Using persistent homology, we can analyze the (topological) shape of a cloud of points using the following three-step method:
- convert the point set into a simplicial complex (and there are a few different ways of doing this) parameterized by a "noise" parameter
- Compute the homology groups of this complex (again parameterized by the parameter)
- look at the evolution of the groups as the parameter evolves.
The "time of life" of the different groups looks like a collection of intervals, which is called the "barcode" of the shape.
Is there an easy explanation of what the barcode looks like if the simplicial complex is merely a 1-skeleton (i.e a graph) ? In other words, suppose we start with a graph (rather than a point set) and then do the remaining two steps as above.