APPNP model on Heterogeneous graph?

I’m trying to reproduce the model in a paper Pre-training on Large-Scale Heterogeneous Graph. Basically, it’s a heterogeneous graph contrastive learning method based on Personalized PageRank. To perform PPNP on heterogeneous, they obtain personalized PageRank Π^𝑅 for a relation 𝑅 by iteratively updating the following:

I don’t understand the following points:

  1. What’t the initial state of the iteration before updating
  2. what’s the dimension of Π^𝑅 and Π^𝑅(-1), in my opinion, it should match the number of nodes of the corresponding A_1 & A_2 type.
  3. Correspondingly, how can the symmetric (by definition) unix matrix I be added to the asymmetric Π^𝑅 (whose dimension should match the number of nodes of the corresponding A_1 & A_2 type)

I email to the authors but haven’t been replied, so I have to seek help here. It would be great if someone familiar with this model could reply to me. Thanks in advance!

Based on the sparsification in Sec. 3.2.2, I think \Pi^{R}\in\mathbb{R}^{N \times M} and \Pi^{R^{-1}}\in\mathbb{R}^{M \times N}, where N and M are number of nodes with type A_1 and A_2, respectively. However, in that case, the equation (10) just doesn’t make any sense in the matrix form.

You may continue to reach out to the authors for more details or dive into their approximation algorithm in Appendix A.