Module ateams.statistics.homology

Functions

def bettis(pairs, essential, breaks)

Computes the betti curves admitted by the provided persistence pairs. The ith betti curve is the piecewise function that records the rank of the ith homology group of the filtration over time.

Args

pairs : np.array

Two-dimensional integer array (e.g. the one returned by InvasionPercolation.proposal()) representing persistence pairs.

esssential : np.array

Integer array of all homological percolation times (e.g. the one returned by InvasionPercolation.proposal()).

breaks : np.array

Two-dimensional integer array where the $i$th row indicates the first and last indices of $i$-cells in the complete complex filtration. Can be constructed from the .breaks property of any Model, or given immediately by the .tranches property of any Model.

Returns

A $k \times N$ array where the $i$th row corresponds to the expected degree of the $i$th homology group at all times $0 \leq t \leq N$, for $N$ the total number of cells in the complex.