ulupo
commented
3 years ago Hi! I am not 100% sure I understand the question but tell me if this helps: Given a set of points in 3D (or in any number of dimensions in that matter) and a set of edges between these points which you can use to define a graph, you have a few options depending on what exactly you want to achieve and how much of the structure in the data you want to use.
Simplest: number of holes in the unweighted graph
Here you wouldn't use persistent homology or even giotto-tda at all! If all you care about is the number of holes in the graph G, then you can use the Euler-Poincaré formula as follows: since you only have a graph and no triangles (no "faces"), the formula says that
V - E = β0 - β1
where V is the total number of vertices, E is the total number of edges, β0 is the number of connected components (zeroth Betti number) and β1 is the number of holes (first Betti number). Since V and E are known, and there are many algorithms for computing β0 (see e.g. https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.csgraph.connected_components.html), you can rearrange this formula to get
β1 = E - V + β0
Number of holes in a simplicial complex constructed from the graph
You can add triangles, tetrahedra, etc to the initial graph (which includes only vertices and edges) in several ways but arguably the simplest is the flag complex: whenever you have three vertices p, q and r and the (undirected) edges {p, q}, {p, r}, {q, r} all exist, then you declare that triangle {p, q, r} also exists! Now you are dealing with an object more general than a graph (a simplicial complex) and you may want to compute Betti numbers β0, β1 and β2, but the Euler-Poincaré formula is no longer sufficient to determine them. You need a real homology algorithm. You can achieve this with giotto-tda using ripser, and it will be very fast, but a bit indirect (I can explain if needed). The pyflagser library (https://github.com/giotto-ai/pyflagser), on the other hand, has a simpler API for this kind of thing. The function flagser_unweighted will return precisely the sequence of Betti numbers β0, β1 and β2. It is documented here.
Note. When you "complete" a graph using the flag complex method (with flagser_unweighted, for example), loops surrounding triangles in the graph no longer contribute to the count of holes, because in the flag complex construction we actually fill the loop with a two-dimensional triangle, so there is no hole.
Persistent homology version: persistence diagram of a filtered flag complex
Here, as above, you use the flag complex idea, but this time you apply it to thresholded versions of the graph, where you imagine that each edge comes with a weight. Once you have weights, you can use this. Euclidean distances between vertices could be weights, for example. giotto-tda is particularly suited for this. I recommend reading this tutorial if you are interested in this route.
Final remarks
I am not sure what you mean by "continuous two-dimensional code".
tanjia123456
Hello, thank you for your works! I currently have three-dimensional node coordinates and triangle, so I can use the tri_ surf builds a graph, I want to use this library for topology data analysis of my data. My topic is to segment the brain by graph neural network. The goal is to get continuous two-dimensional code and Betti number to process the holes on the graph. Which examples are more suitable?