1 Introduction
Graph classification, the problem of predicting a label to each graph in a given set, is of significant interest in the bio and chemoinformatics domains, among others; with typical applications in predicting chemical properties [Li and Zemel2016], drug effectiveness [Neuhaus et al.2009], protein functions [Shervashidze et al.2009], and classification of segmented images [Scarselli et al.2009].
There are two major challenges faced by graph classifiers: First, a problem of ordering, i.e. the ability to recognise isomorphic graphs as equivalent when the order of their nodes/edges are permuted, and second, in how to handle instances of varying dimensions, i.e. graphs with different numbers of nodes/edges. In image classification, the ordering of pixels is given, and instances differing in size may be scaled; however, for graphs the ordering of nodes/edges is typically arbitrary, and finding the analogous transformation to scaling an image is evidently nontrivial.
Typical approaches to solving these challenges include kernel methods, in which implicit kernel spaces circumvent the need to map each instance to a fixed size, ordered representation for classification [Zhang et al.2018]
, and deep learning architectures with some explicit feature extraction method whereby a fixed size representation is constructed for each graph and passed to a CNN (or similar) classification model
[Niepert et al.2016]. While the deep learning approaches often outperform the kernel methods with respect to scalability in the number of graphs, they require suitable fixed size representation for each graph, and typically cannot guarantee that isomorphic graphs will be interpreted as the same.In order to address these issues, we propose PiNet; an endtoend deep learning graph convolution architecture with guaranteed invariance to permutation of nodes in the input graphs. To present our model, we first review relevant literature, then present the architecture with proof of its invariance to permutation. We conduct three experiment to evaluate PiNet’s effectiveness: We verify the utility in its invariance to permutation with a graph isomorphism classification task, we then test its ability to learn appropriate message passing matrices, and we perform a benchmark test against existing classifiers on a set of standard molecule classification datasets. Finally, we draw our conclusions and suggest further work.
2 Background
2.1 Graph Kernels
A graph kernel is a positive semidefinite function that maps graphs belonging to a space to an inner product in some Hilbert space , . Graph classification can be performed in the mapped space with standard classification algorithms, or using the kernel trick (with SVMs, for example) the mapped feature space may be exploited implicitly. In this sense, kernel methods are well suited to deal with the high and variable dimensions of graph data, where explicit computation of such a feature space may not be possible.
Despite the large number of graph kernels found in the literature, they typically fall into just three distinct classes [Shervashidze et al.2011]: Graph kernels based on random walks and paths [Borgwardt et al.2005], graph kernels based on frequencies of limited size subgraphs or graphlets [Shervashidze et al.2009], and graph kernels based on subtree patterns where a similarity matrix between two graphs is defined by the number of matching subtrees in each graph [Harchaoui and Bach2007].
Although kernels are well suited to varying dimensions of graphs, their scalability is limited. In many cases they scale poorly to large graphs [Shervashidze et al.2010] and given their reliance on SVMs or full computation of a kernel matrix, they become intractable for large numbers of graphs.
2.2 Graph Neural Networks
Convolutional and recurrent neural networks, while successful in many domains, struggle with a graph inputs because of the arbitrary order in which the nodes of each instance may appear
[Ying et al.2018]. For node level tasks (i.e. node classification and link prediction) graph neural networks (GNNs) [Scarselli et al.2009] handle this issue well by integrating the neural network structure with that of the graph. In each layer, state associated with each node is propagated to its neighbours via a learned filter and then a non linear function is applied. Thus the network’s inner layers learn a latent representation for each node. For example, the GCN [Kipf and Welling2016] uses a normalised version of the graph adjacency matrix to propagate node features while learning spectral filters. The GCN model has received significant attention in recent years with several extensions already in the literature [Hamilton et al.2017, Atwood and Towsley2016, Romero2018].However, for graph level tasks the GNN model and its variants do not handle permutations of the nodes well. For example, for a pair of isomorphic graphs, corresponding nodes would receive corresponding outputs, but for the graph as whole, the node level outputs will not necessarily be given in the same order, thus two graphs may be given shuffled representations presenting an obvious challenge for a downstream classifier. One approach to solving this problem is proposed by [Verma and Zhang2018] where a permutation invariant layer based on computing the covariance of the data is added to the GCN architecture; however, computation of the covariance matrix in the number of nodes, thus not an attractive solution for large graphs.
2.3 Mixed Models
Combining elements of kernel methods and neural networks, mixed models use an explicit method in order to generate vectorized representations of the graphs which are then passed to a conventional neural network. One benefit to this approach being that explicit kernels are typically more efficient when dealing with large numbers of graphs
[Kriege et al.2015].For example, PATCHYSAN [Niepert et al.2016] extracts fixed size localized patches by applying a graph labelling procedure given by the WL algorithm [Weisfeiler and Lehman1968] and a canonical labeling procedure from [McKay1981]
to order the nodes. It then uses these patches to form a 3dimensional tensor for each graph that is passed to a standard CNN for classification.
A similar procedure is presented in [Gutierrez Mallea et al.2019] where, in order to overcome the potential loss of information associated with the CNN convolution operation, a Capsule network is used to perform the classification.
3 PiNet
Formally, we consider the problem of predicting a label for an unseen test graph, given a training set with corresponding labels . Each graph is defined as the pair , where is the graph adjacency matrix, and is a corresponding matrix of dimensional node features. We fix to be the maximum number of nodes in each of the graphs in
, padding empty rows and columns of
and with zeros. Note, however, that these zero entries do not form part of the final graph representations used by the model.3.1 Model Architecture
PiNet is an endtoend deep neural network architecture that utilizes the permutation equivariance of graph convolutions in order to learn graph representations that are invariant to permutation.
As shown in Figure 1, our model consists of a pair of doublestacked message passing layers combined by a matrix product. Let and
denote the rectified linear unit and softmax activations functions,
the preprocessed adjacency matrix, and and the dimensions per node of the latent representation of the features and attention stacks respectively. The features and attention stacks each output a tensor ( and ) with each element corresponding to an input graph given by the functions andrespectively, where(1) 
(2) 
and
(3) 
where
is the identity matrix of order
, is the degree matrix such that(4) 
and .
The use of softmax activation on the attention stack applies the constraint that outputs sum to , thus preventing all node attention weightings from dropping to and keeping the resulting products within a reasonable range.
The trainable parameters and offer an extra attention mechanism that enables the model to weigh the importance of symmetric normalisation of the adjacency matrix, and the addition of self loops. Table 1 shows the four matrices given by the extreme cases of and ; however, intermediate combinations are of course possible. Also note that and may be different for each message passing layer.
Matrix  Definition  

Adjacency  
with S.L.  
Sym Norm Adjacency  
Sym Norm with S.L. 
As seen in Figure 2, the weighting on the self loops given by allows the model to include each node’s own state in the previous layer as an input to the next layer.
The final output of the model is the matrix , where each row is the predicted label given by the function
(5) 
where is a reshape function. Note that since has been transposed, the columns of each matrix (corresponding directly to the nodes of the graph) align exactly with the rows of each matrix in , thus the product aligns the weights for each node learned by the attention stack with the latent features learned by the features stack, resulting in a weighted sum of the features of each node without being affected by permutations of the nodes at input.
Finally, to train the model, we minimise the categorical crossentropy
(6) 
where is the target labels, and the number of classes.
3.2 Permutation Invariance
Here, following the necessary definitions, we provide proof of our model’s invariance to permutation of the nodes in the input graphs.
Definition 3.1 (Permutation).
A permutation is a bijection of a set onto itself, such that it moves an object at position to position . For example, a permutation of to the vector would be .
Definition 3.2 (Permutation Matrix).
Definition 3.3 (Graph Permutation).
We define the permutation of to a graph as a mapping of the node indices, i.e.
(10) 
Definition 3.4 (Permutation Invariance).
Let be the set of all valid permutation matrices of order , then a function is invariant to row permutation iff
(11) 
and is invariant to column permutation iff
(12) 
Definition 3.5 (Permutation Equivariance).
Let be the set of all valid permutation matrices of order , then a function is equivariant to row permutation iff
(13) 
similarly, for column permutation .
Lemma 3.1.
For any matrices and , and any permutation , the product remains unchanged by a permutation of applied to the rows of and , i.e.
(14) 
Proof.
Consider the vector product
(15) 
We permute the rows of and the columns of
(16) 
and observe a reordering of terms, in which the factor pairs remain in correspondence. Since addition is commutative, then
(17) 
(18) 
By the same logic, we see that
(19)  
(20)  
(21)  
(22) 
∎
Lemma 3.2.
If is the preprocessed version of , then is the preprocessed version of , i.e. If
(23) 
then
(24) 
where is the diagonal degree matrix of as defined in Equation 4.
Proof.
From Equation 4,
(25)  
(26) 
Considering each factor of Equation 24 (RHS), we observe that
(27) 
(28)  
(29)  
and since the matrix is diagonal  
(30) 
We also have
(31) 
By Equation 30 and 31,
(32) 
and since is orthogonal, , so
(33) 
∎
Theorem 3.3.
For any input graph , and any permutation applied to , the output of PiNet is equal, i.e.
(34) 
where is the forward pass function of PiNet given in Equation 5.
Proof.
By Equation 1, Definition 3.3 and Lemma 3.2,
(35)  
(36) 
is orthogonal, so , giving
(37) 
Since is an elementwise operation,
(38) 
then
(39) 
(40)  
(41)  
(42)  
(43) 
By the same logic as Equation 35 to 40,
(44) 
(45)  
(46) 
where is a columnwise softmax
(47) 
When the rows of its input are permuted by ,
(48) 
we observe that rows of the output are also permuted by , thus by Equation 46 and 48
(49) 
From Equation 5
(50) 
and by Equation 43 and 49 we see that
(51) 
which by Lemma 3.1
(52)  
(53)  
(54) 
Finally, by Equation 50 and 51 to 54,
(55)  
(56) 
∎
3.3 Implementation
To implement PiNet we use Keras + Tensorflow. The model operates on batches and uses a mixture of Scipy sparse matrices and Numpy arrays to represent the graphs and their features. Full source code for PiNet is available at LINK(reveals authors).
4 Experiments
We conduct three experiments: We empirically verify the utility of our model’s invariance to permutation with a graph isomorphism classification task, we evaluate the effect of different message passing matrices and the model’s ability to select an appropriate message passing configuration, and we compare the model’s classification performance against existing graph classifiers on a set of standard molecule classification datasets. We next describe the data used followed by a description of each experiment.
4.1 Datasets
For the isomorphism test, we generate a dataset of 500 graphs. To create sufficient challenge, in each randomly sampled Erdos Reny [Erdõs and Rényi1960] graph, we fix the number of nodes, and the node degree distributions, to be constant. We generate the dataset according to Algorithm 1, with the parameters: , , , and .
For the final two experiments we use the binary classification molecule datasets detailed in Table 2.
MUTAG  NCI1  NCI109  PTC  PROTEINS  
188  4110  4127  344  1113  
Max.  28  111  111  109  620 
Mean  18  29.8  29.6  25.56  39.06 
7  37  38  18  3  
% of +ve  66.49  50.05  50.38  39.51  59.57 
4.2 Isomorphism Test
We test PiNet’s ability to recognise isomorphic graphs  specifically, unseen permutations of given training graphs. For a baseline, we test against two variants of the GCN [Kipf and Welling2016], one in which the graph level representation is given by a sum of the node representations, and the other in which we apply a dense layer directly to the node level outputs of the GCN. We also compare against two state of the art graph classifiers: the WL Kernel [Shervashidze et al.2011] and PATCHYSAN [Niepert et al.2016]. We perform 10 trials for every training sample size.
4.3 Message Passing Mechanisms
We study the impact on classification accuracy of the four matrices shown in Table 1 that facilitate message passing between nodes, alongside the parametrised matrix shown in Equation 3, in which the extent to which to normalise neighbours’ values and include the node’s own state as input are controlled by the learned parameters and . Note that and are learned for each message passing layer and are not required to be the same, however, for each of the matrices from Table 1 we test, we use the same matrix in all four message passing layers.
For each matrix, we perform a 10fold cross validation and record the classification accuracy. For all runs we use the following hyperparameters: batch size = 50, epochs = 200, first layer latent feature size
= = 100, second layer latent feature size = = 64, and learning rate = .4.4 Comparison Against Existing Methods
MUTAG  NCI1  NCI109  PROTEINS  PTC  
GCN + Dense  
GCN + Sum  
PATCHYSAN  
WLKernel  
PiNet (Manual and )  
PiNet (Learned and ) 
As with the isomorphism test, we compare against the GCN with a dense layer applied directly to the node outputs as well as with a sum of the node representations, the Weisfeiler Lehman graph kernel, and PATCHYSAN. For each model we perform 10fold cross validation on each dataset.
For PiNet, we use the same hyperparameters as described in Section 4.3. We test the model with and as trainable values, and also search over the space . For the GCN we use two layers of sizes and hidden units, with a learning rate of , and for PATCHYSAN we search over two labelling procedures, betweenness centrality [Brandes2001] and NAUTY [McKay1981] canonical labelling.
5 Results
5.1 Isomorphism Test
As seen in Figure 3, PiNet outperforms all competitors tested. Using an independent twosample test we observe statistical significance (value ) in all cases except a single point (circled in red). PiNet fails to achieve 100% accuracy since the neural network learns a surjective function, thus with so few training examples in some cases complete multiple classes become indistinguishable.
5.2 Message Passing Mechanisms
In Figure 4 we observe that the optimal message passing matrix (when and are fixed for all layers, and ) varies depending on the particular set of graphs given. With and as trainable parameters (, and may be different for each layer), we see that for the MUTAG and PROTEINS datasets the model learns values that outperform those found with our manual search. For the others, however, the model is unable to find the optimal s and s, suggesting that the model finds only local minima. We note however, that in every case tested, the model is able to learn the values and that give better than average classification performance when compared with our manual search space.
5.3 Comparison Against Existing Methods
As shown in Table 3, PiNet achieves competitive classification performance on all datasets tested. An independent twosample test indicates PiNet achieves a statistically significant (value ) gain in only a few cases (); however, with competitive performance on all datasets it demonstrates a robustness to the properties of the different sets of graphs.
6 Conclusion
We have proposed PiNet, an endtoend deep neural network graph classifier invariant to permutations of nodes in the input graphs. We have provided theoretical proof of its invariance to permutation, and demonstrated the utility in such a property empirically with a graph isomorphism classification task against a set of existing graph classifiers, achieving a statistically significant gain in classification accuracy on a range of small training set sizes. The permutation invariance is achieved through a differentiable attention mechanism in which the model learns the weight by which the states associated with each node should be aggregated into the final graph representation.
We have demonstrated that PiNet is able to learn an effective parametrisation of a message passing matrix that enables it to adapt to different types of graphs with a flexible state propagation and diffusion mechanism. Finally, we have shown PiNet’s robustness to the properties of different sets of graphs in achieving consistently competitive classification performance against a set of existing techniques on five commonly used molecule datasets.
For future work we plan to explore more advanced aggregation mechanisms by which the latent representations learned for each node of the graph may be combined.
Acknowledgements
We thank Braintree Ltd. for providing the full funding for this work.
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