GNN trained on S can achieve comparable performance to one trained on the much larger T . Thus, the objective can be formulated as the following bi-level problem, min L (GNN θ s (A, X), Y)     s.t    θ s = arg min L (GNN θ (A', X'), Y'), (1) where GNN θ denotes the GNN model parameterized with θ, θ S denotes the parameters of the model trained on S , and L denotes the loss function used to measure the difference between model predictions and ground truth, i.e. cross-entropy loss. However, optimizing the above objective can lead to overfitting on a specific model initialization. To generate condensed data that generalizes to a distribution of random initializations P θ 0 , we rewrite the objective as follows: s.t.     θ S = arg min L(GNN θ(θω)(A', X'), Y'). min Eoo~poo [L (GNNos (A, X), Y)] (2) where θ(θ 0 ) indicates that θ is a function acting on θ 0 . Note that the setting discussed above is for inductive learning where all the nodes are labeled and test nodes are unseen during training. We can easily generalize graph condensation to transductive setting by assuming Y is partially labeled.
3.1
GRAPH CONDENSATION VIA GRADIENT MATCHING
To tackle the optimization problem in Eq. ② , one strategy is to compute the gradient of L w.r.t S and optimize S via gradient descent, as in dataset distillation (Wang et al.) (2018). However, this requires solving a nested loop optimization and unrolling the whole training trajectory of the inner problem, which can be prohibitively expensive. To bypass the bi-level optimization, we follow the gradient matching method proposed in  (Zhao et al.) (2021) which aims to match the network parameters w.r.t. large-real and small-synthetic training data by matching their gradients at each training step. In this way, the training trajectory on small-synthetic data S can mimic that on the large-real data T , i.e.,
(2021) that the distance between θ ξ and θ ξ is typically small, we can simplify the objective as a
GT ∈ R d 1 × d 2 at a specific layer, the distance dis( · , · ) used for condensation is defined as follows,
we are able to achieve parameter matching through an efficient strategy of gradient matching. We note that jointly learning the three variables A ′ , X ′ and Y ′ is highly challenging, as they are interdependent. Hence, to simplify the problem, we fix the node labels Y ′ while keeping the class distribution the same as the original labels Y. Graph Sampling.  GNNs are often trained in a full-batch manner (Kipf & Welling) [2017] [Wu (et al.) (2019b). However, as suggested by previous works that reconstruct data from gradients (Zhu et al., (2019), large batch size tends to make reconstruction more difficult because more variables are involved during optimization. To make things worse, the computation cost of GNNs gets expensive on large graphs as the forward pass of GNNs involves the aggregation of enormous neighboring nodes. To address the above issues, we sample a fixed-size set of neighbors on the original graph in each aggregation layer of GNNs and adopt a mini-batch training strategy. To further reduce memory usage and ease optimization, we calculate the gradient matching loss for nodes from different classes separately, as matching the gradients w.r.t. the data from a single class is easier than that from all classes.  Specifically, for a given class c , we sample a batch of nodes of class c together with a portion of their neighbors from large-real data T . We denote the process as ( A c, X c, Y c ) ～ T . For the condensed graph A ′ , we sample a batch of synthetic nodes of class c but do not sample their neighbors. In other words, we use all of their neighbors, i.e., all other nodes, during the aggregation process, since we need to learn the connections with other nodes. We denote the process as (A', X', Y', ~ S. 3.2 M ODELING C ONDENSED G RAPH D ATA One essential challenge in the graph condensation problem is how to model the condensed graph data and resolve dependency among nodes. The most straightforward way is to treat both A ′ and X ′ as free parameters. However, the number of parameters in A ′ grows quadratically as N ′ increases. The increased model complexity can pose challenges in optimizing the framework and increase the risk of overfitting. Therefore, it is desired to parametrize the condensed adjacency matrix in a way where the number of parameters does not grow too fast. On the other hand, treating A ′ and X ′ as independent parameters overlooks the implicit correlations between graph structure and features, which have been widely acknowledged in the literature (III et al.) (2014) Shalizi & Thomas) (2011); e.g., in social networks, users interact with others based on their interests, while in e-commerce, users purchase products due to certain product attributes. Hence, we propose to model the condensed
In Eq. ( Д , we intentionally control A ′ j = A ′ j i to make the condensed graph structure symmetric since we are mostly dealing with symmetric graphs.  It can also adjust to asymmetric graphs by setting A ij = Sigmoid(MLP φ ([ x i ; x i ]). Then we rewrite our objective as
Note that there are two clear benefits of the above formulation over the naïve one (free parameters). Firstly, the number of parameters for modeling graph structure no longer depends on the number of nodes, hence avoiding jointly learning O( N ′ 2 ) parameters; as a result, when N ′ gets larger, GC OND suffers less risk of overfitting. Secondly, if we want to grow the synthetic graph by adding more synthetic nodes condensed from real graph, the trained MLP φ can be employed to infer the connections of new synthetic nodes, and hence we only need to learn their features. Alternating Optimization Schema. Jointly optimizing X' and Φ is often challenging as they are directly affecting each other. Instead, we propose to alternatively optimize X ′ and Φ: we update Φ for the first τ 1 epochs and then update X ′ for τ 2 epochs; the process is repeated until the stopping condition is met – we find empirically that this does better as shown in Appendix Cl Sparsification. In the learned condensed adjacency matrix A ′ , there can exist some small values which have little effect on the aggregation process in GNNs but still take up a certain amount of storage (e.g. 4 bytes per float). Thus, we remove the entries whose values are smaller than a given threshold δ to promote sparsity of the learned A ′ . We further justify that suitable choices of δ for sparsification do not degrade performance a lot in Appendix C The detailed algorithm can be found in Algorithm 1 in Appendix [ B ] In detail, we first set the con- densed label set Y ′ to fixed values and initialize X ′ as node features randomly selected from each class. In each outer loop, we sample a GNN model initialization θ from a distribution P θ . Then, for each class we sample the corresponding node batches from T and S , and calculate the gradient matching loss within each class. The sum of losses from different classes are used to update X ′ or Φ . After that we update the GNN parameters for τθ epochs. When finishing the updating of condensed graph parameters, we filter edge weights smaller than δ to obtain the final sparsified graph structure. A "Graphless" Model Variant. We now explore another parameterization for the condensed graph data. We provide a model variant named GC OND - X that only learns the condensed node features X ′ without learning the condensed structure A ′ . In other words, we use a fixed identity matrix I as the condensed graph structure. Specifically, this model variant aims to match the gradients of GNN parameters on the large-real data ( A , X ) and small-synthetic data ( I , X ′ ). Although GC OND-X is unable to learn the condensed graph structure which can be highly useful for downstream data analysis, it still shows competitive performance in Table [2] in the experiments because the features are learned to incorporate relevant information from the graph via the matching loss.
4
E XPERIMENTS
In this section, we design experiments to validate the effectiveness of the proposed framework GC OND . We first introduce experimental settings, then compare GC OND against representative baselines with discussions and finally show some advantages of GC OND .
4.1
EXPERIMENTAL SETUP
Datasets. We evaluate the condensation performance of the proposed framework on three transduc- tive datasets, i.e., Cora, Citeseer (Kipf & Welling) (2017) and Ogbn-arxiv (Hu et al.) (2020), and two inductive datasets, i.e., Flickr (Zeng et al.) (2020) and Reddit (Hamilton et al.) (2017) . We use the pub- lic splits for all the datasets. For the inductive setting, we follow the setup in (Hamilton et al., (2017) where the test graph is not available during training. Dataset statistics are shown in Appendix [A] We compare our proposed methods to five baselines:  (i) one graph coarsening Baselines. method (Loukas) [2019] [Huang et al.] (2021b), (ii-iv) three coreset methods ( Random , Herd- ing (Welling) (2009) and K-Center (Farahani & Hekmatfar) (2009) Sener & Savarese) (2018)), and (v) dataset condensation (DC). For the graph coarsening method, we adopt the variation neighbor- hoods method implemented by [Huang et al.] (2021Б). For coreset methods, we first use them to select nodes from the original dataset and induce a subgraph from the selected nodes to serve as the reduced graph. In Random, the nodes are randomly selected. The Herding method, which is often used in continual learning (Rebuffi et al.) (2017) Castro et al.) (2018) , picks samples that are closest to the cluster center. K-Center selects the center samples to minimize the largest distance between a sample and its nearest center. We use the implementations provided by (Zhao et al.) (2021) for Herding, K-Center and DC. As vanilla DC cannot leverage any structure information, we develop a variant named DC-Graph, which additionally leverages graph structure during test stage, to replace DC for the following experiments. A comparison between DC, DC-Graph, GC OND and GC OND-X is shown in Table Π and their training details can be found in Appendix [A.3]. Evaluation. We first use the aforementioned baselines to obtain condensed graphs and then eval- uate them on GNNs for both transductive and inductive node classification tasks. For transductive datasets, we condense the full graph with N nodes into a synthetic graph with rN (0 < r < 1) nodes, where r is the ratio of synthetic nodes to original nodes. For inductive datasets, we only condense the training graph since the rest of the full graph is not available during training.  The choices of rP are listed in Table 2. For each r , we generate 5 condensed graphs with different seeds. To evaluate the effectiveness of condensed graphs, we have two stages: (1) a training stage, where we train a GNN model on the condensed graph, and (2) a test stage, where the trained GNN uses the test graph (or full graph in transductive setting) to infer the labels for test nodes. The resulting test performance is compared with that obtained when training on original datasets. All experiments are repeated 10 times, and we report average performance and variance. Hyperparameter settings. As our goal is to generate highly informative synthetic graphs which can benefit GNNs, we choose one representative model, GCN (Kipf & Welling)[2017], for performance evaluation. For the GNN used in condensation, i.e., the GNN g ( - ) in Eq. (8), we adopt SGC (Wul et al.) (2019a) which decouples the propagation and transformation process but still shares similar graph filtering behavior as GCN. Unless otherwise stated, we use 2-layer models with 256 hidden units. The weight decay and dropout for the models are set to 0 in condensation process. More details for hyper-parameter tuning can be found in Appendix A
Comparison with Baselines
4.2
In this subsection, we test the performance of a 2-layer GCN on the condensed graphs, and compare the proposed GC OND and GC OND -X with baselines. Notably, all methods produce both structure. and node features, i.e. A ′ and X ′ , except DC-Graph and GC OND-X . Since DC-Graph and GC OND- X do not produce any structure, we simply use an identity matrix as the adjacency matrix when training GNNs solely on condensed features.  However, during inference, we use the full graph (transductive setting) or test graph (inductive setting) to propagate information based on the trained GNNs.  This training paradigm is similar to the C&S model (Huang et al.) (2021a) which trains an MLP without the graph information and performs label propagation based on MLP predictions. Table [2] reports node classification performance; we make the following observations: Obs 1.  Condensation methods achieve promising performance even with extremely large re- duction rates. Condensation methods, i.e., GC OND , GC OND - X and DC-Graph, outperform coreset methods and graph coarsening significantly at the lowest ratio r for each dataset. This shows the importance of learning synthetic data using the guidance from downstream tasks. Notably, GC OND
Table 2: GC OND and GC OND -X achieves promising performance in comparison to baselines even with extremely large reduction rates. We report transductive performance on Citeseer, Cora, Ogbn- arxiv; inductive performance on Flickr, Reddit. Performance is reported as test accuracy (%).
achieves 79.8%, 80.1% and 79.3% at 1.3%, 2.6% and 5.2% condensation ratios at Cora, while the whole dataset performance is 81.2%. The GC OND variants also show promising performance on Cora, Flickr and Reddit at all coarsening ratios. Although the gap between whole-dataset Ogbn-arxiv and our methods is larger, they still outperform baselines by a large margin. Obs 2. Learning X' instead of (A',X') as the condensed graph can also lead to good results. GC OND -X achieves close performance to GC OND on 11 of 15 cases. Since our objective in graph condensation is to achieve parameter matching through gradient matching, training a GNN on the learned features X ′ with identity adjacency matrix is also able to mimic the training trajectory of GNN parameters. One reason could be that X ′ has already encoded node features and structural in- formation of the original graph during the condensation process. However, there are many scenarios where the graph structure is essential such as the generalization to other GNN architectures (e.g., GAT) and visualizing the patterns in the data. More details are given in the following subsections. Obs 3. Condensing node features and structural information simultaneously can lead to better performance. In most cases, GC OND and GC OND-X obtain much better performance than DC- Graph. One key reason is that GCond and GCond-x can take advantage of both node features and structural information in the condensation process. We notice that DC-Graph achieves a highly comparable result (90.5%) on Reddit at 0.2% condensation ratio to the whole dataset performance (93.9%). This may indicate that the original training graph structure might not be useful. To verify this assumption, we train a GCN on the original Reddit dataset without using graph structure (i.e., setting A gain = 1), but allow using the test graph structure for inference using the trained model. The obtained performance is 92.5%, which is very close to the original performance 93.9%, indicating that training without graph structure can still achieve comparable performance. We also note that learning X 7 , A ′ simultaneously creates opportunities to absorb information from graph structure directly into learned features, lessening reliance on distilling graph properties reliably while still achieving good generalization performance from features. Obs 4.  Larger condensed graph size does not strictly indicate better performance.  Although larger condensed graph sizes allow for more parameters which can potentially encapsulate more. information from original graph, it simultaneously becomes harder to optimize due to the increased
model complexity.  We observe that once the condensation ratio reaches a certain threshold, the performance becomes stable. However, the performance of coreset methods and graph coarsening is much more sensitive to the reduction ratio. Coreset methods only select existing samples while graph coarsening groups existing nodes into super nodes. When the reduction ratio is too low, it becomes extremely difficult to select informative nodes or form representative super nodes by grouping.
GENERALIZABILITY OF CONDENSED GRAPHS
4.3
Next, we illustrate the generalizability of condensed graphs from the following three perspectives. Different Architectures. Next, we show the generalizability of the graph condensation proce- dure. Specifically, we show test performance when using a graph condensed by one GNN model to train different GNN architectures. Specifically, we choose APPNP (Klicpera et al.) (2019), GCN, SGC (Wu et al.) (2019a), GraphSAGE (Hamilton et al.) (2017), Cheby (Defferrard et al.) (2016) and GAT (Velickovic et al.[2018]. We also include MLP and report the results in Table 3] From the ta- ble, we find that the condensed graphs generated by GC OND show good generalization on different architectures. We may attribute such transferability across different architectures to similar filtering behaviors of those GNN models, which have been studied in [Ma et al.](2020); [Zhu et al.](2021a). Versatility of GC OND . The proposed GC OND is highly composable in that we can adopt various GNNs inside the condensation network. We investigate the performances of various GNNs when us- ing different GNN models in the condensation process, i.e., GNN θ( · ) in Eq. ( δ ). We choose APPNP, Cheby, GCN, GraphSAGE and SGC to serve as the models used in condensation and evaluation. Note that we omit GAT due to its deterioration under large neighborhood sizes (Ma et al.) [2021). We choose Cora and Ogbn-arxiv to report the performance in Table円where C and T denote con- densation and test models, respectively. The graphs condensed by different GNNs all show strong transfer performance on other architectures. Neural Architecture Search. We also perform experiments on neural architecture search, detailed in Appendix [C.2] We search 480 architectures of APPNP and perform the search process on Cora, Citeseer and Ogbn-arxiv. Specifically, we train each architecture on the reduced graph for epochs on as the model converges faster on the smaller graph. We observe reliable correlation of performances between condensed dataset training and whole-dataset training as shown in Table 9: 0.76/0.79/0.64 for Cora/Citeseer/Ogbn-arxiv.
Analysis on Condensed Data
4.4
Statistics of Condensed Graphs. In Table 51 we compare several properties between condensed graphs and original graphs. Note that a widely used homophily measure is defined in (Zhu et al.) (2020) but it does not apply to weighted graphs. Hence, when computing homophily, we binarize the graphs by removing edges whose weights are smaller than 0.5. We make the following observations. First, while achieving similar performance for downstream tasks, the condensed graphs contain fewer nodes and take much less storage.  Second, the condensed graphs are less sparse than their Table 4: Cross-architecture performance is shown in test accuracy (%). SAGE: GraphSAGE. Graphs condensed by different GNNs all show strong transfer performance on other architectures.
Table 5: Comparison between condensed graphs and original graphs. The condensed graphs have fewer nodes and are more dense.
larger counterparts.  Since the condensed graph is on extremely small scale, there would be almost no connections between nodes if the condensed graph maintains the original sparsity.  Third, for Citeseer, Cora and Flickr, the homophily information are well preserved in the condensed graphs. Visualization. We present the visualization results for all datasets in Figure 41 where nodes with the same color are from the same class. Notably, as the learned condensed graphs are weighted graphs, we use black lines to denote the edges with weights larger than 0.5 and gray lines to denote the edges with weights smaller than 0.5. From Figure 4 we can observe some patterns in the con- densed graphs, e.g., the homophily patterns on Cora and Citeseer are well preserved. Interestingly, the learned graph for Reddit is very close to a star graph where almost all the nodes only have con- nections with very few center nodes. Such a structure can be meaningless for GNNs because almost all the nodes receive the information from their neighbors. In this case, the learned features X' play a major role in training GNN parameters, indicating that the original training graph of Reddit is not very informative, aligning with our observations in Section 4.2.
5
C ONCLUSION
The prevalence of large-scale graphs poses great challenges in training graph neural networks. Thus, we study a novel problem of graph condensation which targets at condensing a large-real graph into a small-synthetic one while maintaining the performances of GNNs. Through our proposed framework, we are able to significantly reduce the graph size while approximating the original per- formance.  The condensed graphs take much less space of storage and can be used to efficiently train various GNN architectures. Future work can be done on (1) improving the transferability of condensed graphs for different GNNs. (2) studying graph condensation for other tasks such as graph classification and (3) designing condensation framework for multi-label datasets.
ACKNOLWEDGEMENT
Wei Jin and Jiliang Tang are supported by the National Science Foundation (NSF) under grant num- bers IIS1714741, CNS1815636, IIS1845081, IIS1907704, IIS1928278, IIS1955285, IOS2107215, and IOS2035472, the Army Research Office (ARO) under grant number W911NF-21-1-0198, the Home Depot, Cisco Systems Inc. and SNAP Inc.
ETHICS STATEMENT
To the best of our knowledge, there are no ethical issues with this paper.
REPRODUCIBILITY STATEMENT
To ensure reproducibility of our experiments, we provide our source code at https://github. Com/ChandlerBang/GCondl The hyper-parameters are described in details in the appendix. We also provide a pseudo-code implementation of our framework in the appendix.
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