GRAPH 1

https://towardsdatascience.com/how-to-do-deep-learning-on-graphs-with-graph-convolutional-networks-7d2250723780

https://github.com/dmlc/dgl

https://github.com/dglai/DGL-GTC2019/blob/master/slides.pptx

http://tkipf.github.io/misc/SlidesCambridge.pdf

GCN

$G(V, E)$

Input feature matrix

$H ^ 0 = X \leftarrow [N \cdot F ^ 0]$

N = # of nodes
F0 = # of features of each node

Adjacency matrix

representation of the graph structure

$A \leftarrow [N \cdot N]$

Output

$H ^{l+1} = f(H^l \cdot A) \leftarrow [N \cdot F^{l+1}]$

f = propagation

Propagation

Sum Rule

sum up feature representations of the neighbors of the ith node

$f(H^l \cdot A) = \sigma(A \cdot H^l \cdot W^l)$

$aggregate(A, X) = \sum_{j=1}^{N} A_{i,j} \cdot X_j$

W = weight matrix, dimension = [F_l, F_l+1]

Mean Rule: Self-loop & normalization

$\hat{A} = A + I$

$f(X \cdot A) = D^{-1} \cdot \hat{A} \cdot X$

$aggregate(A, X) = \sum_{j=1}^{N} \frac{A_{i,j}}{D_{i, i}} \cdot X_j$

D = degree matrix (for normalization)

Spectral Rule

$f(X \cdot A) = D^{-0.5} \cdot \hat{A} \cdot D^{-0.5} \cdot X$

$aggregate(A, X) = \sum_{j=1}^{N} \frac{1}{D_{i, i}^{0.5}} \cdot A_{i,j} \cdot \frac{1}{D_{j,j}^{0.5}} \cdot X_j$

aggregate feature of the ith node = the degree of the ith node, and also the degree of the jth node.

Inductive = supervised

Transductive = unsupervised