topobench.transforms.liftings.graph2simplicial.vietoris_rips_lifting module#

This module implements the SimplicialVietorisRipsLifting class, which lifts graphs to simplicial complexes.

class Graph2SimplicialLifting(complex_dim=2, **kwargs)#

Bases: GraphLifting

Abstract class for lifting graphs to simplicial complexes.

Parameters:

complex_dimint, optional: The maximum dimension of the simplicial complex to be generated. Default is 2.
**kwargsoptional: Additional arguments for the class.

__init__(complex_dim=2, **kwargs)#

class SimplicialComplex(simplices=None, **kwargs)#

Bases: Complex, Generic[ElementType]

Class representing a simplicial complex.

Class for construction boundary operators, Hodge Laplacians, higher order (co)adjacency operators from a collection of simplices.

A simplicial complex is a topological space of a specific kind, constructed by “gluing together” points, line segments, triangles, and their higher-dimensional counterparts. It is a generalization of the notion of a triangle in a triangulated surface, or a tetrahedron in a tetrahedralized 3-dimensional manifold. Simplicial complexes are the basic objects of study in combinatorial topology.

For example, a triangle is a simplicial complex because it is a collection of three points that are connected to each other in a specific way. Similarly, a tetrahedron is a simplicial complex because it is a collection of four points that are connected to each other in a specific way. These simplices can be thought of as the “building blocks” of a simplicial complex, and the complex itself is constructed by combining these building blocks in a specific way. For example, a 2-dimensional simplicial complex could be a collection of triangles that are connected to each other to form a surface, while a 3-dimensional simplicial complex could be a collection of tetrahedra that are connected to each other to form a solid object.

The SimplicialComplex class is a class for representing simplicial complexes, which are a type of topological space constructed by “gluing together” points, line segments, triangles, and higher-dimensional counterparts. The class provides methods for computing boundary operators, Hodge Laplacians, and higher-order adjacency operators on the simplicial complex. It also allows for compatibility with NetworkX and the GUDHI library.

Features:

The SimplicialComplex class allows for the dynamic construction of simplicial complexes,
enabling users to add or remove simplices from the complex after its initial creation.
The class provides methods for computing boundary operators, Hodge Laplacians,
and higher-order adjacency operators on the simplicial complex.
The class is compatible with the gudhi library, allowing users to leverage the powerful
algorithms and data structures provided by this package.
The class supports the attachment of arbitrary attributes and data to simplices,
enabling users to store and manipulate additional information about these objects.
The class has robust error handling and input validation, ensuring reliable and easy use of the class.

Parameters:

simplicesiterable, optional: Iterable of maximal simplices that define the simplicial complex.
**kwargskeyword arguments, optional: Attributes to add to the complex as key=value pairs.

Attributes:

complexdict: A dictionary that can be used to store additional information about the complex.

Notes

A simplicial complex is determined by its maximal simplices, simplices that are not contained in any other simplices. If a maximal simplex is inserted, all faces of this simplex will be inserted automatically.

Examples

Define a simplicial complex using a set of maximal simplices:

>>> SC = tnx.SimplicialComplex([[1, 2, 3], [2, 3, 5], [0, 1]])

TopoNetX is also compatible with NetworkX, allowing users to create a simplicial complex from a NetworkX graph. Existing node and edge attributes are copied to the simplicial complex:

>>> G = nx.Graph()
>>> G.add_edge(0, 1, weight=4)
>>> G.add_edges_from([(0, 3), (0, 4), (1, 4)])
>>> SC = tnx.SimplicialComplex(simplices=G)
>>> SC.add_simplex([1, 2, 3])
>>> SC.simplices
SimplexView([(0,), (1,), (3,), (4,), (2,), (0, 1), (0, 3), (0, 4), (1, 4), (1, 2), (1, 3), (2, 3), (1, 2, 3)])
>>> SC[(0, 1)]["weight"]
4

__init__(simplices=None, **kwargs)#

Initialize a new instance of the Complex class.

Parameters:

**kwargskeyword arguments, optional: Attributes to add to the complex as key=value pairs.

add_elements_from_nx_graph(G)#

Add elements from a networkx graph to this simplicial complex.

Parameters:

Gnetworkx.Graph: A networkx graph instance.

add_node(node, **kwargs)#

Add node to simplicial complex.

Parameters:

nodeHashable or Simplex: The node to be added to the simplicial complex.
**kwargskeyword arguments, optional: Additional attributes to be associated with the node.

add_simplex(simplex, **kwargs)#

Add simplex to simplicial complex.

In case sub-simplices are missing, they are added without attributes to the simplicial complex to fulfill the simplex requirements.

If the given simplex is already part of the simplicial complex, its attributes will be updated by the provided ones.

Parameters:

simplexCollection

The simplex to be added to the simplicial complex.

If a Simplex object is given, its attributes will be copied to the simplicial complex. kwargs take precedence over the attributes of the Simplex object.

**kwargskeyword arguments, optional

Additional attributes to be associated with the simplex.

add_simplices_from(simplices)#

Add simplices from iterable to simplicial complex.

Parameters:

simplicesiterable: Iterable of simplices to be added to the simplicial complex.

adjacency_matrix(rank, signed=False, weight=None, index=False)#

Compute the adjacency matrix of the simplicial complex.

Parameters:

rankint: Rank of the adjacency matrix.
signedbool: Whether to return the signed or unsigned adjacency matrix.
weightstr, optional: The name of the simplex attribute to use as weights for the hodge laplacian matrix. If None, the matrix is binary.
indexbool, default=False: Whether to return the indices of the rows and columns of the adjacency matrix.

Returns:

indicesdict: Dictionary mapping each row and column of the coadjacency matrix to a simplex. Only returned if index is True.
adjacency_matrixscipy.sparse.csr.csr_matrix: The adjacency matrix of rank rank of this simplicial complex.

Examples

>>> SC = tnx.SimplicialComplex()
>>> SC.add_simplex([1, 2, 3, 4])
>>> SC.add_simplex([1, 2, 4])
>>> SC.add_simplex([3, 4, 8])
>>> adj1 = SC.adjacency_matrix(1)

clone()#

Return a copy of the simplicial complex.

The clone method by default returns an independent shallow copy of the simplicial complex. Use Python’s copy.deepcopy for new containers.

Returns:

SimplicialComplex: A shallow copy of this simplicial complex.

coadjacency_matrix(rank, signed=False, weight=None, index=False)#

Compute the coadjacency matrix of the simplicial complex.

Parameters:

rankint: Rank of the coadjacency matrix.
signedbool: Whether to return the signed or unsigned coadjacency matrix.
weightstr, optional: The name of the simplex attribute to use as weights for the hodge laplacian matrix. If None, the matrix is binary.
indexbool, default=False: Whether to return the indices of the rows and columns of the coadjacency matrix.

Returns:

indiceslist: List identifying the rows and columns of the coadjacency matrix with the simplices of the simplicial complex. Only returned if index is True.
coadjacency_matrixscipy.sparse.csr.csr_matrix: The coadjacency matrix of this simplicial complex.

coincidence_matrix(rank, signed=True, weight=None, index=False)#

Compute coincidence matrix of the simplicial complex.

This is also called the coboundary matrix.

Parameters:

rankint: For which rank of simplices to compute the coincidence matrix.
signedbool, default=True: Whether to return the signed or unsigned coincidence matrix.
weightstr, optional: The name of the simplex attribute to use as weights for the coincidence matrix. If None, the matrix is binary.
indexbool, default=False: Whether to return the indices of the rows and columns of the coincidence matrix.

Returns:

row_indices, col_indicesdict: Dictionary assigning each row and column of the coincidence matrix to a simplex. Only returned if index is True.
coincidence_matrixscipy.sparse.csr.csr_matrix: The coincidence matrix.

dirac_operator_matrix(signed=True, weight=None, index=False)#

Compute dirac operator matrix matrix.

Parameters:

signedbool, default=False: Whether the returned dirac matrix should be signed (i.e., respect orientations) or unsigned.
weightstr, optional: The name of the simplex attribute to use as weights for the dirac operator matrix. If None, the matrix is binary.
indexbool, default=False: Whether to return the indices of the rows and columns of the dirac operator matrix.

Returns:

row_indices, col_indicesdict: List identifying rows and columns of the dirac operator matrix. Only returned if index is True.
scipy.sparse.csr.csc_matrix: The dirac operator matrix of this simplicial complex.

Examples

>>> SC = tnx.SimplicialComplex()
>>> SC.add_simplex([1, 2, 3, 4])
>>> SC.add_simplex([1, 2, 4])
>>> SC.add_simplex([3, 4, 8])
>>> SC.dirac_operator_matrix()

down_laplacian_matrix(rank, signed=True, weight=None, index=False)#

Compute the down Laplacian matrix of the simplicial complex.

Parameters:

rankint: Rank of the down Laplacian matrix.
signedbool: Whether to return the signed or unsigned down laplacian.
weightstr, optional: The name of the simplex attribute to use as weights for the hodge laplacian matrix. If None, the matrix is binary.
indexbool, default=False: Whether to return the indices of the rows and columns of the laplacian.

Returns:

indicesdict: Dictionary assigning each row and column of the laplacian matrix to a simplex. Only returned if index is True.
down_laplacianscipy.sparse.csr.csr_matrix: The down laplacian matrix of this simplicial complex.

classmethod from_gudhi(tree)#

Import from gudhi.

Parameters:

treegudhi.SimplexTree: The input gudhi simplex tree.

Returns:

SimplicialComplex: The resulting SimplicialComplex.

Examples

>>> from gudhi import SimplexTree
>>> tree = SimplexTree()
>>> _ = tree.insert([1, 2, 3, 5])
>>> SC = tnx.SimplicialComplex.from_gudhi(tree)

classmethod from_nx(G)#

Convert from netwrokx graph.

Parameters:

Gnx.Graph: The graph to convert to a simplicial complex.

Returns:

SimplicialComplex: The simplicial complex.

Examples

>>> G = nx.Graph()  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_edge(1, 2, weight=2)
>>> G.add_edge(3, 4, weight=4)
>>> SC = tnx.SimplicialComplex.from_nx(G)
>>> SC[(1, 2)]["weight"]
2

classmethod from_spharapy(mesh)#

Import from sharpy.

Parameters:

meshspharapy.trimesh.TriMesh: The input spharapy object.

Returns:

SimplicialComplex: The resulting SimplicialComplex.

Examples

>>> import spharapy.trimesh as tm
>>> import spharapy.spharabasis as sb
>>> import spharapy.datasets as sd
>>> mesh = tm.TriMesh(
...     [[0, 1, 2]], [[1.0, 0.0, 0.0], [0.0, 2.0, 0.0], [0.0, 0.0, 3.0]]
... )

>>> SC = tnx.SimplicialComplex.from_spharapy(mesh)

classmethod from_spharpy(mesh)#

Import from sharpy.

Parameters:

meshspharapy.trimesh.TriMesh: The input spharapy object.

Returns:

SimplicialComplex: The resulting SimplicialComplex.

classmethod from_trimesh(mesh)#

Import from trimesh.

Parameters:

meshtrimesh.Trimesh: The input trimesh object.

Returns:

SimplicialComplex: The resulting SimplicialComplex.

Examples

>>> import trimesh
>>> mesh = trimesh.Trimesh(
...     vertices=[[0, 0, 0], [0, 0, 1], [0, 1, 0]],
...     faces=[[0, 1, 2]],
...     process=False,
... )
>>> SC = tnx.SimplicialComplex.from_trimesh(mesh)
>>> print(SC.nodes)
>>> print(SC.simplices)
>>> SC[(0)]["position"]

get_all_maximal_simplices()#

Get all maximal simplices of this simplicial complex.

A simplex is maximal if it is not a face of any other simplex in the complex.

Returns:

list of tuples: List of maximal simplices in this simplicial complex.

Examples

>>> SC = tnx.SimplicialComplex([(1, 2), (1, 2, 3), (1, 2, 4), (2, 5)])
>>> SC.get_all_maximal_simplices()
[(2, 5), (1, 2, 3), (1, 2, 4)]

static get_boundaries(simplices, min_dim=None, max_dim=None)#

Get boundaries of the given simplices.

Parameters:

simplicesIterable: Iterable of simplices for which to compute the boundaries.
min_dimint, optional: Constrain the max dimension of faces.
max_dimint, optional: Constrain the max dimension of faces.

Returns:

set of frozensets: Set of simplices that are boundaries of the given simplices. If min_dim or max_dim are given, only simplices with dimension in the given range are returned.

get_cofaces(simplex, codimension)#

Get cofaces of simplex.

Parameters:

simplexIterable of AtomType: The simplex for which to compute the cofaces.
codimensionint: The codimension of the returned cofaces. If zero, all cofaces are returned.

Returns:

list of tuples: The cofaces of the given simplex.

static get_edges_from_matrix(matrix)#

Get edges from matrix.

Parameters:

matrixnumpy or scipy array: The matrix to get the edges from.

Returns:

list of int: List of indices where the operator is not zero.

Notes

Most operaters (e.g. adjacencies/(co)boundary maps) that describe connectivity of the simplicial complex can be described as a G whose nodes are the simplices used to construct the operator and whose edges correspond to the entries in the matrix where the operator is not zero.

This property implies that many computations on simplicial complexes can be reduced to G computations.

get_maximal_simplices_of_simplex(simplex)#

Get maximal simplices of simplex.

Parameters:

simplexIterable: The simplex for which to compute the maximal simplices.

Returns:

set of Simplex: Set of maximal simplices of the given simplex.

get_node_attributes(name)#

Get node attributes from simplicial complex.

Parameters:

namestr: Attribute name.

Returns:

dict: Dictionary mapping each node to the value of the given attribute name.

Examples

>>> SC = tnx.SimplicialComplex()
>>> SC.add_simplex([1, 2, 3, 4])
>>> SC.add_simplex([1, 2, 4])
>>> SC.add_simplex([3, 4, 8])
>>> SC.set_simplex_attributes({1: "red", 2: "blue", 3: "black"}, name="color")
>>> SC.get_node_attributes("color")
{1: 'red', 2: 'blue', 3: 'black'}

get_simplex_attributes(name, rank=None)#

Get node attributes from simplical complex.

Parameters:

namestr: Attribute name.
rankint, optional: Restrict the returned attribute values to simplices of a specific rank.

Returns:

dict[tuple[Hashable, …], Any]: Dictionary mapping each simplex to the value of the given attribute name.

Examples

>>> SC = tnx.SimplicialComplex()
>>> SC.add_simplex([1, 2, 3, 4])
>>> SC.add_simplex([1, 2, 4])
>>> SC.add_simplex([3, 4, 8])
>>> d = {(1, 2): "red", (2, 3): "blue", (3, 4): "black"}
>>> SC.set_simplex_attributes(d, name="color")
>>> SC.get_simplex_attributes("color")
{(1, 2): 'red', (2, 3): 'blue', (3, 4): 'black'}

get_star(simplex)#

Get star.

Parameters:

simplexIterable of AtomType: The simplex for which to compute the star.

Returns:

list of tuples: The star of the given simplex.

Notes

This function is equivalent to calling get_cofaces(simplex, 0).

graph_skeleton()#

Return the graph-skeleton of this simplicial complex.

The graph-skeleton consists of the 0 and 1-simplices (i.e., nodes and edges) of the simplicial complex.

Returns:

nx.Graph: The graph-skeleton of this simplicial complex.

hodge_laplacian_matrix(rank, signed=True, weight=None, index=False)#

Compute hodge-laplacian matrix for the simplicial complex.

Parameters:

rankint: Dimension of the Laplacian matrix.
signedbool, default=True: Whether to return the signed or unsigned hodge laplacian.
weightstr, optional: The name of the simplex attribute to use as weights for the hodge laplacian matrix. If None, the matrix is binary.
indexbool, default=False: Indicates whether to return the indices that define the hodge laplacian matrix.

Returns:

indexlist: List assigning each row and column of the laplacian matrix to a simplex. Only available when index is True.
laplacianscipy.sparse.csr.csr_matrix: The hodge laplacian matrix of rank rank.

Examples

>>> SC = tnx.SimplicialComplex()
>>> SC.add_simplex([1, 2, 3, 4])
>>> SC.add_simplex([1, 2, 4])
>>> SC.add_simplex([3, 4, 8])
>>> L1 = SC.hodge_laplacian_matrix(1)

incidence_matrix(rank, signed=True, weight=None, index=False)#

Compute incidence matrix of the simplicial complex.

Getting the matrix that correpodnds to the boundary matrix of the input SC.

Parameters:

rankint: For which rank of simplices to compute the incidence matrix.
signedbool, default=True: Whether to return the signed or unsigned incidence matrix.
weightstr, optional: The name of the simplex attribute to use as weights for the incidence matrix. If None, the matrix is binary.
indexbool, default=False: Whether to return the indices of the rows and columns of the incidence matrix.

Returns:

row_indices, col_indicesdict: Dictionary assigning each row and column of the incidence matrix to a simplex. Only returned if index is True.
incidence_matrixscipy.sparse.csr.csr_matrix: The incidence matrix.

Raises:

ValueError: If rank is negative or larger than the dimension of the simplicial complex.

Examples

>>> SC = tnx.SimplicialComplex()
>>> SC.add_simplex([1, 2, 3, 4])
>>> SC.add_simplex([1, 2, 4])
>>> SC.add_simplex([3, 4, 8])
>>> B1 = SC.incidence_matrix(1)
>>> B2 = SC.incidence_matrix(2)

is_connected()#

Check if the simplicial complex is connected.

Returns:

bool: True if the simplicial complex is connected, False otherwise.

Notes

A simplicial complex is connected iff its 1-skeleton is connected.

is_maximal(simplex)#

Check if simplex is maximal, i.e., not a face of any other simplex in the complex.

Parameters:

simplexIterable: The Simplex to check.

Returns:

bool: True if the given simplex is maximal in this simplicial complex, False otherwise.

Raises:

ValueError: If simplex is not in the simplicial complex.

Examples

>>> SC = tnx.SimplicialComplex([[1, 2, 3]])
>>> SC.is_maximal([1, 2, 3])
True
>>> SC.is_maximal([1, 2])
False

is_triangular_mesh()#

Check if the simplicial complex is a triangular mesh.

Returns:

bool: True if the simplicial complex can be converted to a triangular mesh, False otherwise.

laplace_beltrami_operator(mode='inv_euclidean')#

Compute a laplacian matrix for a triangular mesh.

The method creates a laplacian matrix for a triangular mesh using different weighting function.

Parameters:

mode{‘unit’, ‘inv_euclidean’, ‘half_cotangent’}, optional: The methods for determining the edge weights. Using the option ‘unit’ all edges of the mesh are weighted by unit weighting function, the result is an adjacency matrix. The option ‘inv_euclidean’ results in edge weights corresponding to the inverse Euclidean distance of the edge lengths. The option ‘half_cotangent’ uses the half of the cotangent of the two angles opposed to an edge as weighting function. the default weighting function is ‘inv_euclidean’.

Returns:

numpy.ndarray, shape (n_vertices, n_vertices): Matrix, which contains the discrete laplace operator for data defined at the vertices of a triangular mesh. The number of vertices of the triangular mesh is n_points.

classmethod load_mesh(file_path, process=False)#

Load a mesh.

Parameters:

file_pathstr or pathlib.Path: The source of the data to be loaded.
processbool: Whether trimesh should try to process the mesh before loading it.

Returns:

SimplicialComplex: The output simplicial complex stores the same structure stored in the mesh input file.

Notes

mesh files supported : obj, off, glb

Examples

>>> SC = tnx.SimplicialComplex.load_mesh("C:/temp/stanford-bunny.obj")
>>> SC.nodes

normalized_laplacian_matrix(rank, weight=None)#

Return the normalized hodge Laplacian matrix of simplicial complex .

The normalized hodge Laplacian is the matrix

\[N_d = D^{-1/2} L_d D^{-1/2}\]

where L is the simplicial complex Laplacian and D is the diagonal matrix of simplices of rank d.

Parameters:

rankint: Rank of the hodge laplacian matrix.
weightstr, optional: The name of the simplex attribute to use as weights for the hodge laplacian matrix. If None, the matrix is binary.

Returns:

Scipy sparse matrix: The normalized hodge Laplacian matrix.

Examples

>>> SC = tnx.SimplicialComplex([[1, 2, 3], [2, 3, 5], [0, 1]])
>>> SC.normalized_laplacian_matrix(1)

remove_maximal_simplex(simplex)#

Remove maximal simplex from simplicial complex.

Parameters:

simplexCollection: The simplex to be removed from the simplicial complex.

Raises:

KeyError: If simplex is not in simplicial complex.
ValueError: If simplex is not maximal.

Examples

>>> SC = tnx.SimplicialComplex()
>>> SC.add_simplex((1, 2, 3, 4), weight=1)
>>> SC.add_simplex((1, 2, 3, 4, 5))
>>> SC.remove_maximal_simplex((1, 2, 3, 4, 5))

remove_nodes(node_set)#

Remove the given nodes from the simplicial complex.

Any simplices that become invalid due to the removal of nodes are also removed.

Parameters:

node_setIterable: The nodes to be removed from the simplicial complex.

Examples

>>> SC = tnx.SimplicialComplex([(1, 2), (2, 3), (3, 4)])
>>> SC.remove_nodes([1])
>>> SC.simplices
SimplexView([(2,), (3,), (4,), (2, 3), (3, 4)])

restrict_to_nodes(node_set)#

Construct a new simplicial complex by restricting the simplices.

The simplices are restricted to the nodes referenced by node_set.

Parameters:

node_setiterable of hashables: A subset of nodes of the simplicial complex to restrict to.

Returns:

SimplicialComplex: A new simplicial complex restricted to the nodes in node_set.

Examples

>>> SC = tnx.SimplicialComplex([(1, 2, 3), (1, 2, 4), (1, 2, 5)])
>>> new_complex = SC.restrict_to_nodes([1, 2, 3, 4])
>>> new_complex.simplices
SimplexView([(1,), (2,), (3,), (4,), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (1, 2, 3), (1, 2, 4)])

restrict_to_simplices(cell_set)#

Construct a simplicial complex using a subset of the simplices.

Parameters:

cell_setiterable of hashables or simplices: A subset of elements of the simplicial complex that should be in the new simplicial complex.

Returns:

SimplicialComplex: New simplicial complex restricted to the simplices in cell_set.

Examples

>>> c1 = tnx.Simplex((1, 2, 3))
>>> c2 = tnx.Simplex((1, 2, 4))
>>> c3 = tnx.Simplex((1, 2, 5))
>>> SC = tnx.SimplicialComplex([c1, c2, c3])
>>> SC1 = SC.restrict_to_simplices([c1, (2, 4)])
>>> SC1.simplices
SimplexView([(1,), (2,), (3,), (4,), (1, 2), (1, 3), (2, 3), (2, 4), (1, 2, 3)])

set_simplex_attributes(values, name=None)#

Set simplex attributes.

Parameters:

valuesdict: Either provide a mapping from simplices to values or a dict of dicts. In the former case, the attribute name for each simplex is set to the corresponding value. In the latter case, the entire dictionary is used to update the attributes of the simplices.
namestr, optional: The name of the attribute to set.

Notes

If the dict contains simplices that are not in self.simplices, they are silently ignored.

Examples

After computing some property of the simplex of a simplicial complex, you may want to assign a simplex attribute to store the value of that property for each simplex:

>>> SC = tnx.SimplicialComplex()
>>> SC.add_simplex([1, 2, 3, 4])
>>> SC.add_simplex([1, 2, 4])
>>> SC.add_simplex([3, 4, 8])
>>> d = {(1, 2, 3): "red", (1, 2, 4): "blue"}
>>> SC.set_simplex_attributes(d, name="color")
>>> SC[(1, 2, 3)]["color"]
'red'

If you provide a dictionary of dictionaries as the second argument, the entire dictionary will be used to update simplex attributes:

>>> SC = tnx.SimplicialComplex()
>>> SC.add_simplex([1, 3, 4])
>>> SC.add_simplex([1, 2, 3])
>>> SC.add_simplex([1, 2, 4])
>>> d = {
...     (1, 3, 4): {"color": "red", "attr2": 1},
...     (1, 2, 4): {"color": "blue", "attr2": 3},
... }
>>> SC.set_simplex_attributes(d)
>>> SC[(1, 3, 4)]["color"]
'red'

classmethod simplicial_closure_of_hypergraph(H)#

Compute the simplicial complex closure of a hypergraph.

Parameters:

Hhyernetx hypergraph: The hypergraph to compute the simplicial complex closure of.

Returns:

SimplicialComplex: Simplicial complex closure of the hypergraph.

Examples

>>> import hypernetx as hnx
>>> hg = hnx.Hypergraph([[1, 2, 3, 4], [1, 2, 3]], static=True)
>>> sc = tnx.SimplicialComplex.simplicial_closure_of_hypergraph(hg)
>>> sc.simplices
SimplexView([(1,), (2,), (3,), (4,), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4), (1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4), (1, 2, 3, 4)])

skeleton(rank)#

Compute the rank-skeleton of the simplicial complex containing simplices of given rank.

Parameters:

rankint: The rank of the skeleton to compute.

Returns:

list of Simplex: Simplices of rank rank in the simplicial complex.

to_cell_complex()#

Convert a simplicial complex to a cell complex.

Returns:

toponetx.classes.CellComplex: The cell complex corresponding to this simplicial complex.

Examples

>>> c1 = tnx.Simplex((1, 2, 3))
>>> c2 = tnx.Simplex((1, 2, 4))
>>> c3 = tnx.Simplex((2, 5))
>>> SC = tnx.SimplicialComplex([c1, c2, c3])
>>> SC.to_cell_complex()

to_combinatorial_complex()#

Convert a simplicial complex to a combinatorial complex.

Returns:

toponetx.classes.CombinatorialComplex: The combinatorial complex equivalent to this simplicial complex.

Examples

>>> c1 = tnx.Simplex((1, 2, 3))
>>> c2 = tnx.Simplex((1, 2, 3))
>>> c3 = tnx.Simplex((1, 2, 4))
>>> SC = tnx.SimplicialComplex([c1, c2, c3])
>>> CCC = SC.to_combinatorial_complex()

to_hasse_graph()#

Create the hasse graph corresponding to this simplicial complex.

Returns:

nx.DiGraph: A NetworkX Digraph representing the Hasse graph of the Simplicial Complex.

Examples

>>> SC = tnx.SimplicialComplex()
>>> SC.add_simplex([0, 1, 2])
>>> G = SC.to_hasse_graph()

to_hypergraph()#

Convert a simplicial complex to a hypernetx hypergraph.

Returns:

Hypergraph: The hypergraph corresponding to this simplicial complex.

Raises:

RuntimeError: If hypernetx is not installed

Examples

>>> c1 = tnx.Simplex((1, 2, 3))
>>> c2 = tnx.Simplex((1, 2, 4))
>>> c3 = tnx.Simplex((2, 5))
>>> SC = tnx.SimplicialComplex([c1, c2, c3])
>>> SC.to_hypergraph()
Hypergraph({'e0': [1, 2], 'e1': [1, 3], 'e2': [1, 4], 'e3': [2, 3], 'e4': [2, 4], 'e5': [2, 5], 'e6': [1, 2, 3], 'e7': [1, 2, 4]},name=)

to_spharapy(vertex_position_name='position')#

Convert to spharapy.

Parameters:

vertex_position_namestr, default=”position”: The simplex attribute name that contains the vertex positions.

Returns:

spharapy.trimesh.TriMesh: The spharapy object corresponding to this simplicial complex.

Raises:

RuntimeError: If package spharapy is not installed.

Examples

>>> import spharapy.trimesh as tm
>>> import spharapy.spharabasis as sb
>>> import spharapy.datasets as sd
>>> mesh = tm.TriMesh([[0, 1, 2]], [[0, 0, 0], [0, 0, 1], [0, 1, 0]])
>>> SC = tnx.SimplicialComplex.from_spharapy(mesh)
>>> mesh2 = SC.to_spharapy()
>>> mesh2.vertlist == mesh.vertlist
>>> mesh2.trilist == mesh.trilist

to_trimesh(vertex_position_name='position')#

Convert simplicial complex to trimesh object.

Parameters:

vertex_position_namestr, default=”position”: The simplex attribute name that contains the vertex positions.

Returns:

trimesh.Trimesh: The trimesh object corresponding to this simplicial complex.

up_laplacian_matrix(rank, signed=True, weight=None, index=False)#

Compute the up Laplacian matrix of the simplicial complex.

Parameters:

rankint: Rank of the up Laplacian matrix.
signedbool: Whether to return the signed or unsigned up laplacian.
weightstr, optional: The name of the simplex attribute to use as weights for the hodge laplacian matrix. If None, the matrix is binary.
indexbool, default=False: Whether to return the indices of the rows and columns of the laplacian.

Returns:

row_indices, col_indiceslist: List identifying the rows and columns of the laplacian matrix with the simplices of this complex. Only returned if index is True.
up_laplacianscipy.sparse.csr.csr_matrix: The upper laplacian matrix of this simplicial complex.

Examples

>>> SC = tnx.SimplicialComplex([[1, 2, 3], [2, 3, 5], [0, 1]])
>>> SC.up_laplacian_matrix(1)

property dim: int#

Maximal dimension of the simplices in this complex.

Returns:

int: The dimension of the simplicial complex.

Examples

>>> SC = SimplicialComplex([[0, 1], [1, 2, 3], [2, 3, 4]])
>>> SC.dim
2

property maxdim: int#

Maximum dimension of the simplicial complex.

This is the highest dimension of any simplex in the complex.

Returns:

int: The maximum dimension of the simplicial complex.

property nodes: NodeView#

Return the list of nodes in the simplicial complex.

Returns:

NodeView: A NodeView object representing the nodes of the simplicial complex.

property shape: tuple[int, ...]#

Shape of simplicial complex, i.e., the number of simplices for each dimension.

Returns:

tuple of ints: This gives the number of cells in each rank.

Examples

>>> SC = SimplicialComplex([[0, 1], [1, 2, 3], [2, 3, 4]])
>>> SC.shape
(5, 5, 2)

property simplices: SimplexView#

Set of all simplices in the simplicial complex.

Returns:

SimplexView: A SimplexView object representing the set of all simplices in the simplicial complex.

class SimplicialVietorisRipsLifting(distance_threshold=1.0, max_simplices=5, **kwargs)#

Bases: Graph2SimplicialLifting

Lift graphs to simplicial complex domain using the Vietoris-Rips complex based on pairwise distances.

Parameters:

distance_thresholdfloat: The maximum distance between vertices to form a simplex.
max_simplicesint: Max number of simplices to create for a given node and rank.
**kwargsoptional: Additional arguments for the class.

__init__(distance_threshold=1.0, max_simplices=5, **kwargs)#

lift_topology(data)#

Lift topology of a graph to a simplicial complex using the Vietoris-Rips complex.

Parameters:

datatorch_geometric.data.Data: The input data to be lifted.

Returns:

dict: The lifted topology.

class tqdm(*_, **__)#

Bases: Comparable

Decorate an iterable object, returning an iterator which acts exactly like the original iterable, but prints a dynamically updating progressbar every time a value is requested.

Parameters:

iterableiterable, optional

Iterable to decorate with a progressbar. Leave blank to manually manage the updates.

descstr, optional

Prefix for the progressbar.

totalint or float, optional

The number of expected iterations. If unspecified, len(iterable) is used if possible. If float(“inf”) or as a last resort, only basic progress statistics are displayed (no ETA, no progressbar). If gui is True and this parameter needs subsequent updating, specify an initial arbitrary large positive number, e.g. 9e9.

leavebool, optional

If [default: True], keeps all traces of the progressbar upon termination of iteration. If None, will leave only if position is 0.

fileio.TextIOWrapper or io.StringIO, optional

Specifies where to output the progress messages (default: sys.stderr). Uses file.write(str) and file.flush() methods. For encoding, see write_bytes.

ncolsint, optional

The width of the entire output message. If specified, dynamically resizes the progressbar to stay within this bound. If unspecified, attempts to use environment width. The fallback is a meter width of 10 and no limit for the counter and statistics. If 0, will not print any meter (only stats).

minintervalfloat, optional

Minimum progress display update interval [default: 0.1] seconds.

maxintervalfloat, optional

Maximum progress display update interval [default: 10] seconds. Automatically adjusts miniters to correspond to mininterval after long display update lag. Only works if dynamic_miniters or monitor thread is enabled.

minitersint or float, optional

Minimum progress display update interval, in iterations. If 0 and dynamic_miniters, will automatically adjust to equal mininterval (more CPU efficient, good for tight loops). If > 0, will skip display of specified number of iterations. Tweak this and mininterval to get very efficient loops. If your progress is erratic with both fast and slow iterations (network, skipping items, etc) you should set miniters=1.

asciibool or str, optional

If unspecified or False, use unicode (smooth blocks) to fill the meter. The fallback is to use ASCII characters “ 123456789#”.

disablebool, optional

Whether to disable the entire progressbar wrapper [default: False]. If set to None, disable on non-TTY.

unitstr, optional

String that will be used to define the unit of each iteration [default: it].

unit_scalebool or int or float, optional

If 1 or True, the number of iterations will be reduced/scaled automatically and a metric prefix following the International System of Units standard will be added (kilo, mega, etc.) [default: False]. If any other non-zero number, will scale total and n.

dynamic_ncolsbool, optional

If set, constantly alters ncols and nrows to the environment (allowing for window resizes) [default: False].

smoothingfloat, optional

Exponential moving average smoothing factor for speed estimates (ignored in GUI mode). Ranges from 0 (average speed) to 1 (current/instantaneous speed) [default: 0.3].

bar_formatstr, optional

Specify a custom bar string formatting. May impact performance. [default: ‘{l_bar}{bar}{r_bar}’], where l_bar=’{desc}: {percentage:3.0f}%|’ and r_bar=’| {n_fmt}/{total_fmt} [{elapsed}<{remaining}, ‘

‘{rate_fmt}{postfix}]’

Possible vars: l_bar, bar, r_bar, n, n_fmt, total, total_fmt,: percentage, elapsed, elapsed_s, ncols, nrows, desc, unit, rate, rate_fmt, rate_noinv, rate_noinv_fmt, rate_inv, rate_inv_fmt, postfix, unit_divisor, remaining, remaining_s, eta.

Note that a trailing “: “ is automatically removed after {desc} if the latter is empty.

initialint or float, optional

The initial counter value. Useful when restarting a progress bar [default: 0]. If using float, consider specifying {n:.3f} or similar in bar_format, or specifying unit_scale.

positionint, optional

Specify the line offset to print this bar (starting from 0) Automatic if unspecified. Useful to manage multiple bars at once (eg, from threads).

postfixdict or *, optional

Specify additional stats to display at the end of the bar. Calls set_postfix(**postfix) if possible (dict).

unit_divisorfloat, optional

[default: 1000], ignored unless unit_scale is True.

write_bytesbool, optional

Whether to write bytes. If (default: False) will write unicode.

lock_argstuple, optional

Passed to refresh for intermediate output (initialisation, iterating, and updating).

nrowsint, optional

The screen height. If specified, hides nested bars outside this bound. If unspecified, attempts to use environment height. The fallback is 20.

colourstr, optional

Bar colour (e.g. ‘green’, ‘#00ff00’).

delayfloat, optional

Don’t display until [default: 0] seconds have elapsed.

guibool, optional

WARNING: internal parameter - do not use. Use tqdm.gui.tqdm(…) instead. If set, will attempt to use matplotlib animations for a graphical output [default: False].

Returns:

outdecorated iterator.

__init__(iterable=None, desc=None, total=None, leave=True, file=None, ncols=None, mininterval=0.1, maxinterval=10.0, miniters=None, ascii=None, disable=False, unit='it', unit_scale=False, dynamic_ncols=False, smoothing=0.3, bar_format=None, initial=0, position=None, postfix=None, unit_divisor=1000, write_bytes=False, lock_args=None, nrows=None, colour=None, delay=0.0, gui=False, **kwargs)#

clear(nolock=False)#: Clear current bar display.

close()#: Cleanup and (if leave=False) close the progressbar.

display(msg=None, pos=None)#

Use self.sp to display msg in the specified pos.

Consider overloading this function when inheriting to use e.g.: self.some_frontend(**self.format_dict) instead of self.sp.

Parameters:

msgstr, optional. What to display (default: repr(self)).
posint, optional. Position to moveto: (default: abs(self.pos)).

classmethod external_write_mode(file=None, nolock=False)#: Disable tqdm within context and refresh tqdm when exits. Useful when writing to standard output stream

static format_interval(t)#

Formats a number of seconds as a clock time, [H:]MM:SS

Parameters:

tint: Number of seconds.

Returns:

outstr: [H:]MM:SS

static format_meter(n, total, elapsed, ncols=None, prefix='', ascii=False, unit='it', unit_scale=False, rate=None, bar_format=None, postfix=None, unit_divisor=1000, initial=0, colour=None, **extra_kwargs)#

Return a string-based progress bar given some parameters

Parameters:

nint or float

Number of finished iterations.

totalint or float

The expected total number of iterations. If meaningless (None), only basic progress statistics are displayed (no ETA).

elapsedfloat

Number of seconds passed since start.

ncolsint, optional

The width of the entire output message. If specified, dynamically resizes {bar} to stay within this bound [default: None]. If 0, will not print any bar (only stats). The fallback is {bar:10}.

prefixstr, optional

Prefix message (included in total width) [default: ‘’]. Use as {desc} in bar_format string.

asciibool, optional or str, optional

If not set, use unicode (smooth blocks) to fill the meter [default: False]. The fallback is to use ASCII characters “ 123456789#”.

unitstr, optional

The iteration unit [default: ‘it’].

unit_scalebool or int or float, optional

If 1 or True, the number of iterations will be printed with an appropriate SI metric prefix (k = 10^3, M = 10^6, etc.) [default: False]. If any other non-zero number, will scale total and n.

ratefloat, optional

Manual override for iteration rate. If [default: None], uses n/elapsed.

bar_formatstr, optional

‘{rate_fmt}{postfix}]’

Possible vars: l_bar, bar, r_bar, n, n_fmt, total, total_fmt,: percentage, elapsed, elapsed_s, ncols, nrows, desc, unit, rate, rate_fmt, rate_noinv, rate_noinv_fmt, rate_inv, rate_inv_fmt, postfix, unit_divisor, remaining, remaining_s, eta.

Note that a trailing “: “ is automatically removed after {desc} if the latter is empty.

postfix*, optional

Similar to prefix, but placed at the end (e.g. for additional stats). Note: postfix is usually a string (not a dict) for this method, and will if possible be set to postfix = ‘, ‘ + postfix. However other types are supported (#382).

unit_divisorfloat, optional

[default: 1000], ignored unless unit_scale is True.

initialint or float, optional

The initial counter value [default: 0].

colourstr, optional

Bar colour (e.g. ‘green’, ‘#00ff00’).

Returns:

outFormatted meter and stats, ready to display.

static format_num(n)#

Intelligent scientific notation (.3g).

Parameters:

nint or float or Numeric: A Number.

Returns:

outstr: Formatted number.

static format_sizeof(num, suffix='', divisor=1000)#

Formats a number (greater than unity) with SI Order of Magnitude prefixes.

Parameters:

numfloat: Number ( >= 1) to format.
suffixstr, optional: Post-postfix [default: ‘’].
divisorfloat, optional: Divisor between prefixes [default: 1000].

Returns:

outstr: Number with Order of Magnitude SI unit postfix.

classmethod get_lock()#: Get the global lock. Construct it if it does not exist.

moveto(n)#

classmethod pandas(**tqdm_kwargs)#

Registers the current tqdm class with: pandas.core. ( frame.DataFrame | series.Series | groupby.(generic.)DataFrameGroupBy | groupby.(generic.)SeriesGroupBy ).progress_apply

A new instance will be created every time progress_apply is called, and each instance will automatically close() upon completion.

Parameters:

tqdm_kwargsarguments for the tqdm instance

References

<https://stackoverflow.com/questions/18603270/ progress-indicator-during-pandas-operations-python>

Examples

>>> import pandas as pd
>>> import numpy as np
>>> from tqdm import tqdm
>>> from tqdm.gui import tqdm as tqdm_gui
>>>
>>> df = pd.DataFrame(np.random.randint(0, 100, (100000, 6)))
>>> tqdm.pandas(ncols=50)  # can use tqdm_gui, optional kwargs, etc
>>> # Now you can use `progress_apply` instead of `apply`
>>> df.groupby(0).progress_apply(lambda x: x**2)

refresh(nolock=False, lock_args=None)#

Force refresh the display of this bar.

Parameters:

nolockbool, optional: If True, does not lock. If [default: False]: calls acquire() on internal lock.
lock_argstuple, optional: Passed to internal lock’s acquire(). If specified, will only display() if acquire() returns True.

reset(total=None)#

Resets to 0 iterations for repeated use.

Consider combining with leave=True.

Parameters:

totalint or float, optional. Total to use for the new bar.

set_description(desc=None, refresh=True)#

Set/modify description of the progress bar.

Parameters:

descstr, optional
refreshbool, optional: Forces refresh [default: True].

set_description_str(desc=None, refresh=True)#: Set/modify description without ‘: ‘ appended.

classmethod set_lock(lock)#: Set the global lock.

set_postfix(ordered_dict=None, refresh=True, **kwargs)#

Set/modify postfix (additional stats) with automatic formatting based on datatype.

Parameters:

ordered_dictdict or OrderedDict, optional
refreshbool, optional: Forces refresh [default: True].
kwargsdict, optional

set_postfix_str(s='', refresh=True)#: Postfix without dictionary expansion, similar to prefix handling.

static status_printer(file)#: Manage the printing and in-place updating of a line of characters. Note that if the string is longer than a line, then in-place updating may not work (it will print a new line at each refresh).

unpause()#: Restart tqdm timer from last print time.

update(n=1)#

Manually update the progress bar, useful for streams such as reading files. E.g.: >>> t = tqdm(total=filesize) # Initialise >>> for current_buffer in stream: … … … t.update(len(current_buffer)) >>> t.close() The last line is highly recommended, but possibly not necessary if t.update() will be called in such a way that filesize will be exactly reached and printed.

Parameters:

nint or float, optional: Increment to add to the internal counter of iterations [default: 1]. If using float, consider specifying {n:.3f} or similar in bar_format, or specifying unit_scale.

Returns:

outbool or None: True if a display() was triggered.

classmethod wrapattr(stream, method, total=None, bytes=True, **tqdm_kwargs)#

stream : file-like object. method : str, “read” or “write”. The result of read() and

the first argument of write() should have a len().

>>> with tqdm.wrapattr(file_obj, "read", total=file_obj.size) as fobj:
...     while True:
...         chunk = fobj.read(chunk_size)
...         if not chunk:
...             break

classmethod write(s, file=None, end='\n', nolock=False)#: Print a message via tqdm (without overlap with bars).

property format_dict#: Public API for read-only member access.

monitor = None#

monitor_interval = 10#

nodes_search(nodes, max_num_nodes, current_idx, path_lengths, intersection, distance_threshold)#

Find nodes that all have distance less than distance_threshold.

Parameters:

nodeslist: Current list of nodes.
max_num_nodesint: Target number of nodes to return.
current_idxint: Index of the node from intersection to add. Can be None.
path_lengthsdict: Dictionary with path lengths between nodes in the graph.
intersectionlist: List of possible nodes to add.
distance_thresholdint: Max distance to consider for nodes.

Returns:

list: List of nodes where each pair of nodes had distance less or equal to the threshold. Returns None if there was no set of nodes satisfying the condition.