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Persistent homology of a weighted network or a distance matrix

Source: R/simplicial_wrappers.R
hg_homology.Rd

Tracks the homology classes (components, loops, voids) of a filtration: a weighted clique filtration of a similarity matrix (type = "clique", strongest edges first) or a Vietoris-Rips filtration of a distance matrix (type = "vr"). Every class is paired with the simplex that creates it (birth) and the one that destroys it (death), by exact boundary-matrix reduction over Z/2.

Usage

hg_homology(x, n_steps = 20L, max_dim = 3L, type = "clique", max_scale = NULL)

Arguments

x

A square matrix, a network object carrying one (netobject, tna), points for type = "vr" (see simplicial()), a simplicial_complex built with type = "vr" (its stored filtration is used as is), or a window complex from simplicial(type = "window"). For a window complex the filtration is the count filtration (Petri et al. 2013): the complex at count t holds the sets of actions observed in at least t windows and their faces, so a class is born at the count where it appears and dies at a lower count; birth, death and the Betti-curve threshold are counts, and n_steps is not used (every observed count is a threshold).

n_steps

Grid points of the reported Betti curve. Default 20; the persistence diagram itself is exact.

max_dim

Highest simplex dimension tracked. Default 3.

type

"clique" (default) or "vr".

max_scale

For type = "vr": the longest edge admitted. NULL uses the largest distance.

Value

A persistent_homology object. Read it with hg_get(): the persistence diagram (what = "persistence", the default) or the Betti curves ("betti").

References

Edelsbrunner, H., Letscher, D., & Zomorodian, A. (2002). Topological persistence and simplification. Discrete & Computational Geometry, 28, 511–533. doi:10.1007/s00454-002-2885-2

Petri, G., Scolamiero, M., Donato, I., & Vaccarino, F. (2013). Topological strata of weighted complex networks. PLoS ONE, 8(6), e66506. doi:10.1371/journal.pone.0066506

See also

hg_landscape(), hg_bottleneck(), hg_wasserstein()

Examples

set.seed(1)
w <- matrix(runif(36), 6, 6, dimnames = list(letters[1:6], letters[1:6]))
w <- (w + t(w)) / 2
diag(w) <- 0
ph <- hg_homology(w, n_steps = 10)
hg_get(ph, sort_by = "persistence")
#>    dimension     birth     death persistence
#> 1          0 0.7427237 0.0000000  0.74272370
#> 2          3 0.2344513 0.0000000  0.23445130
#> 3          3 0.2344513 0.0000000  0.23445130
#> 4          0 0.7427237 0.5838612  0.15886254
#> 5          3 0.1558863 0.0000000  0.15588635
#> 6          3 0.1558863 0.0000000  0.15588635
#> 7          3 0.1558863 0.0000000  0.15588635
#> 8          1 0.5170309 0.3655044  0.15152644
#> 9          0 0.7427237 0.6583996  0.08432412
#> 10         0 0.7427237 0.6902349  0.05248880
#> 11         0 0.7427237 0.7162022  0.02652146
#> 12         1 0.6441215 0.6299381  0.01418338