  Topological data analysis can extract effective information from
higher-dimensional data. Its mathematical basis is persistent homology. The
persistent homology can calculate topological features at different
spatiotemporal scales of the dataset; that is, establishing the integrated
taxonomic relation among points, lines and simplices. Here, the simplicial
network composed of all-order simplices in a simplicial complex is essential.
Because the sequence of nested simplicial subnetworks can be regarded as a
discrete Morse function from the simplicial network to real values, a method
based on the concept of critical simplices can be developed by searching
all-order spanning trees. Employing this new method, not only the Morse
function values with the theoretical minimum number of critical simplices can
be obtained, but also the Betti numbers and composition of all-order cavities
in the simplicial network can be calculated quickly. Finally, this method is
used to analyze some examples and compared with other methods, showing its
effectiveness and feasibility.
