Graph neural network

In deep learning, a graph neural network (GNN) is a subarea of neural network, which is designed to process graph structured data or data that is able to be formulated as a graph potentially^[1][2] (e.g. social network, polygon mesh, point cloud). Since graph data is non-Euclidean, relations between data points cannot be easily represented by their ordering when recording them, and hence standard CNN is not able to be directly applied to graph data. On the other hand, GNN not only be able to applied to non-Euclidean data but also Euclidean data such as sentences, images or videos since such data can be represented as graph data if organized properly.

Graph with 5 nodes.

Pipeline of a GNN model^[1]

The design pipeline for a GNN model can be generally derived as four steps: find graph structure, specify graph, design loss functions, and build model .

 

The general design pipeline for a GNN model.

Find graph structure

In graph theory, a graph is denoted as ${\displaystyle G=(V,E)}$ , where:

• ${\displaystyle V}$ , a set of vertices (also called nodes or points);
• ${\displaystyle E}$ , a set of edges (either directed or undirected, also called links or lines);
• ${\displaystyle A}$ , the given graph's adjacency matrix;
• ${\displaystyle D}$ , the given graph's degree matrix.

If the input data is already in graph structure, then this step is done. Otherwise, you need to observe the data first and reorganize it to be a graph according to your requirement, while not destroying the data's property (so that your model won't face the "garbage in, garbage out" problem).

Specify graph

 

An example of scene graph.

After a graph structure is found in the given data, the type of this graph should also be specified. A graph can be simply categorize as directed/undirected or homogeneous/heterogeneous. Note that for heterogeneous graphs, each edge may differ to the others by its property. For example, each edge in a scene graph^[3] has different meaning to represent the relation between nodes. Sometimes the data's nodes can be merged to obtain graphs of different resolutions, and hence the graph structure may dynamically changed during the learning process. For example, when regarding point cloud as a graph, it is mostly a dynamic graph^[4][5][6][7].

Design loss function

Base on the task you are dealing with, loss functions have to be chosen wisely. For example, for a supervised node-level classification task, cross-entropy might be a reasonable choice.

Build model

• Propagation module: updating information carried by nodes and/or edges by some aggregation methods.
• Sampling module: when a graph is too large, sampling modules are needed for computation preservation.
• Pooling module: when higher level information (sub-graph-level, graph-level) is needed for the task, pooling modules are needed to aggregate low-level information and provide hierarchical propagation.

Architecture

Convolution based methods

The main idea of this type of methods is to generalize standard CNN or attentional methods to graph structured data, and that is, to define some receptive fields with respect to given nodes and propagate information within. Based on the operation ___domain, we can further divide these methods into two groups:

Spectral approaches

For spectral approaches, a graph signal ${\displaystyle x}$  is first transformed to spectral ___domain by the graph Fourier Transform ${\displaystyle {\mathcal {F}}}$ , then the convolution operation can be conducted. After that, we can transform the result back to the original ___domain (spatial ___domain) by the inverse graph Fourier Transform ${\displaystyle {\mathcal {F^{-1}}}}$ . ${\displaystyle {\mathcal {F}}}$  and ${\displaystyle {\mathcal {F^{-1}}}}$  are defined as:

• ${\displaystyle {\mathcal {F}}(x)=U^{T}x}$ 
• ${\displaystyle {\mathcal {F^{-1}}}(x)=Ux}$ ,

where ${\displaystyle U}$  is the matrix of eigenvectors of the symmetric normalized graph Laplacian ${\displaystyle L^{\text{sym}}}$ . ${\displaystyle L^{\text{sym}}}$  is denoted as:

${\displaystyle L^{\text{sym}}:=D^{-{\frac {1}{2}}}LD^{-{\frac {1}{2}}}=I-D^{-{\frac {1}{2}}}AD^{-{\frac {1}{2}}}=U\Lambda U^{T}}$  ,

where ${\displaystyle L}$  is the original graph Laplacian, ${\displaystyle I}$  is an identity matrix, and ${\displaystyle \Lambda }$  is a diagonal matrix.

Therefore, based on the convolution's property, the convolution of the signal ${\displaystyle x}$  and a learnable kernel function ${\displaystyle g}$  is defined as:

${\displaystyle g\star x={\mathcal {F^{-1}}}({\mathcal {F}}(g)\odot {\mathcal {F}}(x))=U(U^{T}g\odot U^{T}x)}$ ,

and if we set the learnable kernel function to be a diagonal one ${\displaystyle g_{\theta }}$ , this operation is further simplified to:

${\displaystyle g_{\theta }\star x=Ug_{\theta }U^{T}x}$ .

References

1. "Graph neural networks: A review of methods and applications". AI Open. 1: 57–81. 2020-01-01. doi:10.1016/j.aiopen.2021.01.001. ISSN 2666-6510.
2. Zhang, Si; Tong, Hanghang; Xu, Jiejun; Maciejewski, Ross (2019-11-10). "Graph convolutional networks: a comprehensive review". Computational Social Networks. 6 (1): 11. doi:10.1186/s40649-019-0069-y. ISSN 2197-4314.
3. Johnson, Justin; Krishna, Ranjay; Stark, Michael; Li, Li-Jia; Shamma, David A.; Bernstein, Michael S.; Fei-Fei, Li. "Image retrieval using scene graphs". 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR): 3668–3678. doi:10.1109/CVPR.2015.7298990.
4. Wang, Yue; Sun, Yongbin; Liu, Z.; Sarma, Sanjay E.; Bronstein, M.; Solomon, J. (2019). "Dynamic Graph CNN for Learning on Point Clouds". ACM Trans. Graph. doi:10.1145/3326362.
5. Thomas, Hugues; Qi, Charles R.; Deschaud, Jean-Emmanuel; Marcotegui, Beatriz; Goulette, François; Guibas, Leonidas. "KPConv: Flexible and Deformable Convolution for Point Clouds". 2019 IEEE/CVF International Conference on Computer Vision (ICCV): 6410–6419. doi:10.1109/ICCV.2019.00651.
6. Lin, Zhi-Hao; Huang, Sheng-Yu; Wang, Yu-Chiang Frank. "Convolution in the Cloud: Learning Deformable Kernels in 3D Graph Convolution Networks for Point Cloud Analysis". 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR): 1797–1806. doi:10.1109/CVPR42600.2020.00187.
7. Lin, Zhi-Hao; Huang, Sheng Yu; Wang, Yu-Chiang Frank (2021). "Learning of 3D Graph Convolution Networks for Point Cloud Analysis". IEEE Transactions on Pattern Analysis and Machine Intelligence: 1–1. doi:10.1109/TPAMI.2021.3059758. ISSN 1939-3539.