1 Introduction
Graph Neural Networks (GNNs) are deep learningbased methods that have been successfully applied in graph analysis. It is one of the most important machine learning tools for solving graph problems. Unlike other machine learning data, graphs are nonEuclidean data. Many realworld problems can be modeled as graphs, such as knowledge graphs, proteinprotein interaction networks, social networks, etc. The neural networks like Recurrent Neural Networks (RNNs) or Convolutional Neural Networks (CNNs) cannot directly apply to graph data. Hence, GNNs have received more and more attention. Some GNN models have been proposed and obtain promising results on some graph tasks, such as node classification
[8, 7, 20, 14], link prediction [23] and clustering [22].However, most of the GNNs suffer the low expressive power problem due to their shallow architectures. Some works [13, 19]
have been proposed to solve this problem. The design of deep GNNs requires a huge amount of human effort for neural architecture tuning. GNN models are usually very sensitive to the hyperparameters, for different tasks, we might also need to adjust the hyperparameters to obtain the optimal result. For example, the activation function needs to be carefully selected to avoid features degradation
[13], the number of attention heads of GAT [20] needs to be carefully selected for different data, etc. The variants of GNNs may have a better performance in some specific problems. It is impossible to explore all possibilities manually.We notice that the Neural Architecture Search (NAS) has archived great success in designing the CNNs and RNNs for many computer vision and language modeling tasks
[26, 17, 11]. Many NAS methods for CNNs and RNNs have been proposed recently. For example, Zoph et al. [26]apply reinforcement learning to design CNNs for image classification problems. They use a recurrent network controller to generate CNN models and use the validation result of the CNN models as a reward to update the controller. Real et al.
[17] design an evolutionary algorithm to evolve the CNN models from scratch and obtain stateoftheart results. However, these works cannot be applied to GNNs directly.Inspired by the success of NAS in designing CNNs and RNNs, recent works [6, 25] are tried to apply NAS methods to design GNN models for citation networks. They propose to use reinforcement learning to design the GNN models. However, their proposed method can only generate fixedlength GNN models, and the generated GNN models only have shallow architectures. The deep GNNs generated by their methods will suffer the oversmoothing problem.
To overcome the abovementioned problem, we propose a new AutoGraph method that applies an evolutionary algorithm to automatically generate deep GNNs. We first design a new search space and schema for the GNN model, which allows GNN with various layers and covers most of the stateoftheart models. Then we apply evolutionary algorithm and mutation operations to evolve the initial GNN models. Next, we demonstrate a method to search for the best hyperparameters for the new GNN models which allow us to fairly compare the generated models and improve the robustness of our method. Finally, we conduct experiments on both transductive and inductive learning tasks and compare our method with baseline GNNs and the models generated by other reinforcement learning and random search strategies. The results show that we can generate stateoftheart models for all test data efficiently. In summary, our contributions are:

To the best of our knowledge, we are the first to study deep GNNs by using NAS. Our method can automate the architecture engineering process for deep GNNs, which can save many human efforts.

Experiment results show that our proposed method can search for deep GNN models for different tasks efficiently.

The GNN models generated by our method can outperform the handcrafted stateoftheart GNN models.
2 Related Work
Inspired by CNNs [10, 9] and graph embedding [2, 4], GNNs are proposed to collectively aggregate information from graph structure. It is first proposed in [18]. GNNs have been widely applied for graph analysis [24, 21] recently. The target of GNNs is to learn a representation of each node which contains information for its neighborhood. The also called a state embedding of a node. It can be used to produce an output , e.g., the node labels. They can defined as follows [24]:
(1)  
(2) 
where is the transition function that updates the node state according to the neighborhood, is the output function that generates output from the node state and features. ,,, are the features of , the features of its edges, the features and the states of its neighborhood, respectively.
Let , , and
be the stacked vectors of
, , all features (node features, edge features, neighborhood features, etc.) and all the node features. Then the state embedding and output can be defined as:(3)  
(4) 
Due to the shallow learning mechanisms of most GNNs, one major problem of GNNs is the low expressive power limit. The main challenge of this problem is that most of the deep GNNs would suffer from the oversmoothing issue, i.e., the deep model would aggregate more and more node and edge information from neighbors which would lead to the representation of node and edge indistinguishable. Some works have been proposed to solve this problem recently. For example, in the work of [13], the authors show that the Tanh activation function may be more suitable for deep GNNs and they also propose a DenseNet like architecture to alleviate the vanishgradient problem.
To automate neural network exploration, some NAS methods have been proposed. Due to the substantial effort of human experts for discovering the stateoftheart neural network architectures, there has been a growing interest in developing an automatic algorithm to design the neural network architecture automatically. Recently, the architectures generated by NAS have achieved stateoftheart results in tasks like image classification, object detection or semantic segmentation. Most of the NAS methods are based on Reinforcement Learning (RL) [26, 27, 15] and Evolutionary Algorithm (EA) [17, 16].
Although the aforementioned NAS methods have successfully designed CNN or RNN architectures for image and language modeling tasks, the GNN is very different from CNN or RNN. Thus they cannot be directly applied to the GNN architecture search. Gao et al. [6] and Zhou et al. [25] propose a new schema to encode the GNN architecture and apply reinforcement learning to search for GNN models, but their methods cannot generate deep GNNs and their methods are not efficient and robust enough.
3 Method
In this section, we first define the AutoGraph problem. Then we describe our search space and schema to represent GNN architectures. Next, we show our evolutionary algorithm for the AutoGraph. Finally, we show a method to improve the robustness of the search process.
3.1 Problem Statement
The AutoGraph problem can be formally defined as follows. Given search space , the target of our algorithm is to search the optimal GNN architecture which minimizes the validation loss . It can be written as follows:
(5)  
s.t.  (6) 
where denotes the optimal parameters learned for the architecture in the training set. This is a bilevel optimization problem [6].
We propose an efficient method to solve this problem based on the evolutionary algorithm. Each generated architecture is trained and obtains the optimal weight of in the training set, then it is evaluated in the validation set. At last, the best architecture in the validation set is reported. The following sections explain the process in more detail.
3.2 Search Space
Many stateoftheart GNNs would suffer from the oversmoothing problem which makes the representation of even distant nodes indistinguishable [24]. The recent work [13]
shows that Tanh is better than ReLU for keeping linear independence among column features for GNNs. They propose a denselyconnected graph network which is similar to DenseNet as follows:
(7)  
(8)  
(9) 
where and are activation functions; , and are learnable parameters, is the number of input channels in layer . This architecture stacks all the outputs of previous layers as the input of current layers. It can increase the variety of features for each layer, encourage the feature reuse, alleviate the vanishing gradient problem. However, concatenating all the outputs of previous layers will cause the parameters of the GNNs to increase exponentially.
Inspired by this, we allow each layer of our generated GNN models to connect to a various number of previous layers. To generate deep GNNs, we also allow our method to add a new layer to the GNN model during the searching process. So we define the search space and schema of our method as follows. We first apply the same setting of Attention Function, Attention Head, Hidden Dimension, Aggregation Function and Activation Function in [6]. Then we introduce two new states:

Skip Connection. It has been observed that most GNN models deeper than two layers could not perform well because of the noisy information from expanding neighbors. This problem usually can be addressed by skip connection. Inspired by Luan et al. [13], we allow skip connections between any previous layers to the current layer. For each previous layer, represents no skip connection, represents there is a skip connection between that layer to the current layer, e.g., Fig. 2.

Layer Add^{1}^{1}1“Layer Add” state is only used in the evolutionary process. This state is only used during the mutation process. When this state is selected, we duplicate the current layer and add the new layer after the current layer. This state allows our method to extend the depth of GNNs automatically.
Noted that most of the GNN layers can be represented by the above first six states, as shown in Fig. 1. The above search space can cover a wide variety of stateoftheart GNN models. If the skip connections are applied then the input dimension of the current layer would be the sum of all the output dimensions of the connected layers.
3.3 Evolutionary Algorithm
Inspired by Real et al. [16], we apply the Aging Evolution Algorithm to search for the deep GNNs. Similar to most of the evolutionary algorithms, our algorithm can be divided into three stages, i.e., initialization, mutation and updating. In the initialization stage, we randomly generate GNN models with two layers. is the size of the population. The initial models are trained and evaluated. Then they are added to the population.
In the mutation stage, we sample candidates from the population. The candidate with the highest score in the sample set is selected to apply mutation. We randomly select one state in the search space and change it to a new value in the state set. Then the newly generated candidate is trained and evaluated. Next, the new candidate needs to be added to the population. Since we need to keep the population size unchanged, we would select the oldest candidate in the population and remove it before we add the new candidate to the population. This is the main difference between the Aging Evolution Algorithm and other evolutionary algorithms.
We allow multiple skip connections for each layer. The skip connection between the previous layer to the current layer can be represented by binary . Since there is always a connection between layer to layer , we only need to consider ( represents the input of the network). Thus, the skip connections state of layer can be represented as
(10) 
Then the possible state of is [,). When , i.e., the current layer is the first layer, the skip connection state would be always . Figure 2 shows an example of skip connection representation. To avoid a significant change of the GNN model, each mutation operation will only change one state of the model. During the search process, every evaluated GNN model is added to the history list. After the whole search process is finished, the model with the highest score in the history list will be reported.
3.4 GNNs Evaluation
We notice that the GNN model is sensitive to change in hyperparameters, such as the learning rate and weight decay. The best performance of a GNN architecture can be achieved at different learning rates, weight decay and iteration number. If we use the same hyperparameters to train and evaluate different GNN architectures, we may miss the best GNN model because the hyperparameters are not set properly. To fairly compare the architecture, we apply the hyperparameters tuning for each generated GNN model.
The work of Bergstra et al. [1]
shows that the Treestructured Parzen Estimator Approach (TPE) performs well on the hyperparameter search. We use the TPE algorithm to search the hyperparameters for each GNN model. To avoid overfitting and speed up the search process. We allow early stops during the training process. For each GNN architecture, we will use the best performance reported by the TPE algorithm as the performance of the architecture. The comparison between different GNN models is based on the performance of their best hyperparameter settings.
4 Experiments
We conduct experiments in both transductive and inductive learning tasks. For the transductive learning task, we test our method on the Cora, Citeseer and Pubmed datasets. For the inductive learning task, we test on the proteinprotein interaction (PPI) dataset. Our method is evaluated in the following aspects:

Performance. We evaluate the performance of our AutoGraph method by comparing the generated GNN model with the handcrafted stateoftheart GNN models.

Efficiency. We analyze the efficiency of our method by comparing it with other search strategies, i.e., GraphNAS (a reinforcement learningbased method) and random search.

Scalability. We analyze the scalability of our method by comparing the performance of GNN models with different layers.
4.1 Experimental Setup
The configuration of our method in the experiments is set as follows. The population size is 100. The max evaluation architecture is 2,000. The maximum training iterations is 1,000. As described in the Methods, the mutation probabilities are uniform. The generated GNN architecture is trained with the ADAM optimizer. The maximum hyperparameters search number for the TPE algorithm is 50. We run the search algorithm in four RTX 2080 Ti GPU cards. For each task, the best model which has the lowest validation loss is selected as our GNN model to compare with other baseline models.
4.2 Datasets
Transductive Learning. In transductive learning tasks, the same graphs are observed during training and testing. The experiment datasets for the transductive learning are Cora, Citeseer and Pubmed. In these datasets, the nodes represent the documents and the edges (undirected) represent citations. The features of the nodes are got by the bagofwords representation of the documents. The Cora dataset contains 2,708 nodes and 5,429 edges. We will use 140 nodes for training, 500 nodes for validation and 1,000 nodes for testing. The Citeseer dataset contains 3,327 nodes and 4,732 edges. The training, validation and test set separations are the same as the setup of [20].
Inductive Learning. In inductive learning tasks, the graphs in training and testing are different. The experiment dataset for inductive learning is the proteinprotein interaction (PPI). The graphs in this dataset represent different human tissues. There are 20 graphs in the training set, two in the validation set and two in the test set. The data in the test set is completely unobserved during training.
The statistical detail of transductive learning and inductive learning datasets is shown in Table 1. The Cora, Citeseer and Pubmed datasets are classification problems. The PPI dataset is a multilabel problem.
Cora  Citeseer  Pubmed  PPI  
Task  Transductive  Transductive  Transductive  Inductive 
# Nodes  2,708 (1 graph)  3,327 (1 graph)  19,717 (1 graph)  56,944 (24 graphs) 
# Edges  5,429  4,732  44,338  818,716 
# Features/Node  1,433  3,703  500  50 
# Classes  7  6  3  121 (multilabel) 
# Training Nodes  140  120  60  44,906 (20 graphs) 
# Validation Nodes  500  500  500  6,514 (2 graphs) 
# Test Nodes  1,000  1,000  1,000  5,524 (2 graphs) 
4.3 Baseline Methods
We compare the GNN model generated by our approach with the following stateofthearts methods:

Chebyshev [3]
. This method removes the need to compute the eigenvectors of the Laplacian by using
localized convolution to define a graph convolutional neural network. 
GCN [8]. This method alleviates the problem of overfitting by limiting the layerwise convolution operation to .

GAT [20]. This method introduces the attention mechanism to GNN. It obtains good results in many graph tasks.

LGCN [5]. It introduces regular convolutional operations to GNN.

GraphSAGE [7]. This method can be applied to inductive tasks. It samples and aggregates features from a node’s neighborhood.

GeniePath [12]. It uses an adaptive path layer which consists of two complementary functions.
We use the public released implementations of these methods to do the comparisons. The evaluation metric for transductive learning tasks is accuracy. For the inductive learning task, we use the microF1 score.
To evaluate the efficiency of our method, we also compare our method with GraphNAS and random search. GraphNAS applies a reinforcement learning controller to generate GNN models. For the random search baseline, we randomly sample GNN models from the same search space in our approach.
Models  Cora  Citeseer  Pubmed 

Chebyshev  
GCN  
GAT  
LGCN  
GraphNAS  
AutoGraph 
Models  microF1 

GraphSAGE (lstm)  
GeniePath  
GAT  
LGCN  
GraphNAS  
AutoGraph 
4.4 Results
After our algorithm generates 2,000 GNN models, the model which has the lowest loss in the validation set is selected and tested on the test set. The experiment results of transductive learning datasets are summarized in Table 2. The results of the inductive learning dataset are summarized in Table 3.
Performance. For the transductive learning tasks, we compare the classification accuracy with the abovementioned GNN model and GraphNAS. From Table 2 we can see that our generated model can get the stateoftheart result in all transductive datasets.
For the inductive task, we compare the microF1 score with the popular GNN models and GraphNAS. The result shows that our method also performs well in the inductive dataset.
Method  Accuracy  Time (GPU hours)  Best GNN Layers 

Random Search  10  2  
GraphNAS  10  2  
AutoGraph 
Efficiency. To evaluate the effectiveness of our search method, we compare our method with different search strategies, i.e., random search and reinforcement learningbased search method—GraphNAS [6]. Since GraphNAS does not do the hyperparameters tuning when evaluating the GNNs, we also disable our hyperparameters tuning during the search process. During the training process, we record the generated architectures and their performance. From the Table 4, we can see that our method can search for a better GNN model with less time and our method can generate deeper GNNs.
Scalability. We know that most of the handcraft GNNs would suffer from the oversmoothing problem. We compare the performance of the GNNs generated by our method with different layers. Figure 3 shows the best performance of the GNNs generated by our method from two layers to nine layers. We can see that our generated GNN models have good performance in deep architectures.
5 Discussion & Conclusion
In this work, we study the problem of AutoGraph. We present an efficient evolutionary algorithm to search for GNN models. We can see that our method can generate deep GNNs which alleviate the oversmoothing problem. The experiments show that the generated models can outperform current handcraft stateoftheart models. In summary, we can see our proposed method has the following advantages:

It can save substantial efforts to explore good GNN models for different graph tasks.

Our generated GNN models can get stateoftheart results.

Our approach can generate deep GNN models which can alleviate the oversmoothing problem.
Although our proposed method can design stateoftheart GNNs for graph tasks, it is remarkable that there are still many improvements that can be made. The first problem is that the search process is timeconsuming. We notice that some approaches to reduce the search time have been proposed in NAS for CNNs. However, most of them cannot be directly applied to GNNs, we need to design a proper improvement method for GNNs. The second problem is that the search space in our method is still limited, we can try to design a better search space to explore more novel GNNs. We will focus on these two problems in our future works.
References
 [1] Bergstra, J., Bardenet, R., Bengio, Y., Kégl, B.: Algorithms for hyperparameter optimization. In: Advances in Neural Information Processing Systems 24: 25th Annual Conference on Neural Information Processing Systems 2011. Proceedings of a meeting held 1214 December 2011, Granada, Spain. pp. 2546–2554 (2011)
 [2] Cui, P., Wang, X., Pei, J., Zhu, W.: A survey on network embedding. IEEE Trans. Knowl. Data Eng. 31(5), 833–852 (2019)
 [3] Defferrard, M., Bresson, X., Vandergheynst, P.: Convolutional neural networks on graphs with fast localized spectral filtering. In: Advances in Neural Information Processing Systems 29: Annual Conference on Neural Information Processing Systems 2016, December 510, 2016, Barcelona, Spain. pp. 3837–3845 (2016)
 [4] Fu, X., Zhang, J., Meng, Z., King, I.: MAGNN: metapath aggregated graph neural network for heterogeneous graph embedding. In: WWW ’20: The Web Conference 2020, Taipei, Taiwan, April 2024, 2020. pp. 2331–2341 (2020)
 [5] Gao, H., Wang, Z., Ji, S.: Largescale learnable graph convolutional networks. In: Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, KDD 2018, London, UK, August 1923, 2018. pp. 1416–1424 (2018)
 [6] Gao, Y., Yang, H., Zhang, P., Zhou, C., Hu, Y.: Graphnas: Graph neural architecture search with reinforcement learning. CoRR abs/1904.09981 (2019)
 [7] Hamilton, W.L., Ying, Z., Leskovec, J.: Inductive representation learning on large graphs. In: Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, 49 December 2017, Long Beach, CA, USA. pp. 1024–1034 (2017)
 [8] Kipf, T.N., Welling, M.: Semisupervised classification with graph convolutional networks. In: 5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 2426, 2017, Conference Track Proceedings (2017)

[9]
Krizhevsky, A., Sutskever, I., Hinton, G.E.: Imagenet classification with deep convolutional neural networks. In: Advances in Neural Information Processing Systems 25: 26th Annual Conference on Neural Information Processing Systems 2012. Proceedings of a meeting held December 36, 2012, Lake Tahoe, Nevada, United States. pp. 1106–1114 (2012)
 [10] LeCun, Y., Bottou, L., Bengio, Y., Haffner, P., et al.: Gradientbased learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998)

[11]
Li, Y., King, I.: Architecture search for image inpainting. In: Lu, H., Tang, H., Wang, Z. (eds.) Advances in Neural Networks  ISNN 2019  16th International Symposium on Neural Networks, ISNN 2019, Moscow, Russia, July 1012, 2019, Proceedings, Part I. Lecture Notes in Computer Science, vol. 11554, pp. 106–115. Springer (2019)

[12]
Liu, Z., Chen, C., Li, L., Zhou, J., Li, X., Song, L., Qi, Y.: Geniepath: Graph neural networks with adaptive receptive paths. In: The ThirtyThird AAAI Conference on Artificial Intelligence, AAAI 2019, The ThirtyFirst Innovative Applications of Artificial Intelligence Conference, IAAI 2019, The Ninth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2019, Honolulu, Hawaii, USA, January 27  February 1, 2019. pp. 4424–4431 (2019)
 [13] Luan, S., Zhao, M., Chang, X.W., Precup, D.: Break the ceiling: Stronger multiscale deep graph convolutional networks. In: Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, 814 December 2019, Vancouver, BC, Canada. pp. 10943–10953 (2019)

[14]
Manessi, F., Rozza, A., Manzo, M.: Dynamic graph convolutional networks. Pattern Recognit.
97 (2020)  [15] Pham, H., Guan, M.Y., Zoph, B., Le, Q.V., Dean, J.: Efficient neural architecture search via parameter sharing. In: Proceedings of the 35th International Conference on Machine Learning, ICML 2018, Stockholmsmässan, Stockholm, Sweden, July 1015, 2018. pp. 4092–4101 (2018)

[16]
Real, E., Aggarwal, A., Huang, Y., Le, Q.V.: Regularized evolution for image classifier architecture search. In: The ThirtyThird AAAI Conference on Artificial Intelligence, AAAI 2019, The ThirtyFirst Innovative Applications of Artificial Intelligence Conference, IAAI 2019, The Ninth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2019, Honolulu, Hawaii, USA, January 27  February 1, 2019. pp. 4780–4789 (2019)
 [17] Real, E., Moore, S., Selle, A., Saxena, S., Suematsu, Y.L., Tan, J., Le, Q.V., Kurakin, A.: Largescale evolution of image classifiers. In: Proceedings of the 34th International Conference on Machine Learning, ICML 2017, Sydney, NSW, Australia, 611 August 2017. pp. 2902–2911 (2017)
 [18] Scarselli, F., Gori, M., Tsoi, A.C., Hagenbuchner, M., Monfardini, G.: The graph neural network model. IEEE Trans. Neural Networks 20(1), 61–80 (2009)
 [19] Sun, K., Lin, Z., Zhu, Z.: Adagcn: Adaboosting graph convolutional networks into deep models. CoRR abs/1908.05081 (2019)
 [20] Velickovic, P., Cucurull, G., Casanova, A., Romero, A., Liò, P., Bengio, Y.: Graph attention networks. In: 6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30  May 3, 2018, Conference Track Proceedings (2018)
 [21] Wu, Z., Pan, S., Chen, F., Long, G., Zhang, C., Philip, S.Y.: A comprehensive survey on graph neural networks. IEEE Transactions on Neural Networks and Learning Systems (2020)
 [22] Ying, Z., You, J., Morris, C., Ren, X., Hamilton, W.L., Leskovec, J.: Hierarchical graph representation learning with differentiable pooling. In: Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, 38 December 2018, Montréal, Canada. pp. 4805–4815 (2018)
 [23] Zhang, J., Shi, X., Zhao, S., King, I.: STARGCN: stacked and reconstructed graph convolutional networks for recommender systems. In: Proceedings of the TwentyEighth International Joint Conference on Artificial Intelligence, IJCAI 2019, Macao, China, August 1016, 2019. pp. 4264–4270 (2019)
 [24] Zhou, J., Cui, G., Zhang, Z., Yang, C., Liu, Z., Sun, M.: Graph neural networks: A review of methods and applications. CoRR abs/1812.08434 (2018)
 [25] Zhou, K., Song, Q., Huang, X., Hu, X.: Autognn: Neural architecture search of graph neural networks. CoRR abs/1909.03184 (2019)
 [26] Zoph, B., Le, Q.V.: Neural architecture search with reinforcement learning. In: 5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 2426, 2017, Conference Track Proceedings (2017)
 [27] Zoph, B., Vasudevan, V., Shlens, J., Le, Q.V.: Learning transferable architectures for scalable image recognition. In: 2018 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2018, Salt Lake City, UT, USA, June 1822, 2018. pp. 8697–8710 (2018)
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