Package 'SimplicialComplex'

Compute Euler characteristic for an abstract simplicial complex

Description

Compute Euler characteristic for an abstract simplicial complex

Usage

abstract_simplicial_complex(simplices, dimension, tol = NULL)

Arguments

simplices	A list of simplices (each a numeric vector).
dimension	Optional max dimension to compute up to.
tol	Optional numerical tolerance to pass to rankMatrix().

Value

The Euler characteristic $\chi$.

Examples

simplices <- list(c(1, 2), c(3, 4), c(2, 1, 3), c(4, 2))
abstract_simplicial_complex(simplices, 2)

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Compute the boundary operator for a simplicial complex

Description

Compute the boundary operator for a simplicial complex

Usage

boundary(simplices, bound_dim)

Arguments

simplices	A list of simplices (each a numeric vector).
bound_dim	The dimension k of the boundary operator $\partial_{k}$.

Details

$\partial_k \sigma = \sum_i (-1)^i [v_0 v_1 \ldots \hat{v}_i \ldots v_k]$

Value

A sparse matrix representing $\partial_{k}$.

Examples

simplices <- list(c(1, 2), c(3, 4), c(2, 1, 3), c(4, 2))
boundary(simplices, 0)

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Get the boundary matrix and its reduction information in matrix form

Description

Get the boundary matrix and its reduction information in matrix form

Usage

boundary_info(filist)

Arguments

filist

Filtration list, each element includes simplex and time.

Value

A list containing the boundary matrix, the last boundary row, and the pivot owner for persistence extraction.

Examples

points <- matrix(c(0, 1, 1, 0, 0, 0, 1, 1), ncol = 2)
filtration <- build_vr_filtration(points, eps_max=1.2)
res <- boundary_info(filtration)

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Vietoris-Rips Filtration: Get the boundary matrix and its reduction information in matrix form

Description

Vietoris-Rips Filtration: Get the boundary matrix and its reduction information in matrix form

Usage

build_vr_filtration(points, eps_max)

Arguments

points	Data point input.
eps_max	Maximum scale (epsilon).

Value

A list of simplices with their information.

Examples

points <- matrix(c(0, 1, 1, 0, 0, 0, 1, 1), ncol = 2)
filtration <- build_vr_filtration(points, eps_max=1.2)

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Compute the Euler characteristic $\chi$ of a simplicial complex

Description

Compute the Euler characteristic $\chi$ of a simplicial complex

Usage

euler_characteristic(simplices, tol)

Arguments

simplices	A list of simplices (each a numeric vector).
tol	Optional numerical tolerance to pass to rankMatrix().

Details

The Euler characteristic is computed as:

$\chi = \sum_{k=0}^{k_{\max}} (-1)^k \beta_k$

where $\beta_k$ is the $k$th Betti number, and $k_{\max}$ is the highest dimension of any simplex in the complex.

Interpretation of values:

• $\chi = 2$: Sphere-like surfaces

• $\chi = 1$: Disk-like spaces

• $\chi = 0$: Torus-like or circle-like spaces

• $\chi < 0$: Surfaces with multiple handles or genus

Value

An integer representing the Euler characteristic $\chi$.

See Also

betti_number

Examples

simplices <- list(c(1, 2), c(3, 4), c(2, 1, 3), c(4, 2))
euler_characteristic(simplices, tol=0.1)

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This function extracts the persistence from combining the boundary matrix and its filtration

Description

This function extracts the persistence from combining the boundary matrix and its filtration

Usage

extract_persistence_pairs(filist, last_1, pivot_owner)

Arguments

filist	Filtration list, each element includes simplex and time.
last_1	The last 1 row index for each column in boundary matrix.
pivot_owner	The column index owning the pivot row.

Value

A data frame with columns: dimension, birth, and death.

Examples

points <- matrix(c(0, 1, 1, 0, 0, 0, 1, 1), ncol = 2)
filtration <- build_vr_filtration(points, eps_max=1.2)
res <- boundary_info(filtration)
pairs <- extract_persistence_pairs(filtration, res$last_1, res$pivot_owner)

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Generate all unique faces of a given dimension from simplices

Description

Generate all unique faces of a given dimension from simplices

Usage

faces(simplices, target_dim)

Arguments

simplices	A list of simplices (each a numeric vector).
target_dim	The target dimension $k$ for the faces (e.g., 0 for vertices, 1 for edges, etc.).

Details

The function generates all possible subsets (combinations) of each simplex, removes duplicates, and filters them to only include those of length target_dim + 1.

For example, a 2-simplex c(1, 2, 3) has three 1-dimensional faces (edges): c(1,2), c(1,3), and c(2,3), and three 0-dimensional faces (vertices): 1, 2, and 3.

Value

A list of faces (each a numeric vector) of dimension target_dim.

Examples

simplices <- list(c(1, 2), c(3, 4), c(2, 1, 3), c(4, 2))
faces(simplices, target_dim=0)

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Plot Persistence Diagram

Description

Plot Persistence Diagram

Usage

plot_persistence(df)

Arguments

df	Dataframe from plot_persistence.

Value

A ggplot2 object representing the persistence diagram.

Examples

points <- matrix(c(0, 1, 1, 0, 0, 0, 1, 1), ncol = 2)
filtration <- build_vr_filtration(points, eps_max=1.2)
res <- boundary_info(filtration)
pairs <- extract_persistence_pairs(filtration, res$last_1, res$pivot_owner)
plot_persistence(pairs)

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Construct a Vietoris–Rips Complex (1-skeleton + maximal simplices)

Description

Construct a Vietoris–Rips Complex (1-skeleton + maximal simplices)

Usage

VietorisRipsComplex(points, epsilon)

Arguments

points	A numeric matrix or data.frame with one point per row (columns are coordinates).
epsilon	A positive numeric threshold; connect points with distance < $\epsilon$.

Details

The Vietoris–Rips complex at scale $\epsilon$ includes a simplex for every finite set of points with pairwise distances < $\epsilon$. This function constructs the 1-skeleton (edges only) and then uses maximal cliques in that graph as the maximal simplices.

Value

A list with:

network

An igraph object representing the 1-skeleton.

simplices

A list of integer vectors, each the vertex indices of a maximal simplex.

Examples

points <- matrix(c(0, 1, 1, 0, 0, 0, 1, 1), ncol = 2)
epsilon <- 1.5
vr_complex <- VietorisRipsComplex(points, epsilon)

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