Topological-Similarity Based Canonical Representations for Biological Link Prediction

Abstract

Graph machine learning algorithms are being commonly applied to a broad range of prediction tasks in systems biology. An important design criterion in this regard is the definition of “topological similarity” between two nodes in a network, which is used to design convolution matrices for graph convolution or loss functions to evaluate node embeddings. Many measures of topological similarity exist in network science literature (e.g., random walk based proximity, shared neighborhood) and recent comparative studies show that the choice of topological similarity can have a significant effect on the performance and reliability of graph machine learning models.

We propose GRAPHCAN, a framework for computing canonical representations for biological networks using a similarity-based Graph Convolutional Network (GCN). GRAPHCAN integrates multiple node similarity measures (Common Neighbor, Adamic Adar, Random Walk with Restart, Von Neumann, Resource Allocation, Hub-Depressed Index, Hub-Promoted Index, and adjacency matrix) to compute canonical node embeddings for a given network. The resulting embeddings can be utilized directly for downstream machine learning tasks.

We comprehensively evaluate GRAPHCAN in the context of various link prediction tasks in systems biology. Our results show that the integration of multiple similarity measures improves the robustness of the framework, especially when the input networks are sparse. GRAPHCAN is available as open source at https://github.com/Meng-zhen-Li/Similarity-based-GCN.git.

I. INTRODUCTION

BIOLOGICAL systems are composed of interacting cellular components, e.g. genes, proteins, or metabolites. The associations among the cellular components are modeled as biological networks [1], including protein-protein interaction networks [2], [3], drug-drug interaction networks [4], etc. Mining knowledge from large-scale biological networks has proved useful in a broad range of applications in systems biology and molecular medicine.

With the breakthrough in machine learning in the last decade, development of computational methods to utilize biological networks in predictive tasks have been soaring. To this end, in recent years, network embedding techniques [5]–[8] have been useful in applying sophisticated machine learning algorithms to biological network data. In essence, network embedding techniques map the network to a low-dimensional space in such a way that proximity in the network is preserved in the low-dimensional representations (a vector for each node). Then, these vectors are used as an input to the machine learning algorithms for various down-stream tasks, e.g., link prediction [9]–[11], identification of cell types [12], prioritization of candidate disease genes [13], and prediction of disease-gene associations [14].

In state-of-the-art applications, network embedding or graph convolution based machine learning techniques take the raw input network as an input and perform learning on this raw network. An important drawback of the raw input networks is the noise and missing edges in the input network [15], skewness of the degree distribution [16], and the distribution of network density across different localities of the network [11]. In graph analysis tasks, this causes imbalance in the distribution of flow across the network, creating or enhancing bias in the resulting models [17]. A solution that is commonly applied to address these issues is to use an alternate graph representations to represent the topological information of input network(s), e.g. similarity graphs for graph convolution [11].

Despite the effectiveness of topological similarity measures in designing convolution matrices [11], a single similarity measure may over- or under-emphasize certain topological patterns. Such false interpretations from a single similarity measure can largely reduce the quality of the embeddings for down-stream tasks. However, the false or missing information with respect to one similarity measure may be corrected by another similarity measure. Hence, a promising approach to overcome the limitation of single similarity-based embedding is to utilize the compatible and complementary information in multiple topological similarity measures.

We propose GRAPHCAN [18], which learns canonical graph representations for biological networks. GRAPHCAN (i) integrates a multitude of topological similarity measures, (ii) computes low-dimensional embeddings representing these similarity measures using a Graph Convolutional Network (GCN), (iii) computes consensus embeddings from these embeddings using canonical correlation. The computed canonical representation can be utilized directly for downstream machine learning tasks.

In this article, we implement and test GRAPHCAN with a comprehensive set of similarity measures, and systematically evaluate GRAPHCAN in the context of various link prediction tasks in systems biology. We show that the integration of multiple similarity measures help increase the accuracy and robustness of GRAPHCAN. For each application and dataset, we identify the optimal combination of similarities. We also investigate the effect of network densities and training ratios on predictive performance. Our results show that GRAPHCAN outperforms other network embedding methods that directly use the adjacency matrix of the input network.

II. BACKGROUND

A. Network Embedding

Network embedding aims to learn a low-dimensional representation of nodes in networks [19]. To be more specific, for a given graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, a node embedding is a $f:\mathcal{V}\to {ℝ}^d$ that maps each node $v\in \mathcal{V}$ to a vector in ${ℝ}^d$ where $d\ll |\mathcal{V}|$. The representation is learned such that the proximity in the embedding space reflects the proximity/similarity in the network. That is, nodes that are similar in the network are close to each other in the embedding space. There are different types of approaches to define the proximity of the nodes, and thus different mapping functions from network topology to embedding space.

B. Graph Convolutional Networks and Graph Auto-Encoders

Graph Convolutional Network(GCN) is a scalable approach for semi-supervised learning on graph-structured data that is based on an efficient variant of convolutional neural networks which operate directly on graphs [5]. Graph structures are encoded as a neural network model $f(X,A)$, where $X\in {ℝ}^{n\times k}$ is a matrix of node feature vectors (k is the number of features) and $A\in {ℝ}^{n\times n}$ is the adjacency matrix of the graph. If no node features are available, $X$ is an identity matrix of the same size as $A$.

Graph Autoencoder(GAE) is a framework for unsupervised learning on graph-structured data based on the variational autoencoder (VAE) [6]. Given an unweighted and undirected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, GAE learns an embedding $Z\in {ℝ}^{n\times d}$ using a graph-based neural network with a two-layer GCN encoder and an inner-product decoder. The inner product of the embedding $Z$ reconstructs the adjacency matrix: $\widehat{A}=\sigma \left(Z{Z}^T\right)$ where $\sigma$ is the sigmoid activation function. The learned GAE embedding can be applied to graph learning tasks like link prediction.

III. METHODS

We propose GRAPHCAN, a graph convolutional network (GCN) based framework to compute a canonical representation for an input network, by integrating multiple measures of topological similarity between pairs of nodes.

A. Workflow of GRAPHCAN

Figure 1 shows the workflow of the proposed framework, which can be roughly divided into three categories: (i) the GCN-based encoder, (ii) the inner-product decoder, and (iii) computation of consensus embedding. The encoder consists of two GCN layers, and a fully-connected neural network layer. The decoder computes the inner products of the encoded embeddings to reconstruct similarity matrices that capture node similarities based on different measures of topological similarity. Once separate embeddings representing each similarity measure are computed, a consensus embedding (Section III-E) is computed to integrate the learned embeddings.

Fig. 1. Illustration of GRAPHCAN.

The input is the adjacency matrix of the network. The general embedding is generated using a two-layer GCN, and the separate embeddings are generated by a hidden neural network layer. The reconstructed similarity matrices are the inner products of the separate embeddings. The final output embedding is the consensus embedding of the optimized separate embeddings. The loss function is the Mean Square Error(MSE) between true similarities and reconstructed similarities.

The input to GRAPHCAN is the adjacency matrix $A\in {ℝ}^{n\times n}$ of the input network $\mathcal{G}=(\mathcal{V},\mathcal{E})$, a dimensionality parameter $d$, and a set of topological similarity measures. An example list of the similarity measures that are used in our experiments are shown in Figure 1. Using the input similarity measures, we compute “true” similarity matrices $S_1$, $S_2,\dots ,S_k\in {ℝ}^{n\times n}$ from $A$, each representing a different similarity measure. Using these $k$ similarity measures, GRAPHCAN learns $d$-dimensional embeddings for the nodes in $\mathcal{V}$, such that proximity in the embedding space is associated with similarity according to the multiple similarity measures that are specified in the input.

To compute individual embeddings, GRAPHCAN uses the featureless version of GCNs. The two GCN hidden layers generate an embedding $Z\in {ℝ}^{n\times d}$, and the separate embeddings $Z_1$, $Z_2,\dots ,Z_k\in {ℝ}^{n\times d}$ for respective similarity matrices $S_1$, $S_2,\dots ,S_k$ are computed with a fully-connected hidden layer. The inner product decoder uses the inner product $Z_i{Z}_i^T$ of as the reconstruction of the similarity matrices. The loss function for the neural network is the mean square error between these inner products and similarity matrices, thus the separate embeddings are optimized to minimize the difference between the “true” and reconstructed similarity matrices. The consensus embedding of the separate embeddings is the learned canonical representation $Z_{GraphCan}\in {ℝ}^{n\times d}$.

B. Topological Similarity Measures

Node similarity measures aim to quantify the similarity between a pair of nodes in a network in terms of their topological properties and/or location in the network [24]. There is a multitude of topological similarity measures in the literature, used for tasks ranging from link prediction and denoising to community detection and network visualization [25], [26]. Each topological measure captures a different aspect of the topological relationship between a pair of nodes, including the overlap between neighbors, distance in the network, multiplicity of the paths connecting the two nodes, or the association between their proximity profiles [27].

Our objective here is to generate a comprehensive view of the pairwise relationships between the nodes of a network - one that can be reliably used for downstream machine learning applications. For this purpose, we propose to integrate multiple similarity measures, thereby enabling multiple similarity measures to complement each other to extract reliable relationships between pairs of nodes. We consider a broad set of measures that span both direct neighbor based (local) and propagation-based (global) approaches:

• Common Neighbor measures the overlap between the neighbors of two nodes:

$$CN(x,y)=|N(x)\cap N(y)|$$ (1)

where $N(x)$ and $N(y)$ respectively denote the sets of neighbors of $x$ and $y$.

• Adamic Adar [20] is based on the premise that common neighbors with many neighbors are less informative on the relationship between two nodes as compared to common neighbors with a small number of neighbors. It refines the notion of common neighbor by assigning more weight to less-connected common neighbors:

$$AA(x,y)=\sum _{u\in N(x)\cap N(y)}\frac{1}{log|N(u)|}$$ (2)

• Random Walk Based Proximity is a network propagation method that considers a random particle that transmits to neighbors with the probability that is proportional to their edge weights [21], and there is a probability for the particle to restart at each stage. Starting from a node $x$, the probability distribution of the particle being at each node at time $t$ can be computed as:

$$RW{R}_{t+1}^{(x)}=(\alpha W)RW{R}_t+(1-\alpha )RW{R}_0^{(x)}$$ (3)

where $W\in {ℝ}^{|V|\times |V|}$ denotes the matrix of transition probabilities of a random walk between each pair of nodes, obtained by colum-normalizing $A$. $RW{R}_0^{(x)}$ is the restart vector with all entries zero except for $RW{R}_0^{(x)}(x)=1$ and $\alpha$ denotes the probability of restart. The random-walk based proximity between nodes $x$ and $y$ is computed as $\left(RW{R}^{(x)}(y)+RW{R}^{(y)}(x)\right)/2$.

• Von Neumann Proximity is also a network propagation method. For a given maximum path length $t$, it can be computed by the following expression:

$$A_s=A\oslash \sqrt{S_r\odot D_c}$$ (4)

$$VN=\sum _{i=1}^t{\alpha }^i{A}_s^i$$ (5)

where $D_r\odot D_c$ indicates the element-wise dot product of the row and column degrees in matrix form $\left(D_r(u,v)=\right.\left.|N(u)|,D_c(u,v)=|N(v)|\right)$, and $\oslash$ is the element-wise divide operation. $A$ is the adjacency matrix and $\alpha$ is the diffusion factor [22]. The Von Neumann proximity between nodes $x$ and $y$ is defined as $VN(x,y)=VN(y,x)$.

• Resource Allocation [23] aims to reduce the effect of highly connected common neighbors, and it is the sum of inverse of the degree of the common neighbors of pairs of nodes:

$$RA(x,y)=\sum _{u\in N(x)\cap N(y)}\frac{1}{|N(u)|}$$ (6)

• Hub-Depressed Index aims to normalize the overlap between neighbors of two nodes based on the degrees of the nodes, and it focuses on the node with higher degree:

$$HDI(x,y)=\frac{|N(x)\cap N(y)|}{\max (|N(x)|,|N(y)|)}$$ (7)

• Hub-Promoted Index is similar to Hub-Depressed Index, but it focuses on the node with lower degree:

$$HPI(x,y)=\frac{|N(x)\cap N(y)|}{\min (|N(x)|,|N(y)|)}$$ (8)

• The Adjacency Matrix can also be considered as a similarity measure by itself (i.e., two nodes are considere similar if they are connected in the network). An identity matrix is added to the adjacency matrix when used as a similarity matrix.

For a given input network, we compute five similarity matrices using these five similarity measures. We then use the similarity matrices as the “labels” for the graph convolutional network, i.e., we train the GCN to learn embeddings that can reconstruct these similarity matrices.

C. Similarity-Based Graph Convolutional Network

For generality, let $k$ denote the number of similarity matrices that are utilized by GRAPHCAN.

The graph convolutional network of GRAPHCAN can be considered as a variational graph autoencoder with one input and multiple outputs. The encoder of GRAPHCAN consists of a two-layer GCN:

$$GCN(X,A)=\text{ReLU}\left(\tilde{A}\text{ReLU}\left(\tilde{A}X{W}_0\right)W_1\right)$$ (9)

where $X$ denotes feature matrix for the nodes, which is an identity matrix of the same size as $A$ since our networks are featureless. $\tilde{A}=S^{-\frac{1}{2}}A{D}^{-\frac{1}{2}}$ is the symmetrically normalized adjacency matrix, where $D$ is the degree matrix of $A$. The output of the encoder is a general embedding $Z\in {ℝ}^{n\times d}$ where $d$ is the size of the second GCN layer. $W_0$ and $W_1$ are the weights of the two GCN layers.

The decoder consists of two parts, a hidden layer and an inner-product decoder. The hidden layer is a fully-connected neural network layer:

$$Z_s=Z{W}_2$$ (10)

where $W_2\in {ℝ}^{d\times kd}$ denotes the weight matrix of the hidden layer. Then the output $Z_s\in {ℝ}^{n\times kd}$ is splitted to $k$ blocks, $Z_1$, $Z_2,\dots ,Z_k\in {ℝ}^{n\times d}$, each of which denote the separate embeddings to reconstruct the $k$ similarity matrices. The inner product decoder computes the inner products of the separate embeddings, which are also the reconstructed similarities for $1\le i\le k$:

$$\widehat{S_i}=Z_i{Z}_i^T.$$ (11)

D. Loss Function and Optimization

To optimize the separate embeddings, the model minimizes the Mean Squared Error (MSE) between each reconstructed similarity and the true similarity:

$$\mathcal{L}=\sum _{i=1}^k\sum _{u,v\in V}{\left(S_i(u,v)-{\widehat{S}}_i(u,v)\right)}^2$$ (12)

Because of the sizes of the weights, optimizing all the weights together is expensive and time-consuming, especially when there are many similarity measures. If we split $W_2$ into multiple parts and optimize separately, the optimization will be faster and distributed(parallel) learning can be applied(computing the gradients of all blocks at the same time and compute the average of the gradients for $W_0$ and $W_1$.

Therefore, the loss of each pair of $S_i$ and ${\widehat{S}}_i$ is computed and optimized separately:

$$\underset{W_0,W_1,W_{2i}}{\min }\sum _{u,v\in \mathcal{V}}{\left(S_i(u,v)-{\widehat{S}}_i(u,v)\right)}^2$$ (13)

where $W_{2i}\in {ℝ}^{d\times d}$ is computed by dividing $W_2$ (Equation 10) into $k$ blocks. In each iteration, we compute the gradients for $W_0$, $W_1$, and $W_{2i}$ for each $i$ so that we get $k$ gradients for $W_0$ and $W_1$, and one gradient for each $W_{2i}$. After we compute all gradients for all $i$, the gradients of $W_{2i}$ can be concatenated to construct the gradient of $W_2$, and the gradients of $W_0$ and $W_1$ are computed as the average of the $k$ gradients of $W_0$ and $W_1$.

E. Consensus Embedding

Once a separate embedding for each similarity matrix is computed, we compute a consensus embedding from these separate embeddings to obtain a canonical embedding for the input network. For this purpose, we use dimensionality reduction on $Z_i\in {ℝ}^{n\times d}$ to compute a d-dimensional node embedding $Z_c$ for $\mathcal{G}$. In the context of network integration, Generalized Canonical Correlation Analysis (GCCA) is shown to outperform other dimensionality reduction methods in computing consensus embeddings [28], since it can accurately map shared dimensions in different embedding spaces. The embeddings we aim to integrate here represent similarity measures that are expected to have many common dimensions, thus GRAPHCAN also computes consensus embedding using GCCA:

$$Z_{GraphCan}=GCCA\left(Z_1,Z_2,\dots ,Z_k\right)$$ (14)

where GCCA represents the generalized canonical correlation analysis, which is shown to be effective in representing the integration of network versions. The consensus embedding $Z_{GraphCan}$ is the canonical representation of graph $\mathcal{G}$ and can be used to perform downstream tasks.

IV. EXPERIMENTAL RESULTS

A. Experimental Data and Benchmark

We apply GRAPHCAN to biological link prediction tasks to assess its ability to obtain meaningful representations of biological network. For this purpose, we use the benchmarking framework provided by BioNEV [29]. BioNEV provides four datasets representing three different biomedical link prediction problems (Table I). On each of these networks, for a given network embedding, BioNEV uses this network embedding to train a binary classifier (using logistic regression) for each pair of node representations, which in turn is used for link prediction. The statistics of each network is shown on Table I, including data types and multiple topological features. The average clustering coefficient of bipartite graphs (DDAs) are computed as the average square clustering coefficient [30].

TABLE I.

DESCRIPTION AND STATISTICS OF THE BIOLOGICAL NETWORKS USED IN THE EXPERIMENTS.

Dataset	Network Type	#Nodes	#Edges	Density	Average Degree	Average Clustering Coefficient
CTD-DDA [31]	drug-disease association	12,765	92,813	0.11%	7.27	0.0597
DrugBank-DDI [4]	drug-drug interaction	2,191	242,027	10.08%	110.46	0.5453
NDFRT-DDA [32]	drug-disease association	13,545	56,515	0.06%	4.17	0.1951
STRING-PPI [2]	protein-protein interaction	15,131	359,776	0.31%	23.78	0.4279

In our experiments, we split the input networks into training and testing sets, where 20% of the edges are used for testing. For experiments that are not related to training ratio, the training set contains 80% of the edges. The negative train sets and testing sets contain the same numbers of random node pairs without edges as the training sets. We repeat the train and test splits 5 times for each experiment and report the mean and variance of performance figures across these runs.

B. Baseline Node Embedding Methods

We compare the GRAPHCAN model with three existing node embedding methods of different categories:

• Node2vec [7] is a random-walk based node embedding method. It runs random walk starting from the nodes in the network, and use the random walk paths to preserve the structural proximity of the network.

• GAE [6] is a neural-network based embedding method. It learns node embeddings using a two-layer GCN and a inner-product decoder. Here, we use the featureless version of GAE.

• GraRep [33] is a matrix-factorization based embedding method. It constructs the high-order proximity matrix based on transition probability by a random walk with specific length. The model is optimized by matrix factorization techniques(Singular Value Decomposition).

Default parameters of baseline methods are used in all experiments.

C. Comparison of GRAPHCAN to Baseline Algorithms

We compare the predictive performance of GRAPHCAN against baseline algorithms by applying it to each link prediction problems using different training ratios from 10% to 80%. Since we remove edges from the training input graph, the training graph becomes sparser as the training ratio becomes smaller. The testing ratio of all experiments is fixed to 20%.

The results of this analysis are shown in Figure 2. We observe that across all datasets, GRAPHCAN clearly outperforms baseline methods on the NDFRT-DDA and STRING-PPI datasets, performs best and is most stable for smaller training sets on the DrugBank-DDI dataset, and performs best for larger training sets on the CTD-DDA dataset. Note that the Drugbank-DDI dataset represents the densest network and most algorithms perform reasonably well on this dataset. GRAPHCAN’s performance appears to steadily improve as the training ratio grows for all datasets, thus we next investigate the effect of the sparsity of the network on GRAPHCAN’s performance.

Fig. 2. Link prediction performance of GRAPHCAN compared to baseline methods as a function of the fraction of the edges that are used for training.

Each panel shows the results of a dataset described in Table I. The colors of the bars represent GRAPHCAN and baseline methods, where GRAPHCAN uses the best combination in Table II. The curves show the average and standard deviation of performance figures across five different runs. The number of dimensions is fixed to 32.

D. Effect of Network Density

We investigate the effect of density on GRAPHCAN by constructing networks with different densities using the DrugBank-DDI dataset. The initial density of the network is approximately 10%, and we create sparser networks by removing different percentages of edges from the edgelist. The results of this analysis are shown in Figure 3, where the AUC-ROC and AUC-PR of GRAPHCAN are compared with the baseline methods. From the figure, GRAPHCAN is more robust against low densities compared to the baseline methods. When the density is too small, the baseline methods, especially node2vec, cannot perform as well as GRAPHCAN in link prediction. However, GRAPHCAN performs well even when the network is very sparse, and is relatively stable when the density increases from 1.25% to 10%. Therefore, integrating multiple similarity measures help increase the robustness of the model when the input networks are sparse.

Fig. 3. Link prediction performance of GRAPHCAN and baseline methods as a function of network density.

All eight similarity measures are used in GRAPHCAN. The left panel shows area under ROC curve (AUC-ROC) and the right panel shows area under precision-recall curve (AUC-PR). The x-axis represents network densities from 0.3125% to 10%, shown in log scale, obtained by downsampling the DrugBank-DDI network. The curves show the average and standard deviation of performance figures across five different runs. The number of dimensions is fixed to 32, and the training ratio is fixed to 80%.

E. Integration of Similarity Measures

Since we integrate multiple similarity measures in the GRAPHCAN model, we also investigate the effect of integrating multiple similarities by comparing the performance of GRAPHCAN using single similarity measure with using different combinations of similarity measures. Since the computational cost of considering all combinations of the 8 similarity measures would be prohibitive, we first rank the similarity measures based on the link prediction performance of each measure. We then cumulatively consider combinations of similarity measures where measures are added to the combination in the order of individual performance. In Figure 4, the curve shows the performance of cumulative similarities. The first point on the curve shows the individual performance of the top-performing similarity measure, the second point represents the performance of the combination of top two individual similarity measures,, etc.

Fig. 4. Link prediction performance of GRAPHCAN using each similarity measure and combinations of similarity measures.

Each panel shows the results on a different dataset (Table I). The colors of the bars represent different the similarity measures listed in Figure 1, and the bars are sorted by the AUC-PR provided by each similarity measure alone. Each point of the purple curve shows the average AUC-PR provided by the cumulative combination of the similarity measures to its left, and the shaded area shows the standard deviation across five runs. The number of dimensions is fixed to 32, and the training ratio is fixed to 80%.

As seen in Figure 4, the AUC-PR of most datasets experience a slight increase as more similarity measures are added to the model. However, the performance will no longer increase as more similarity measures are included. In some cases, the performance goes down with the inclusion of more similarity measures. The reason for this behavior might be the difficulty of convergence as the model becomes larger, as it will be harder for the model to reconstruct all similarity matrices accurately. Motivated by this observation, we perform more detailed analysis of convergence in Section IV-G.

Table II summarizes the effects of integrating similarity measures. In most cases, combining multiple similarity measures performs better than using a single similarity measure. Also, GRAPHCAN using multiple or all similarity measures often performs better than single similarity, and using all similarity measures performs almost as good as the best combination of similarity measures (except for NDFTR-DDA, which is the sparsest network considered). Therefore, when optimal similarity measure or optimal combination is not known, integrating multiple similarity measures is likely to improve the robustness of the model.

TABLE II.

COMPARISON OF DIFFERENT OPTIONS FOR INTEGRATING DIFFERENT SIMILARITY MEASURES IN TERMS OF LINK PREDICTION PERFORMANCE (AUC-PR) OF GRAPHCAN.

Datasets	Adjacency Matrix	Average of all Matrices	Best Matrix		Best Combination		All Matrices
CTD-DDA	0.9470	0.9260	CN	0.9610	All	0.9676	0.9676
DrugBank-DDI	0.8762	0.8787	VN	0.8830	VN	0.8830	0.8774
NDFRT-DDA	0.8564	0.8688	AA	0.9230	AA, CN, RWR, VN	0.9360	0.8684
STRING-PPI	0.8622	0.8049	Adj	0.8622	Adj, HDI	0.8700	0.8608

F. The Effect of The Number of Dimensions

We compare the performance of GRAPHCAN with the optimal combinations of similarity measures (Table II) with baseline methods. We compare the AUC-ROC and AUC-PR, and plot them as a function of numbers of embedding dimensions from 8 to 64. The results of these analyses are shown in Figure 5. As seen in the figure, GRAPHCAN outperforms the baseline methods in most cases.

Fig. 5. Link prediction performance of GRAPHCAN compared with three state-of-the-art node embedding methods, as a function of number of embedding dmensions.

Optimal combinations of GRAPHCAN(table II) are used. For each dataset, area under ROC curve (AUROC) is shown on the left, area under Precision-Recall curve (AUPR) is shown on the right. The curves and error bars respectively show the average and standard deviation of performance figures across 5 runs. The training ratio is fixed to 80%.

For CTD-DDA, the overall performance of GRAPHCAN is the best among all embedding methods for all embedding dimensions, with more performance difference for lower number of dimensions. For Drugbank-DDI, GRAPHCAN performs best for lower number of dimensions according to both performance criteria, but GraRep and node2vec catch up with increasing number of dimensions. For NDFRT-DDA, GraRep performs best according to AUROC, while GRAPHCAN performs best according to AUPR. The performance of GRAPHCAN is quite variable on this dataset and it is more dependent on the number of embedding dimensions. For STRING-PPI, we also observe that GRAPHCAN delivers the best AUPR while as good as other methods according to AUROC. In general, GRAPHCAN is highly competitive among network embedding methods in link prediction, and its performance is particularly pronounced when AUC-PR is considered.

G. Convergence of the Model

To see how well GRAPHCAN reconstructs the similarity matrices and how the model converges, r-squared is computed at each epoch of training. r-squared is a commonly used measure in regression models, which is defined as the proportion of total variation of outcomes explained by the model [34]. R-squared can be computed as one minus the sum of squared error(unexplained variances) over the total sum of variances:

$$R^2=1-\frac{\sum {\left(y_i-\widehat{y_i}\right)}^2}{\sum {\left(y_i-\overline{y}\right)}^2}$$ (15)

Figure 6 shows the average r-squared among all similarity measures as a function of training epochs. All 8 similarities in figure 1 are used in the experiments. From the three curves in each plot, the GRAPHCAN model reconstructs the similarity matrices better when the training ratio is larger, and denser networks(e.g. DrugBank-DDI and STRING-PPI) have higher r-squared when the optimization converges. This result indicates that GRAPHCAN reconstructs topological similarities better when the input networks are denser.

Fig. 6. The average r-squared of all similarities at different training epochs.

The curves represent the r-squared for different training ratios from 20% to 80%, and the shaded areas are the standard deviation of multiple runs. All similarities in figure 1 are used in this experiment.

In figures 5 and 2, GRAPHCAN has a larger variance in CTD-DDA when number of dimensions training ratio are small, which might because the optimal combination for it is all similariteis, and the input training graph is sparse. The model converges when the r-squared is small, so that the learned embedding does not reconstructs the similarities well.

V. CONCLUSION

In this article, we propose GRAPHCAN, a similarity-based graph convolutional network(GCN) for learning canonical representation of biological networks. We systematically test GRAPHCAN on biological link prediction tasks, and the experimental results show that GRAPHCAN outperforms other baseline methods in terms of link prediction. We also find out that integrating multiple similarity measures improves the robustness and accuracy of the model, and different datasets work well with different combinations of similarities. We also use simulated data to test the effect of network density on GRAPHCAN, and the results show that it is more stable than baseline methods on sparse networks. Our analysis also shows that GRAPHCAN is robust against low dimensions and training ratios.

A potential limitation of the model is that it does not reconstruct all similarities well when too many similarities are used or the input graph is sparse, and thus the link prediction performance would decrease. Directions of the future work include improving the convergence speed and the reconstruction accuracy of the model. Another limitation of GRAPHCAN is that GCN does not support large-scale input networks, so our future research of canonical representation can be extended to similarity-based models based on other machine learning techniques.