# American Institute of Mathematical Sciences

March  2020, 2(1): 19-33. doi: 10.3934/fods.2020002

## Semi-supervised classification on graphs using explicit diffusion dynamics

 1 Department of Mathematics and Imperial College Business School, Imperial College London, London SW7 2AZ, UK 2 Department of Mathematics, Imperial College London, London SW7 2AZ, UK

* Corresponding author: Mauricio Barahona

Current address: Blue Brain Project, Éole polytechnique fédérale de Lausanne (EPFL), Campus Biotech, 1202 Geneva, Switzerland.

Published  February 2020

Fund Project: All authors acknowledge funding through EPSRC award EP/N014529/1 supporting the EPSRC Centre for Mathematics of Precision Healthcare at Imperial

Classification tasks based on feature vectors can be significantly improved by including within deep learning a graph that summarises pairwise relationships between the samples. Intuitively, the graph acts as a conduit to channel and bias the inference of class labels. Here, we study classification methods that consider the graph as the originator of an explicit graph diffusion. We show that appending graph diffusion to feature-based learning as an a posteriori refinement achieves state-of-the-art classification accuracy. This method, which we call Graph Diffusion Reclassification (GDR), uses overshooting events of a diffusive graph dynamics to reclassify individual nodes. The method uses intrinsic measures of node influence, which are distinct for each node, and allows the evaluation of the relationship and importance of features and graph for classification. We also present diff-GCN, a simple extension of Graph Convolutional Neural Network (GCN) architectures that leverages explicit diffusion dynamics, and allows the natural use of directed graphs. To showcase our methods, we use benchmark datasets of documents with associated citation data.

Citation: Robert L. Peach, Alexis Arnaudon, Mauricio Barahona. Semi-supervised classification on graphs using explicit diffusion dynamics. Foundations of Data Science, 2020, 2 (1) : 19-33. doi: 10.3934/fods.2020002
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##### References:
Statistics of datasets as reported in [35] and [30]
 Datasets Nodes Edges Classes Features Citeseer $3,327$ $4,732$ $6$ $3,703$ Cora $2,708$ $5,429$ $7$ $1,433$ Pubmed $19,717$ $44,338$ $3$ $500$ Wikipedia $20,525$ $215,056$ $12$ $100$
 Datasets Nodes Edges Classes Features Citeseer $3,327$ $4,732$ $6$ $3,703$ Cora $2,708$ $5,429$ $7$ $1,433$ Pubmed $19,717$ $44,338$ $3$ $500$ Wikipedia $20,525$ $215,056$ $12$ $100$
Percentage classification accuracy before and after application of relabelling by GDR for various classifiers. We present the improvement of GDR on the uniform prediction (which ignores features). We also consider four supervised classifiers (which learn from features without the graph): projection, RF, SVM and MLP. For RF, we use a maximum depth of $20$; for SVM, we set $C = 50$; for MLP, we implement the same architecture as GCN ($d_1 = 16$-unit hidden layer, $0.5$ dropout, $200$ epochs, $0.01$ learning rate, $L^2$ loss function). Finally, we compare with two semi-supervised graph classifiers: GCN [20] and Planetoid [35]. The numbers in brackets record the change in accuracy accomplished by applying GDR on the corresponding prior classifier. Boldface indicates the method with highest accuracy for each dataset
 Method Citeseer Cora Pubmed Wikipedia Uniform 7.7 13.0 18.0 28.7 GDR (Uniform) 50.6 (+42.9) 71.8 (+58.8) 73.2 (+55.2) 31.4 (+2.7) Projection 61.8 59.0 72.0 32.5 RF 60.3 58.9 68.8 50.8 SVM 61.1 58.0 49.9 31.0 MLP 57.0 56.0 70.7 43.0 GDR (Projection) 70.4 (+8.7) 79.7 (+20.7) 75.8 (+3.8) 36.9 (+4.4) GDR (RF) 70.5 (+10.2) 78.7 (+19.8) 72.2 (+3.2) 50.8 (+0.0) GDR (SVM) 70.3 (+9.2) 81.2 (+23.2) 52.4 (+2.5) 41.9 (+10.8) GDR (MLP) 69.7(+12.7) 78.5 (+22.5) 75.5 (+4.8) 40.5 (-2.5) Planetoid 64.7 75.7 72.2 - GCN 70.3 81.1 79.0 39.2 GDR (GCN) 70.8 (+0.5) 82.2 (+1.1) 79.4 (+0.4) 39.5 (+0.3)
 Method Citeseer Cora Pubmed Wikipedia Uniform 7.7 13.0 18.0 28.7 GDR (Uniform) 50.6 (+42.9) 71.8 (+58.8) 73.2 (+55.2) 31.4 (+2.7) Projection 61.8 59.0 72.0 32.5 RF 60.3 58.9 68.8 50.8 SVM 61.1 58.0 49.9 31.0 MLP 57.0 56.0 70.7 43.0 GDR (Projection) 70.4 (+8.7) 79.7 (+20.7) 75.8 (+3.8) 36.9 (+4.4) GDR (RF) 70.5 (+10.2) 78.7 (+19.8) 72.2 (+3.2) 50.8 (+0.0) GDR (SVM) 70.3 (+9.2) 81.2 (+23.2) 52.4 (+2.5) 41.9 (+10.8) GDR (MLP) 69.7(+12.7) 78.5 (+22.5) 75.5 (+4.8) 40.5 (-2.5) Planetoid 64.7 75.7 72.2 - GCN 70.3 81.1 79.0 39.2 GDR (GCN) 70.8 (+0.5) 82.2 (+1.1) 79.4 (+0.4) 39.5 (+0.3)
Percentage classification accuracy of GCN and its extension diff-GCN, which has an explicit diffusion operator (16)
 Model Citeseer Cora Pubmed Wikipedia GCN 70.3 81.1 79.0 34.1 diff-GCN 71.9 82.3 79.3 45.9
 Model Citeseer Cora Pubmed Wikipedia GCN 70.3 81.1 79.0 34.1 diff-GCN 71.9 82.3 79.3 45.9
Accuracy of GDR using the undirected, directed, and reverse directed graphs of the Cora dataset
 Undirected Directed (fw) Directed (bw) Method $A$ $A_\text{dir}$ $A_\text{dir}^T$ GDR (Projection) 79.7 62.1 64.6 GDR (RF) 78.7 58.0 57.6 GDR (SVM) 81.2 63.6 62.1 GDR (MLP) 78.5 57.3 56.4
 Undirected Directed (fw) Directed (bw) Method $A$ $A_\text{dir}$ $A_\text{dir}^T$ GDR (Projection) 79.7 62.1 64.6 GDR (RF) 78.7 58.0 57.6 GDR (SVM) 81.2 63.6 62.1 GDR (MLP) 78.5 57.3 56.4
Accuracy of GCN and diff-GCN using the undirected, directed, reverse directed, and bidirectional (augmented) graphs of the Cora dataset. The highest accuracy is achieved by diff-GCN with the augmented graph (boldface)
 Undirected Directed (fw) Directed (bw) Augmented (fw, bw) Method $A$ $A_\text{dir}$ $A_\text{dir}^T$ $\begin{bmatrix} A_\text{dir} \, A_\text{dir}^T \end{bmatrix}$ GCN 81.1 67.4 79.8 79.9 diff-GCN 82.3 80.3 81.7 83.0
 Undirected Directed (fw) Directed (bw) Augmented (fw, bw) Method $A$ $A_\text{dir}$ $A_\text{dir}^T$ $\begin{bmatrix} A_\text{dir} \, A_\text{dir}^T \end{bmatrix}$ GCN 81.1 67.4 79.8 79.9 diff-GCN 82.3 80.3 81.7 83.0
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