Vietoris‐Rips Lifting (Graph to Simplicial)

Although typically used for point cloud data, this method can be adapted to graphs by treating vertices as points and defining edges based on graph distances. A k-simplex is included if the pairwise distances between its vertices are all below a certain threshold.

To construct a Vietoris-Rips complex:

1. Define a distance metric based on the graph (e.g., shortest path distance).
2. Select a threshold distance.
3. Form simplices for all sets of vertices with pairwise distances less than the threshold distance.

We use nx.all_pairs_shortest_path_length to calculate the shortest path distances between all pairs of nodes in the graph. The VietorisRipsLifting class includes a distance_threshold parameter to specify the maximum allowed distance for simplex formation.

From https://github.com/pyt-team/challenge-icml-2024/pull/7