Dual Self-Paced Graph Convolutional Network: Towards Reducing Attribute Distortions Induced by Topology

1 Introduction

Graph partitioning has contributed tremendously in both the traditional unsupervised node classification techniques and the latest semi-supervised node classification approaches [Kipf and Welling, 2017; Veliˇckovi´c et al., 2018]. From either the spectral or spatial perspectives, many graph convolutional neural networks (GCNNs) have been proposed recently

∗Corresponding author.

[Defferrard et al., 2016; Hamilton et al., 2017]. By incorporating supervised information, GCNN based semi-supervised node classification methods have significantly improved the performances of the traditional unsupervised techniques.

Similar to other semi-supervised learning tasks [Chapelle et al., 2009; Zhu, 2006], semi-supervised node classification requires efficient incorporations of the unsupervised information. When classifying the nodes in the attributed graphs, the attributes of the unlabelled nodes serve as the unsupervised information. Due to the sparsity of the node attributes, prediction only based on the original node attributes cannot fully exploit their relationships and usually results in overfittings. Therefore, many GCNNs [Kipf and Welling, 2017; Veliˇckovi´c et al., 2018] perform operations which are equivalent to the attribute smoothing (propagating) over the graph [Li et al., 2018]. Therefore, GCNNs actually utilizes the unsupervised information by augmenting the node attributes in local neighbourhoods.

Unforunately, this propagation strategy possesses certain weaknesses. Typically, over propagation may degrade the prediction performance [Li et al., 2018]. Since the over propagation tends to amend all the nodes to have the identical augmented attributes which certainly reduces the discriminabilities of the nodes. Therefore, existing GCNNs only employ two stacked graph convolutional layers, i.e., two-hop propagations to prevent the over propagation. Besides, the attributes may be interfered by the utilization of the topology information. This distortion will induce a certain amount of misclassification of the nodes, which can be correctly predicted with only the attributes, as illustrated in Fig. 1.

In this paper, we intend to reduce the negative effects, i.e., the distortions, caused by the utilization of the topology. Thus, the impact of the edges in attribute propagations is analyzed. As elaborated in Fig. 2, there exists two types of edges. If the two nodes vi and vj, whose attributes are similar, are connected by an edge (green line), the attributes of each node will vary slightly after propagations. Since this kind of edges usually contain less information and the network can quickly learn from them, we denote them as the simple edges. If the two nodes vi and vk, whose attributes are quite different, are connected by an edge (red line), the attributes of each node will be significantly changed after propagations. This kind of edges contain rich information and the network usually requires more trainings to learn from them. Therefore, these edges are denoted as the complex edges.

Spontaneously, a question can be raised that whether these two kinds of edges should be learned simultaneously by the network in the training stage? Curriculum Learning (CL [Bengio et al., 2009]) and Self-Paced Learning (SPL [Kumar et al., 2010]), which mimic the learning process of human beings, conclude that by learning from simple concepts to complex ones, the learning of the network can be significantly improved. Based on this philosophy, we believe that the topology information should be evaluated and feed into the network gradually from the simple edges to the complex ones during the training process of GCNNs.

To reduce the distortions generated by utilizing the topology information while exploit more potentials of the attribute information, Dual Self-Paced Graph Convolutional Network (DSP-GCN) is proposed in this paper. Specifically, the unlabelled nodes with confidently predicted labels are gradually included into the training set in the node-level self-paced learning to better exploit the attribute information. To reduce the distortions when utilizing the topology information, edges are gradually, from the simple edges to the complex ones, added into the graph during the training process in the edge-level self-paced learning. Since the nodes and edges are highly correlated, the node-level and edge-level self-paced learnings are also correlated. Therefore, these two learning strategies are designed to mutually reinforce each other by coupling the selections of the edges and unlabelled nodes. If an unlabelled node is included during the training process, its edges will likely to be included. On the contrary, if most of the edges of an unlabelled node is added, this node as well as its predicted label tend to be put into the training with a high probability.

Our contributions are summarized as below:

• We observe that the utilization of the topology in GCNNs tends to interfere the correct prediction results which can be obtained with respect to the attributes only, according to the experiments.
• We propose an edge-level self-paced learning strategy by gradually train the network from the simple edges to the complex ones to reduce the negative effects caused by the utilization of the topology, instead of feeding the entire topology directly into the network.
• We propose a Dual Self-Paced Graph Convolutional Network (DSP-GCN) which jointly exploits the node-

level and edge-level self-paced learning strategies.

• Experimental results on eight real networks indicate that DSP-GCN can successfully reduce the distortion induced by the topology and yield superiority performances with only one graph convolutional layer.

2 Preliminaries

2.1 Notations

A network can be represented by an attributed graph G = (V,E,X). V = {vi|i = 1, ...,N} is a set of |V | = N vertices, where vi is associated with a feature xi ∈ RK . X ∈ RN×K represents the collection of the features. Each row of X corresponds to a node. E stands for a set of edges. Each of the edges connects two vertices in V . The adjacency matrix A = [aij] ∈ RN×N is obtained according to the network topology. If an edge connects the vertices vi and vj, aij = 1, and vice versa. If self-edges are allowed in the network, then ann = 1. Otherwise ann = 0. an, which denotes the nth column of A, can be utilized to represent the local neighbourhood of the vertex vn. dn = ∑j anj stands for the degree of vn while D = diag(d1, d2, ..., dN) is the degree matrix of A. The graph Laplacian and its normalized form are defined as L = D - A and L = D- 12LD- 12 , respectively. For a network, given the labels Y ∈ R|Vl|×F (where F represents the number of classes), which belong to a set of vertices Vl ⊂ V , a typical semi-supervised node classification algorithm classifies other nodes in V - Vl according to the attributed graph. For simplicity, the first l nodes {vi}li=1 are assumed to be labelled.

2.2 Graph Convolutional Network

By simplifying the complex existing models, Graph Convolutional Network (GCN) [Kipf and Welling, 2017] defines the graph convolution operation as

where ˜ A = A+IN and ˜Dnn = ∑j ˜ Anj = dn+1. Then, the to-be-predicted labels can obtained by feeding HGCN into a fully-connected layer as

By minimizing the cross-entropy, which is defined in Eq. (3), between the predictions and given labels, the parameter W can be obtained.

[Li et al., 2018] concludes the mechanism and success of GCN that it is equivalent to perform a symmetric Laplacian smoothing (HGCN = ˜D- 12 ˜ A ˜D- 12 ) operation before the actual predictions. It aims at augmenting the attributes with topology to alleviate the problem of attribute sparsity. The performance of the smoothing based classification obviously outperforms the classification based on the original attributes. Unfortunately, the employment of the topology information also induces certain negative effects to the classification results. As illustrated in Fig. 1, some nodes (i.e. the non-overlapped orange set), which can be correctly classified according to the original attributes, are actually misclassified with respect to the smoothed attributes after the attribute propagations.

2.3 Self-paced Learning

Inspired by the learning process of human beings, i.e., learning from easy concepts to complex ones, [Bengio et al., 2009] proposes a curriculum learning (CL) strategy which gradually includes the training samples from the easy samples to the complex ones during the training process. Self-paced Learning (SPL [Kumar et al., 2010]) adopts this strategy by incorporating the curriculum design as a regularization term into the learning objective function as

where f(xi, w) represents the prediction function with learnable model parameter w, ℓ(yi, f(xi,w)) stands for the loss function between the prediction f (xi, w) and ground truth yi, and qi ∈ {0, 1} is the sample weight to indicate whether the data point xi has been included in the training process. Note that the second term λ ∑Ni=1 qi is the self-paced regularization term, with λ being the pace parameter which controls the learning pace. The model is trained by jointly minimizing the objective function in Eq. (4) with respect to the model parameter w and the sample weights qi's by gradually increasing the pace parameter λ. For a fixed w, the optimal q∗ i is

where ℓi = ℓ(yi, f(xi,w)) represents the loss of predicting the data sample xi as f(xi, w). With a small λ, only the data samples with small losses (i.e. easy to be predicted) are included in the training data. When λ gradually increases, more complex data samples are added.

Self-paced learning has already been extended to semisupervised learning and unsupervised matrix factorization. Self-paced semi-supervised learning augments the training set with the unlabelled data samples based on the confidence predictions [Ma et al., 2017]. The unlabelled data samples are gradually included by combining the self-paced strategy and co-training of two views. Meanwhile, self-paced matrix factorization extends the previous self-paced learning strategy which is designed for the data samples to a self-paced learning for the pairs of data samples and their corresponding attributes [Zhao et al., 2015].

Self-paced Network Embedding

Recently, self-paced learning is applied to network embedding. [Zhou et al., 2018] follows the co-training strategy in [Ma et al., 2017] and extends the co-training of two views to two tasks, i.e., embedding and prediction. However, this node-level self-paced strategy cannot reduce the negative impact to the attributes induced by the topology. [Gao and Huang, 2018] adopts the self-paced strategy to select the negative samples and performs a self-paced training with the pair of data samples similar to [Zhao et al., 2015]. Unfortunately, self-paced negative sampling requires to compute the similarities between almost all the pairs of nodes, which is computationally inefficient. Besides, similar to other semi-supervised node classification methods introduced above, these two methods also ignore the impact of topology in the self-paced node selections.

3 ProposedWork

In this section, Dual Self-Paced Graph Convolutional Network (DSP-GCN), which reduces the topology distortions on the attributes, is proposed. DSP-GCN consists of two interactive self-paced learning strategies, node-level and edge-level strategies. The node-level self-paced strategy gradually includes the unlabelled nodes, whose labels are predicted with high confidence scores, into the training set. The edge-level self-paced strategy gradually incorporates the edges, from the simple edges to the complex ones, into the graph convolutional operation of GCN. Note that both the strategies will mutually reinforce each other, because the nodes and edges in the graph is usually highly correlated. If an unlabelled node is added into the training set, its edges tend to be included. Meanwhile, if most of the edges of an unlabelled node has already been included , this node as well as its predicted label will likely to be added into training set.

3.1 Node-level Self-Paced Learning

Our node-level self-paced learning augments the training set to fully exploit the unsupervised information (i.e., the attributes of the unlabelled nodes). Since we gradually add the unlabelled nodes, which possess the predicted labels with high confidence scores, into the training set, the objective function can be formulated as

where the three terms at the right-hand side of the equation are a supervised loss, an unsupervised loss and a self-paced regularizer, respectively. yi and yk denote the given labels of the labelled nodes and the predicted labels of the unlabelled nodes, respectively. qk ∈ {0, 1} is the weight assigned to the unlabelled node vk to indicate whether it as well as its predicted label yk has been added into the training set. λnode is the pace parameter which controls the learning pace. If λnode increases, more unsupervised nodes will be added into the training set, i.e., more unsupervised nodes satisfy the condition of q∗ k = 1, and vice versa. The prediction model g(xk,A,W) is identical to GCN as shown in Eqs. (1) and (2), and the loss function ℓ(., .) is the cross-entropy as shown in Eq. (3). With a fixed W, minimizing Eq. (6) with respect to qk gives the same optimal solution as Eq. (5) presents.

3.2 Edge-level Self-Paced Learning

Unfortunately, optimizing Eq. (6) with a fixed topology A is equivalent to directly add all the edges (i.e., the entire topology) into the training of GCN, which may induce certain negative effects to the attributes as elaborated in Fig. 1. To alleviate this issue, the negative impact of the topology to the discriminabilities of the node attributes is considered and formulated. Since there exists different kinds of edges, which possess different degrees of training difficulties and have been illustrated in Sec. 1 and Fig. 2, we propose to exploit the self-paced learning strategy [Kumar et al., 2010] to gradually include edges with different degrees of training difficulties (from the simple edges to the complex ones) into the training process. To measure the difficulty of learning from the edge eij, we exploit the attribute variations on the nodes vi and vj after the attribute propagations as the measurement, which also indicates the amount of information possessed by eij . This quantity can be described as the inner product of the attributes of nodes vi and vj as

where B is the learnable parameter to measure the consistency between the corresponding attributes of the two nodes. For example, "learning" and "inference" are two similar attributes in machine learning community. If the attribute 'learning" is propagated to a node vi which possesses the attribute "inference" yet without the attribute 'learning", the attribute amendment occurred on node vi tends to be minor and only slightly increases the entropy of node vi.

Instead of adding a self-paced regularizer as the node-level strategy, our edge-level self-paced learning strategy directly selects the edges which are included in the training process via

where ˆaij is edge weight and λedge is the pace parameter to control the learning pace of the edges. When λedge is small, only the edges eij with high sij , which will not significantly increase the attribute entropy of the connected nodes, is added. When λedge increases, more edges will be included.

3.3 Dual Self-Paced Learning

The node-level self-paced learning in Eq. (6) and edgelevel self-paced learning in Eq. (8) can be respectively tuned with two pace parameters λnode and λedge. Meanwhile, both strategies will mutually reinforce each other.

Intuitively, if the prediction confidence of an unsupervised node is high, this node will be added into the training set according to the self-paced learning strategy. The high prediction confidence of a node is usually induced by the consistency between it and its neighbourhoods. Besides, this consistency indicates that the edges between them are the simple ones, as shown in Fig. 2. Then, these edges are also added into the graph. On the other hand, the addition of certain edges into the training set may also induce the addition of certain nodes in a similar manner to the above situation.

To model the interactions between the two strategies, the weight qk of unlabelled node and weight ˆaij of edge are jointly considered to form the loss of the interactions as

where ˆ dk = ∑j ˆakj represents the current degree of the node vk in graph ˆ A. ˆ dk + 1, which is equivalent to adding a selfloop to each node as GCN, is employed to avoid the zero divisor. Since the edge-level self-paced learning in Eq. (8) is performed as a constraint instead of a loss term as the nodelevel self-paced learning. A simple combination of the two losses in Eqs. (6) and (9) cannot achieve the reinforcement from the node-level learning to the edge-level learning. To alleviate this issue, Eq. (8) is modified as

If the unlabelled node vi has been added to the training set, i.e., qi = 1, the probability of edge eij being included will increase from λedge to λedge +γ qi ˆ di+1 .

3.4 Objective Function and Optimization

By combining Eqs. (6), (9) and (10), the overall objection function is constructed as

with the hyper-parameters γ, λnode and λedge. The parameters to be learned in Eq. (11) are W, B, yk, qk and ˆaij. This objective function can be minimized according to an alternative optimization strategy. The optimization process are shown in Algorithm 1 with the main steps elaborated as follows:

Update qk: This step is equivalent to minimizing L in Eq. (11) without the supervised term and the constraint. Since qk ∈ {0, 1}, its optimal solution is

Here, ℓk = ℓ(yk, g(xk, ˆ

A,W)) represents the difference between the predictions of two consecutive iterations, where yk stands for the predicted label in the previous iteration, and g(xk, ˆ A,W) represents the predicted label in the current iteration according to the new ˆaij, W and B. The difference threshold is λnode+γ ∑j ( ˆakj ˆ dk+1 ), where the first term λnode is the pace parameter. The second term γ ∑j ( ˆakj ˆ dk+1 ) indicates the probability of this node vk being included in the training set, if more edges ˆakj of the unlabelled node vk is included. By updating qk, we can improve the self-paced node selections with our self-paced edge selections.

Update yk: Since only the second term of L in Eq. (11) contains the variable yk, the optimal y∗ k can be obtained by minimizing ℓ(yk, g(xk, ˆ A,W)). Therefore,

where g(xk, ˆ A,W) is identical to GCN as introduced in Eqs. (1) and (2), i.e., label is predicted by GCN with parameters W, B and the topology ˆaij obtained in the previous iteration.

Update ˆaij: The optimal solution of aij has been shown in Eq. (10). By updating aij, the proposed self-paced edge selections can be further enhanced with respect to the selfpaced node selections.

Update W and B: In this step, we only have to optimize the first two terms of L in Eq. (11), i.e.,

whose parameters are W and B. Here, only the unlabelled nodes, which are included in the training set, i.e., qk = 1, will contribute to the second term. Since the cross-entropy in Eq. (3) is adopted as a loss, the gradient back-propagation can be utilized to obtain W and B similar to GCN [Kipf and Welling, 2017] and GAT[Veliˇckovi´c et al., 2018].

Remark 1: Comparing with the existing self-paced network embedding approaches, DSP-GCN possesses two obvious advantages. 1) DSP-GCN considers self-paced learning on edges to reduce the negative effects caused by employing the topology. 2) DSP-GCN considers and models the correlations between the node-level and edge-level self-paced learning to improve the prediction efficiencies.

Remark 2: The graph construction is the updating of ˆaij in Eq. (10) with sij defined in Eq. (7). The computational complexity of each edge sij is O(K), where K is the number of features. Therefore, the overall computational complexity of the graph construction is O(MK) where M is the number of edges in the graph, i.e., the computations of the graph construction will linearly increase when the scale of the network increases.

4 Experimental Results

In this section, we validate the proposed DSP-GCN by empirically evaluating the performances in the transductive semisupervised node classification task. In DSP-GCN, only one graph convolutional layer is employed instead of the two layers utilized in GCN and GAT. Here, we set γ = 1 and initialize λnode = 0.12 and λedge = 0.2 on all the networks. Note that λnode and λedge are augmented via two multipliers ranging from 1.05 to 1.11, respectively.

4.1 Datasets

In the experiments, three common citation networks [Sen et al., 2008], Cora, CiteSeer and PubMed, and a knowledge graph NELL [Carlson et al., 2010] are employed as shown in Table 1. In each citation network, papers and undirected citations are defined as the nodes and edges, respectively. The node content is represented by the bag-of-word representation of the documents. Papers are classified into various categories according to their disciplines. A bipartite graph is extracted from the knowledge graph as in [Yang et al., 2016]. Besides of the entity nodes, each relation tuple (ei, r, ej) is decomposed into two connected relation nodes (ei, r) and (ej , r). The topology is constructed by adding edges between entity node ei and relation node (ei, r). Besides, five more networks, including Cornell, Texas, Washington, Wisconsin and Wiki, are employed. Four of them, i.e., Texas, Cornell, Washington and Wisconsin, are the sub-networks of the WebKB network. Each of them is the collection of the webpages from an university in U.S.. Similarly, nodes in Wiki network are the webpages fromWikipedia.

4.2 Methods

For comparisons, we employ 11 state-of-the-art semisupervised node classification algorithms, including multilayer perceptron (MLP), label propagation (LP) [Zhu et al., 2003], semi-supervised embedding (SemiEmb) [Weston et al., 2012], manifold regularization (ManiReg) [Belkin et al., 2006], graph embedding (DeepWalk) [Perozzi et al., 2014], iterative classification algorithm (ICA) [Lu and Getoor, 2003], graph-based semi-supervised learning framework (Planetoid) [Yang et al., 2016], graph convolution with Chebyshev filters [Defferrard et al., 2016], graph convolutional network (GCN) [Kipf and Welling, 2017], mixture model networks (MoNet) [Monti et al., 2017], and graph attention networks (GAT) [Veliˇckovi´c et al., 2018]. Besides, to give a comprehensive understanding, we also compare DSP-GCN with 5 community detection methods on the attributed networks. Degree-corrected Stochastic Block Model (DCSBM) [Karrer and Newman, 2011] and NetMRF [He et al., 2018] only adopt the network topology, while PCLDC [Yang et al., 2009], SCI [Wang et al., 2016] and NEMBP [He et al., 2017] utilize both the network topology and node attributes. All the results of the baselines are produced by running the codes from the authors with their default settings.

4.3 Results and Analysis

In the experiments, DSP-GCN is compared to 11 baseline methods on three citation networks by following the experiment protocols in [Yang et al., 2016], where 20 nodes per class, 500 nodes and 1000 nodes are employed for training, validation and performance evaluation, respectively. On the extracted bipartite graph from NELL, we follow the settings in [Yang et al., 2016] where the label rate is 0.1%. The performance is measured with accuracy (AC). As shown in Table 2, our DSP-GCN outperforms all the baseline methods. The improvement of DSP-GCN compared to GAT, which achieves the best performance among the baseline methods, is moderately significant. Although the performance gain of accuracy is 1.8% in average, a large proportion of the error rates has been reduced. Specifically, the average error rates of GAT and our proposed DSP-GCN on citation networks are 21.84% and 19.87%, respectively, Then the error rate reduction, which is achieved by the proposed DSP-GCN, is (21.84% - 19.87%)/21.84% = 9.02%. It clearly demonstrates the effectiveness of our DSP-GCN on jointly exploiting the node content and network topology, which reduces the distortions to the attributes induced by the topology.

To comprehensively evaluate the proposed DSP-GCN, we compare it with five methods, which can also be regarded as node classification techniques. In addition to the three citation networks, five webpage networks, which is often adopted to evaluate the community detection methods, are employed. For the four medium webpage networks, including Cornell, Texas, Washington and Wsicsonsin, we adopt 20% labelled nodes for training, 10% labelled nodes for validation, and the other nodes for testing. For the large Wiki network, the percentages of nodes for training and validation are 3% and 3%, respectively. The results are shown in Table 3. The performance is measured with accuracy (AC) and normalized mutual information (NMI). As can be observed, our DSP-GCN gives relatively consistent performances and significant improvements on the five webpage networks.

The extensive results have demonstrated the superiority and effectiveness of the proposed DSP-GCN under various circumstances compared to 16 baseline methods. On the other hand, the original GCN gives unsatisfactory results (comparable/worse performances compared to the community detection methods) on the webpage networks. The unstable performances of GCN have revealed that the simple attribute smoothing based on the topology may cause severe degradations to the attributes, thus induce an unsatisfactory prediction result. Besides, the NMI scores of GCN, which are more sensitive to the prediction results of the classes containing very few nodes than AC, are obviously worse than AC on the webpage networks. This phenomenon indicates that the prediction results of GCN for the categories possessing very few nodes are unsatisfactory [Zhou et al., 2018], which indicates a severe degradations of the attributes on these rare nodes after the attribute smoothing. Meanwhile, DSP-GCN can obtain satisfactory results on rare node classifications.

4.4 Case Study

To further demonstrate the negative effects caused by the topology in classification, a case study on the Citeseer network is carried out. In this experiment, 25% of the misclassified nodes (74 nodes) in GCN can be correctly classified with respect to the attributes only, where the classification is performed by a single fully-connected layer. This number is reduced to 10% (27 nodes) in our proposed DSP-GCN. The significant gain of our DSP-GCN verifies that our DSP-GCN can successfully reduce the attribute distortions caused by the topology.

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5 Conclusions

Topology information is a double-edged sword for GCNN based semi-supervised node classification. It can be exploited to alleviate the problem of attribute sparsity by attribute smoothing over the graph. On the other hand, its employment in the attribute smoothing also degrades the discriminabilities of the attributes. To reduce its distortions while exploit more potentials of the attribute information, we propose Dual SelfPaced Graph Convolutional Network (DSP-GCN) in this paper. It consists of two highly correlated strategies, node-level and edge-level self-paced learning strategies, which gradually add the unlabelled nodes with confidently predicted labels and the simple edges into training, respectively. Extensive experiments on real networks have demonstrated the superiority of the proposed DSP-GCN compared to the baseline methods and its ability to reduces the attribute distortions induced by topology with only one graph convolutional layer.

Acknowledgments

This work was supported in part by the National Key R&D Program of China (No.2017YFC0820106), in part by the National Natural Science Foundation of China under Grant 61503281 and Grant 61802391, in part by the Foundation for Innovative Research Groups through the National Natural Science Foundation of China under Grant 61421003, and in part by the Fundamental Research Funds for the Central Universities.