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Learning Convolutional Neural Networks for Graphs

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Mathias Niepert

This document discusses a method called Patchy for applying convolutional neural networks to graph-structured data. Patchy selects node sequences from graphs using centrality measures and assembles neighborhoods around the nodes. The neighborhoods are normalized and used as receptive fields for a convolutional architecture. Experiments on graph classification benchmarks show Patchy can outperform graph kernels in terms of efficiency and effectiveness while also supporting visualization of learned edge filters. Potential limitations include increased risk of overfitting on small datasets compared to graph kernels.

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Learning Convolutional Neural Networks for Graphs Mathias Niepert Mohamed Ahmed Konstantin Kutzkov NEC Laboratories Europe GA-653449
2 Learning Convolutional Neural Networks for Graphs Representation Learning for Graphs Telecom Safety Transportation Industry Smart cities (Edge) deployment Deep Learning System Intermediate Representation: Graphs ?
3 Learning Convolutional Neural Networks for Graphs Problem Definition ▌ Input: Finite collection of graphs ● Nodes of any two graphs are not necessarily in correspondence ● Nodes and edges may have attributes (discrete and continuous) ▌ Problem: Learn a representation for classification/regression ▌ Example: Graph classification problem … [ ] = ?class
4 Learning Convolutional Neural Networks for Graphs State of the Art: Graph Kernels ▌ Define kernel based on substructures ● Shortest paths ● Random walks ● Subtrees ● ... ▌ Kernel is similarity function on pairs of graphs ● Count the number of common substructures ▌ Use graph kernels with SVMs
5 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs
6 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes)
7 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes)
8 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes)
9 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes)
10 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes)
11 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes)
12 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes) neighborhood assembly (at least k=4 nodes)
13 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes) neighborhood assembly (at least k=4 nodes)
14 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes) neighborhood assembly (at least k=4 nodes)
15 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes) neighborhood assembly (at least k=4 nodes)
16 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes) neighborhood assembly (at least k=4 nodes)
17 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes) neighborhood assembly (at least k=4 nodes)
18 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes) neighborhood normalization (exactly k=4 nodes) 1 2 3 4 1 2 3 4 1 23 4 1 2 3 4 1 2 3 4 neighborhood assembly (at least k=4 nodes) 12 3 4
19 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs neighborhood normalization ● normalized neighborhoods serve as receptive fields ● node and edge attributes correspond to channels Convolutional architecture 1 2 3 4 1 2 3 4 a 1… a m 1 2 3 4 a 1… a n 1 2 3 4 1 2 3 4 a 1… a m 1 2 3 4 a 1… a n 1 2 3 4 1 2 3 4 a 1… a m 1 2 3 4 a 1… a n … 1 2 3 4 1 2 3 4 1 23 4 1 2 3 4 1 2 3 4 12 3 4
20 Learning Convolutional Neural Networks for Graphs Node Sequence Selection ▌ We use centrality measures to generate the node sequences ▌ Nodes with similar structural roles are aligned across graphs node sequence selection A: Betweenness centrality B: Closeness centrality C: Eigenvector centrality D: Degree centrality
21 Learning Convolutional Neural Networks for Graphs ▌ Simple breadth-first expansion until at least k nodes added, or no additional nodes to add Neighborhood Assembly neighborhood assembly
22 Learning Convolutional Neural Networks for Graphs ▌ Nodes of any two graphs should have similar position in the adjacency matrices iff their structural roles are similar ▌ Result: For several distance measure pairs it is possible to efficiently compare labeling methods without supervision ▌ Example: ||A – A’||1 and edit distance on graphs Graph Normalization Problem 1 2 3 4 normalization distance measures in matrix and graph space adjacency matrices under labeling labeling method (centrality etc.)
23 Learning Convolutional Neural Networks for Graphs Graph Normalization Distance to root node 1 2 2 3 3 3 3
24 Learning Convolutional Neural Networks for Graphs Graph Normalization Distance to root node Centrality measures, etc. 1 2 3 5 5 6 4 1 2 2 3 3 3 3
25 Learning Convolutional Neural Networks for Graphs Graph Normalization Distance to root node Centrality measures, etc. Canonicalization (break ties) 1 2 3 6 5 7 4 1 2 3 5 5 6 4 1 2 2 3 3 3 3
26 Learning Convolutional Neural Networks for Graphs Computational Complexity ▌ At most linear in number of input graphs ▌ At most quadratic in number of nodes for each graph (depends on maximal node degree and centrality measure)
27 Learning Convolutional Neural Networks for Graphs Convolutional Architecture 1 2 3 4 a 1… a n 1 2 3 4 a 1… a n 1 2 3 4 a 1… a n 1 2 3 4 a 1… a n 1 2 3 4 a 1… a n … v 1… v M field size: 4, stride: 4, filters: M
28 Learning Convolutional Neural Networks for Graphs Convolutional Architecture a 1… a n a 1… a n a 1… a n a 1… a n a 1… a n … v 1… v M field size: 4, stride: 4, filters: M 1 2 3 4 1 2 3 4 1 2 3 41 2 3 4 1 2 3 4
29 Learning Convolutional Neural Networks for Graphs Convolutional Architecture a 1… a n a 1… a n a 1… a n a 1… a n a 1… a n … v 1… v M field size: 4, stride: 4, filters: M v 1… v M … 1 2 3 4 1 2 3 4 1 2 3 41 2 3 4 1 2 3 4
30 Learning Convolutional Neural Networks for Graphs Convolutional Architecture a 1… a n a 1… a n a 1… a n a 1… a n a 1… a n … field size: 3, stride: 1, N filters v 1… v N field size: 4, stride: 4, filters: M v 1… v M v 1… v M … 1 2 3 4 1 2 3 4 1 2 3 41 2 3 4 1 2 3 4
31 Learning Convolutional Neural Networks for Graphs Convolutional Architecture a 1… a n a 1… a n a 1… a n a 1… a n a 1… a n … field size: 3, stride: 1, N filters v 1… v N field size: 4, stride: 4, filters: Mv 1… v M v 1… v M … 1 2 3 4 1 2 3 4 1 2 3 41 2 3 4 1 2 3 4
32 Learning Convolutional Neural Networks for Graphs Convolutional Architecture a 1… a n a 1… a n a 1… a n a 1… a n a 1… a n … field size: 3, stride: 1, N filters field size: 4, stride: 4, filters: Mv 1… v M v 1… v N v 1… v N v 1… v M … … 1 2 3 4 1 2 3 4 1 2 3 41 2 3 4 1 2 3 4
33 Learning Convolutional Neural Networks for Graphs Experiments - Graph Classification ▌ Finite collection of graphs and their class labels ● Nodes of any two graphs are not necessarily in correspondence ● Nodes and edges may have attributes (discrete and continuous) ▌ Learn a function from graphs to class labels … [ ] = ?class class = 1 class = 0 class = 1
34 Learning Convolutional Neural Networks for Graphs Experiments - Convolutional Architecture 1 2 … 10 a 1… a n 1 2 … 10 a 1… a n 1 2 … 10 a 1… a n 1 2 … 10 a 1… a n 1 2 … 10 a 1… a n … field size: k, stride: k, filters: 16 field size: 10, stride: 1, filters: 8 flatten, dense 128 units softmax 1 … 16 1 … 16 … 1 … 8 1 … 8 …
35 Learning Convolutional Neural Networks for Graphs Classification Datasets ▌ MUTAG: Nitro compounds where classes indicate mutagenic effect on a bacterium (Salmonella Typhimurium) ▌ PTC: Chemical compounds where classes indicate carcinogenicity for male and female rats ▌ NCI: Chemical compounds where classes indicate activity against non-small cell lung cancer and ovarian cancer cell lines ▌ D&D: Protein structures where classes indicate whether structure is an enzyme or not ▌ ...
36 Learning Convolutional Neural Networks for Graphs ▌ Q: How efficient and effective compared to graph kernels? ▌ Apply Patchy to typical graph classification benchmark data Experiments - Graph Classification
37 Learning Convolutional Neural Networks for Graphs Experiments - Visualization ▌ Q: What do learned edge filters look like? ▌ Restricted Boltzmann machine applied to graphs ▌ Receptive field size of hidden layer: 9 small instances of graphs weights of hidden nodes graphs sampled from RBM
38 Learning Convolutional Neural Networks for Graphs Discussion ▌ Pros: ● Graph kernel design not required ● Outperforms graph kernels on several datasets (speed and accuracy) ● Incorporates node and edge features (discrete and continuous) ● Supports visualizations (graph motifs, etc.) ▌ Cons: ● Prone to overfitting on smaller data sets (graph kernel benchmarks) ● Shift from designing graph kernels to tuning hyperparameters ● Graph normalization not part of learning code to be released: patchy.neclab.eu

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Learning Convolutional Neural Networks for Graphs

  • 1. Learning Convolutional Neural Networks for Graphs Mathias Niepert Mohamed Ahmed Konstantin Kutzkov NEC Laboratories Europe GA-653449
  • 2. 2 Learning Convolutional Neural Networks for Graphs Representation Learning for Graphs Telecom Safety Transportation Industry Smart cities (Edge) deployment Deep Learning System Intermediate Representation: Graphs ?
  • 3. 3 Learning Convolutional Neural Networks for Graphs Problem Definition ▌ Input: Finite collection of graphs ● Nodes of any two graphs are not necessarily in correspondence ● Nodes and edges may have attributes (discrete and continuous) ▌ Problem: Learn a representation for classification/regression ▌ Example: Graph classification problem … [ ] = ?class
  • 4. 4 Learning Convolutional Neural Networks for Graphs State of the Art: Graph Kernels ▌ Define kernel based on substructures ● Shortest paths ● Random walks ● Subtrees ● ... ▌ Kernel is similarity function on pairs of graphs ● Count the number of common substructures ▌ Use graph kernels with SVMs
  • 5. 5 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs
  • 6. 6 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes)
  • 7. 7 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes)
  • 8. 8 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes)
  • 9. 9 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes)
  • 10. 10 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes)
  • 11. 11 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes)
  • 12. 12 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes) neighborhood assembly (at least k=4 nodes)
  • 13. 13 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes) neighborhood assembly (at least k=4 nodes)
  • 14. 14 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes) neighborhood assembly (at least k=4 nodes)
  • 15. 15 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes) neighborhood assembly (at least k=4 nodes)
  • 16. 16 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes) neighborhood assembly (at least k=4 nodes)
  • 17. 17 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes) neighborhood assembly (at least k=4 nodes)
  • 18. 18 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs node sequence selection (w=6 nodes) neighborhood normalization (exactly k=4 nodes) 1 2 3 4 1 2 3 4 1 23 4 1 2 3 4 1 2 3 4 neighborhood assembly (at least k=4 nodes) 12 3 4
  • 19. 19 Learning Convolutional Neural Networks for Graphs Patchy: Learning CNNs for Graphs neighborhood normalization ● normalized neighborhoods serve as receptive fields ● node and edge attributes correspond to channels Convolutional architecture 1 2 3 4 1 2 3 4 a 1… a m 1 2 3 4 a 1… a n 1 2 3 4 1 2 3 4 a 1… a m 1 2 3 4 a 1… a n 1 2 3 4 1 2 3 4 a 1… a m 1 2 3 4 a 1… a n … 1 2 3 4 1 2 3 4 1 23 4 1 2 3 4 1 2 3 4 12 3 4
  • 20. 20 Learning Convolutional Neural Networks for Graphs Node Sequence Selection ▌ We use centrality measures to generate the node sequences ▌ Nodes with similar structural roles are aligned across graphs node sequence selection A: Betweenness centrality B: Closeness centrality C: Eigenvector centrality D: Degree centrality
  • 21. 21 Learning Convolutional Neural Networks for Graphs ▌ Simple breadth-first expansion until at least k nodes added, or no additional nodes to add Neighborhood Assembly neighborhood assembly
  • 22. 22 Learning Convolutional Neural Networks for Graphs ▌ Nodes of any two graphs should have similar position in the adjacency matrices iff their structural roles are similar ▌ Result: For several distance measure pairs it is possible to efficiently compare labeling methods without supervision ▌ Example: ||A – A’||1 and edit distance on graphs Graph Normalization Problem 1 2 3 4 normalization distance measures in matrix and graph space adjacency matrices under labeling labeling method (centrality etc.)
  • 23. 23 Learning Convolutional Neural Networks for Graphs Graph Normalization Distance to root node 1 2 2 3 3 3 3
  • 24. 24 Learning Convolutional Neural Networks for Graphs Graph Normalization Distance to root node Centrality measures, etc. 1 2 3 5 5 6 4 1 2 2 3 3 3 3
  • 25. 25 Learning Convolutional Neural Networks for Graphs Graph Normalization Distance to root node Centrality measures, etc. Canonicalization (break ties) 1 2 3 6 5 7 4 1 2 3 5 5 6 4 1 2 2 3 3 3 3
  • 26. 26 Learning Convolutional Neural Networks for Graphs Computational Complexity ▌ At most linear in number of input graphs ▌ At most quadratic in number of nodes for each graph (depends on maximal node degree and centrality measure)
  • 27. 27 Learning Convolutional Neural Networks for Graphs Convolutional Architecture 1 2 3 4 a 1… a n 1 2 3 4 a 1… a n 1 2 3 4 a 1… a n 1 2 3 4 a 1… a n 1 2 3 4 a 1… a n … v 1… v M field size: 4, stride: 4, filters: M
  • 28. 28 Learning Convolutional Neural Networks for Graphs Convolutional Architecture a 1… a n a 1… a n a 1… a n a 1… a n a 1… a n … v 1… v M field size: 4, stride: 4, filters: M 1 2 3 4 1 2 3 4 1 2 3 41 2 3 4 1 2 3 4
  • 29. 29 Learning Convolutional Neural Networks for Graphs Convolutional Architecture a 1… a n a 1… a n a 1… a n a 1… a n a 1… a n … v 1… v M field size: 4, stride: 4, filters: M v 1… v M … 1 2 3 4 1 2 3 4 1 2 3 41 2 3 4 1 2 3 4
  • 30. 30 Learning Convolutional Neural Networks for Graphs Convolutional Architecture a 1… a n a 1… a n a 1… a n a 1… a n a 1… a n … field size: 3, stride: 1, N filters v 1… v N field size: 4, stride: 4, filters: M v 1… v M v 1… v M … 1 2 3 4 1 2 3 4 1 2 3 41 2 3 4 1 2 3 4
  • 31. 31 Learning Convolutional Neural Networks for Graphs Convolutional Architecture a 1… a n a 1… a n a 1… a n a 1… a n a 1… a n … field size: 3, stride: 1, N filters v 1… v N field size: 4, stride: 4, filters: Mv 1… v M v 1… v M … 1 2 3 4 1 2 3 4 1 2 3 41 2 3 4 1 2 3 4
  • 32. 32 Learning Convolutional Neural Networks for Graphs Convolutional Architecture a 1… a n a 1… a n a 1… a n a 1… a n a 1… a n … field size: 3, stride: 1, N filters field size: 4, stride: 4, filters: Mv 1… v M v 1… v N v 1… v N v 1… v M … … 1 2 3 4 1 2 3 4 1 2 3 41 2 3 4 1 2 3 4
  • 33. 33 Learning Convolutional Neural Networks for Graphs Experiments - Graph Classification ▌ Finite collection of graphs and their class labels ● Nodes of any two graphs are not necessarily in correspondence ● Nodes and edges may have attributes (discrete and continuous) ▌ Learn a function from graphs to class labels … [ ] = ?class class = 1 class = 0 class = 1
  • 34. 34 Learning Convolutional Neural Networks for Graphs Experiments - Convolutional Architecture 1 2 … 10 a 1… a n 1 2 … 10 a 1… a n 1 2 … 10 a 1… a n 1 2 … 10 a 1… a n 1 2 … 10 a 1… a n … field size: k, stride: k, filters: 16 field size: 10, stride: 1, filters: 8 flatten, dense 128 units softmax 1 … 16 1 … 16 … 1 … 8 1 … 8 …
  • 35. 35 Learning Convolutional Neural Networks for Graphs Classification Datasets ▌ MUTAG: Nitro compounds where classes indicate mutagenic effect on a bacterium (Salmonella Typhimurium) ▌ PTC: Chemical compounds where classes indicate carcinogenicity for male and female rats ▌ NCI: Chemical compounds where classes indicate activity against non-small cell lung cancer and ovarian cancer cell lines ▌ D&D: Protein structures where classes indicate whether structure is an enzyme or not ▌ ...
  • 36. 36 Learning Convolutional Neural Networks for Graphs ▌ Q: How efficient and effective compared to graph kernels? ▌ Apply Patchy to typical graph classification benchmark data Experiments - Graph Classification
  • 37. 37 Learning Convolutional Neural Networks for Graphs Experiments - Visualization ▌ Q: What do learned edge filters look like? ▌ Restricted Boltzmann machine applied to graphs ▌ Receptive field size of hidden layer: 9 small instances of graphs weights of hidden nodes graphs sampled from RBM
  • 38. 38 Learning Convolutional Neural Networks for Graphs Discussion ▌ Pros: ● Graph kernel design not required ● Outperforms graph kernels on several datasets (speed and accuracy) ● Incorporates node and edge features (discrete and continuous) ● Supports visualizations (graph motifs, etc.) ▌ Cons: ● Prone to overfitting on smaller data sets (graph kernel benchmarks) ● Shift from designing graph kernels to tuning hyperparameters ● Graph normalization not part of learning code to be released: patchy.neclab.eu