Simplex tree reference manual¶

class gudhi.SimplexTree¶

Bases: object

The simplex tree is an efficient and flexible data structure for representing general (filtered) simplicial complexes. The data structure is described in Jean-Daniel Boissonnat and Clément Maria. The Simplex Tree: An Efficient Data Structure for General Simplicial Complexes. Algorithmica, pages 1–22, 2014.

This class is a filtered, with keys, and non contiguous vertices version of the simplex tree.

__init__()¶

SimplexTree constructor.

Parameters: other¶ (SimplexTree (Optional)) – If other is None (default value), an empty SimplexTree is created. If other is a SimplexTree, the SimplexTree is constructed from a deep copy of other.
Returns: An empty or a copy simplex tree.
Return type: SimplexTree
Raises: TypeError – In case other is neither None, nor a SimplexTree.
Note: If the SimplexTree is a copy, the persistence information is not copied. If you need it in the clone, you have to call compute_persistence() on it even if you had already computed it in the original.

assign_filtration()¶

This function assigns a new filtration value to a given N-simplex.

Parameters

simplex¶ (list of int) – The N-simplex, represented by a list of vertex.
filtration¶ (float) – The new filtration value.

Note

Beware that after this operation, the structure may not be a valid filtration anymore, a simplex could have a lower filtration value than one of its faces. Callers are responsible for fixing this (with more assign_filtration() or make_filtration_non_decreasing() for instance) before calling any function that relies on the filtration property, like persistence().

betti_numbers()¶

This function returns the Betti numbers of the simplicial complex.

Returns: The Betti numbers ([B0, B1, …, Bn]).
Return type: list of int
Note: betti_numbers function requires compute_persistence() function to be launched first.

collapse_edges()¶

Assuming the complex is a graph (simplices of higher dimension are ignored), this method implicitly interprets it as the 1-skeleton of a flag complex, and replaces it with another (smaller) graph whose expansion has the same persistent homology, using a technique known as edge collapses (see [21]).

A natural application is to get a simplex tree of dimension 1 from RipsComplex, then collapse edges, perform expansion() and finally compute persistence (cf. rips_complex_edge_collapse_example.py).

Parameters: nb_iterations¶ (int) – The number of edge collapse iterations to perform. Default is 1.

compute_persistence()¶

This function computes the persistence of the simplicial complex, so it can be accessed through persistent_betti_numbers(), persistence_pairs(), etc. This function is equivalent to persistence() when you do not want the list persistence() returns.

Parameters

homology_coeff_field¶ (int) – The homology coefficient field. Must be a prime number. Default value is 11. Max is 46337.
min_persistence¶ (float) – The minimum persistence value to take into account (strictly greater than min_persistence). Default value is 0.0. Sets min_persistence to -1.0 to see all values.
persistence_dim_max¶ (bool) – If true, the persistent homology for the maximal dimension in the complex is computed. If false, it is ignored. Default is false.

Returns

Nothing.

copy()¶

Returns: A simplex tree that is a deep copy of itself.
Return type: SimplexTree
Note: The persistence information is not copied. If you need it in the clone, you have to call compute_persistence() on it even if you had already computed it in the original.

static create_from_array()¶

Creates a new, empty complex and inserts vertices and edges. The vertices are numbered from 0 to n-1, and the filtration values are encoded in the array, with the diagonal representing the vertices. It is the caller’s responsibility to ensure that this defines a filtration, which can be achieved with either:

filtrations[np.diag_indices_from(filtrations)] = filtrations.min(axis=1)

or:

diag = filtrations.diagonal()
filtrations = np.fmax(np.fmax(filtrations, diag[:, None]), diag[None, :])

Parameters

filtrations¶ (numpy.ndarray of shape (n,n)) – the filtration values of the vertices and edges to insert. The matrix is assumed to be symmetric.
max_filtration¶ (float) – only insert vertices and edges with filtration values no larger than max_filtration

Returns

the new complex

Return type

SimplexTree

dimension()¶

This function returns the dimension of the simplicial complex.

Returns: the simplicial complex dimension.
Return type: int

Note

This function is not constant time because it can recompute dimension if required (can be triggered by remove_maximal_simplex() or prune_above_filtration() methods).

expansion()¶

Expands the simplex tree containing only its one skeleton until dimension max_dim.

The expanded simplicial complex until dimension \(d\) attached to a graph \(G\) is the maximal simplicial complex of dimension at most \(d\) admitting the graph \(G\) as \(1\)-skeleton. The filtration value assigned to a simplex is the maximal filtration value of one of its edges.

The simplex tree must contain no simplex of dimension bigger than 1 when calling the method.

Parameters: max_dimension¶ (int) – The maximal dimension.

expansion_with_blocker()¶

Expands the Simplex_tree containing only a graph. Simplices corresponding to cliques in the graph are added incrementally, faces before cofaces, unless the simplex has dimension larger than max_dim or blocker_func returns True for this simplex.

The function identifies a candidate simplex whose faces are all already in the complex, inserts it with a filtration value corresponding to the maximum of the filtration values of the faces, then calls blocker_func with this new simplex (represented as a list of int). If blocker_func returns True, the simplex is removed, otherwise it is kept. The algorithm then proceeds with the next candidate.

Warning

Several candidates of the same dimension may be inserted simultaneously before calling blocker_func, so if you examine the complex in blocker_func, you may hit a few simplices of the same dimension that have not been vetted by blocker_func yet, or have already been rejected but not yet removed.

Parameters

max_dim¶ (int) – Expansion maximal dimension value.
blocker_func¶ (Callable[[List[int]], bool]) – Blocker oracle.

extend_filtration()¶

Extend filtration for computing extended persistence. This function only uses the filtration values at the 0-dimensional simplices, and computes the extended persistence diagram induced by the lower-star filtration computed with these values.

Note

Note that after calling this function, the filtration values are actually modified within the simplex tree. The function extended_persistence() retrieves the original values.

Note

Note that this code creates an extra vertex internally, so you should make sure that the simplex tree does not contain a vertex with the largest possible value (i.e., 4294967295).

This notebook explains how to compute an extension of persistence called extended persistence.

extended_persistence()¶

This function retrieves good values for extended persistence, and separate the diagrams into the Ordinary, Relative, Extended+ and Extended- subdiagrams.

Parameters

homology_coeff_field¶ (int) – The homology coefficient field. Must be a prime number. Default value is 11. Max is 46337.
min_persistence¶ (float) – The minimum persistence value (i.e., the absolute value of the difference between the persistence diagram point coordinates) to take into account (strictly greater than min_persistence). Default value is 0.0. Sets min_persistence to -1.0 to see all values.

Returns

A list of four persistence diagrams in the format described in persistence(). The first one is Ordinary, the second one is Relative, the third one is Extended+ and the fourth one is Extended-. See https://link.springer.com/article/10.1007/s10208-008-9027-z and/or section 2.2 in https://link.springer.com/article/10.1007/s10208-017-9370-z for a description of these subtypes.

Note

This function should be called only if extend_filtration() has been called first!

Note

The coordinates of the persistence diagram points might be a little different than the original filtration values due to the internal transformation (scaling to [-2,-1]) that is performed on these values during the computation of extended persistence.

This notebook explains how to compute an extension of persistence called extended persistence.

filtration()¶

This function returns the filtration value for a given N-simplex in this simplicial complex, or +infinity if it is not in the complex.

Parameters: simplex¶ (list of int) – The N-simplex, represented by a list of vertex.
Returns: The simplicial complex filtration value.
Return type: float

find()¶

This function returns if the N-simplex was found in the simplicial complex or not.

Parameters: simplex¶ (list of int) – The N-simplex to find, represented by a list of vertex.
Returns: true if the simplex was found, false otherwise.
Return type: bool

flag_persistence_generators()¶

Assuming this is a flag complex, this function returns the persistence pairs, where each simplex is replaced with the vertices of the edges that gave it its filtration value.

Returns: First the regular persistence pairs of dimension 0, with one vertex for birth and two for death; then the other regular persistence pairs, grouped by dimension, with 2 vertices per extremity; then the connected components, with one vertex each; finally the other essential features, grouped by dimension, with 2 vertices for birth.
Return type: Tuple[numpy.array[int] of shape (n,3), List[numpy.array[int] of shape (m,4)], numpy.array[int] of shape (l,), List[numpy.array[int] of shape (k,2)]]
Note: flag_persistence_generators requires that persistence() be called first.

get_boundaries()¶

This function returns a generator with the boundaries of a given N-simplex. If you do not need the filtration values, the boundary can also be obtained as itertools.combinations(simplex,len(simplex)-1).

Parameters: simplex¶ (list of int.) – The N-simplex, represented by a list of vertex.
Returns: The (simplices of the) boundary of a simplex
Return type: generator with tuples(simplex, filtration)

get_cofaces()¶

This function returns the cofaces of a given N-simplex with a given codimension.

Parameters

simplex¶ (list of int) – The N-simplex, represented by a list of vertex.
codimension¶ (int) – The codimension. If codimension = 0, all cofaces are returned (equivalent of get_star function)

Returns

The (simplices of the) cofaces of a simplex

Return type

list of tuples(simplex, filtration)

get_filtration()¶

This function returns a generator with simplices and their given filtration values sorted by increasing filtration values.

Returns: The simplices sorted by increasing filtration values.
Return type: generator with tuples(simplex, filtration)

get_simplices()¶

This function returns a generator with simplices and their given filtration values.

Returns: The simplices.
Return type: generator with tuples(simplex, filtration)

get_skeleton()¶

This function returns a generator with the (simplices of the) skeleton of a maximum given dimension.

Parameters: dimension¶ (int) – The skeleton dimension value.
Returns: The (simplices of the) skeleton of a maximum dimension.
Return type: generator with tuples(simplex, filtration)

get_star()¶

This function returns the star of a given N-simplex.

Parameters: simplex¶ (list of int) – The N-simplex, represented by a list of vertex.
Returns: The (simplices of the) star of a simplex.
Return type: list of tuples(simplex, filtration)

initialize_filtration()¶: This function initializes and sorts the simplicial complex filtration vector.

Deprecated since version 3.2.0.

insert()¶

This function inserts the given N-simplex and its subfaces with the given filtration value (default value is ‘0.0’). If some of those simplices are already present with a higher filtration value, their filtration value is lowered.

Parameters

simplex¶ (list of int) – The N-simplex to insert, represented by a list of vertex.
filtration¶ (float) – The filtration value of the simplex.

Returns

true if the simplex was not yet in the complex, false otherwise (whatever its original filtration value).

Return type

bool

insert_batch¶

Inserts k-simplices given by a sparse array in a format similar to torch.sparse. The n-th simplex has vertices vertex_array[0,n], …, vertex_array[k,n] and filtration value filtrations[n]. If a simplex is repeated, the smallest filtration value is used. Simplices with a repeated vertex are currently interpreted as lower dimensional simplices, but we do not guarantee this behavior in the future. Any time a simplex is inserted, its faces are inserted as well if needed to preserve a simplicial complex.

Parameters

vertex_array (numpy.array of shape (k+1,n)) – the k-simplices to insert.
filtrations (numpy.array of shape (n,)) – the filtration values.

insert_edges_from_coo_matrix()¶

Inserts edges given by a sparse matrix in COOrdinate format. If an edge is repeated, the smallest filtration value is used. Missing entries are not inserted. Diagonal entries are currently interpreted as vertices, although we do not guarantee this behavior in the future, and this is only useful if you want to insert vertices with a smaller filtration value than the smallest edge containing it, since vertices are implicitly inserted together with the edges.

Parameters: edges¶ (scipy.sparse.coo_matrix of shape (n,n)) – the edges to insert and their filtration values.