learning convolutional neural networks for graphs

Learning Convolutional Neural Networks for GraphsMathias 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

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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)

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neighborhood assembly (at least k=4 nodes)

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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

1234

1 2 3 4 a 1 … a m

1 2 3 4 a 1 … a n

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1 2 3 4 a 1 … a m

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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 labelingmethods without supervision

▌ Example: ||A – A’||1 and edit distanceon graphs

Graph Normalization Problem

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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

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24 Learning Convolutional Neural Networks for Graphs

Graph Normalization

Distance to root node

Centrality measures, etc.

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Graph Normalization

Distance to root node

Centrality measures, etc.

Canonicalization (break ties)

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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 … 1

6…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