Table 9 The GraRep overall algorithm

GraRep Algorithm

Input

Adjacency matrix S on graph

Maximum transition step K

Log shifted factor β

Dimension of representation vector d

1. Get k-step transition probability matrix A^k

Compute A = D⁻¹S

Calculate A⁻¹, A⁻²,  A⁻³, …,  A^k, respectively

2. Get each k-step representations

For k = 1 to K

 2.1 Get positive log probability matrix

 calculate \( {\Gamma}_1^k,{\Gamma}_2^k,{\Gamma}_3^k,\dots, {\Gamma}_N^k \) (\( {\Gamma}_i^k={\sum}_p{A}_{p,j}^k \)) respectively

 calculate \( \left\{{X}_{i,j}^k\right\} \)

  \( {X}_{i,j}^k \) = log (\( \frac{A_{i,j}^k}{\Gamma_j^k} \)) – log(β)

 assign negative entries of X^k to 0

 2.2 Construct the representation vector W^k

 SVD(X^k) = [U^k, ∑^k, (V^k)^T]

 \( {W}^k={U}_d^k\ {\left({\sum}_d^k\right)}^{\frac{1}{2}} \)

End for

3. Concatenate all the k-step representations

W = [W¹, W², W³, …W^k]

Output

Matrix of the graph representation W