Topology

Functions related to graph topology.

Connectivity

sknetwork.topology.get_connected_components(input_matrix: scipy.sparse._csr.csr_matrix, connection: str = 'weak', force_bipartite: bool = False) numpy.ndarray[source]

Extract the connected components of a graph.

Parameters
  • input_matrix – Input matrix (either the adjacency matrix or the biadjacency matrix of the graph).

  • connection – Must be 'weak' (default) or 'strong'. The type of connection to use for directed graphs.

  • force_bipartite (bool) – If True, consider the input matrix as the biadjacency matrix of a bipartite graph.

Returns

Connected component of each node. For bipartite graphs, rows and columns are concatenated (rows first).

Return type

labels

Example

>>> from sknetwork.topology import get_connected_components
>>> from sknetwork.data import house
>>> get_connected_components(house())
array([0, 0, 0, 0, 0], dtype=int32)
sknetwork.topology.is_connected(input_matrix: scipy.sparse._csr.csr_matrix, connection: str = 'weak', force_bipartite: bool = False) bool[source]

Check whether the graph is connected.

Parameters
  • input_matrix – Input matrix (either the adjacency matrix or the biadjacency matrix of the graph).

  • connection – Must be 'weak' (default) or 'strong'. The type of connection to use for directed graphs.

  • force_bipartite (bool) – If True, consider the input matrix as the biadjacency matrix of a bipartite graph.

Example

>>> from sknetwork.topology import is_connected
>>> from sknetwork.data import house
>>> is_connected(house())
True
sknetwork.topology.get_largest_connected_component(input_matrix: scipy.sparse._csr.csr_matrix, connection: str = 'weak', force_bipartite: bool = False, return_index: bool = False) Union[scipy.sparse._csr.csr_matrix, Tuple[scipy.sparse._csr.csr_matrix, numpy.ndarray]][source]

Extract the largest connected component of a graph. Bipartite graphs are treated as undirected.

Parameters
  • input_matrix – Adjacency matrix or biadjacency matrix of the graph.

  • connection – Must be 'weak' (default) or 'strong'. The type of connection to use for directed graphs.

  • force_bipartite (bool) – If True, consider the input matrix as the biadjacency matrix of a bipartite graph.

  • return_index (bool) – Whether to return the index of the nodes of the largest connected component in the original graph.

Returns

  • output_matrix (sparse.csr_matrix) – Adjacency matrix or biadjacency matrix of the largest connected component.

  • index (array) – Indices of the nodes in the original graph. For bipartite graphs, rows and columns are concatenated (rows first).

Example

>>> from sknetwork.topology import get_largest_connected_component
>>> from sknetwork.data import house
>>> get_largest_connected_component(house()).shape
(5, 5)

Structure

sknetwork.topology.is_bipartite(adjacency: scipy.sparse._csr.csr_matrix, return_biadjacency: bool = False) Union[bool, Tuple[bool, Optional[scipy.sparse._csr.csr_matrix], Optional[numpy.ndarray], Optional[numpy.ndarray]]][source]

Check whether a graph is bipartite.

Parameters
  • adjacency – Adjacency matrix of the graph (symmetric).

  • return_biadjacency – If True, return a biadjacency matrix of the graph if bipartite.

Returns

  • is_bipartite (bool) – A boolean denoting if the graph is bipartite.

  • biadjacency (sparse.csr_matrix) – A biadjacency matrix of the graph if bipartite (optional).

  • rows (np.ndarray) – Index of rows in the original graph (optional).

  • cols (np.ndarray) – Index of columns in the original graph (optional).

Example

>>> from sknetwork.topology import is_bipartite
>>> from sknetwork.data import cyclic_graph
>>> is_bipartite(cyclic_graph(4))
True
>>> is_bipartite(cyclic_graph(3))
False
sknetwork.topology.is_acyclic(adjacency: scipy.sparse._csr.csr_matrix, directed: Optional[bool] = None) bool[source]

Check whether a graph has no cycle.

Parameters
  • adjacency – Adjacency matrix of the graph.

  • directed – Whether to consider the graph as directed (inferred if not specified).

Returns

is_acyclic – A boolean with value True if the graph has no cycle and False otherwise.

Return type

bool

Example

>>> from sknetwork.topology import is_acyclic
>>> from sknetwork.data import star, grid
>>> is_acyclic(star())
True
>>> is_acyclic(grid())
False

Core decomposition

class sknetwork.topology.get_core_decomposition

Get the k-core decomposition of a graph.

Parameters

adjacency – Adjacency matrix of the graph.

Returns

Core value of each node.

Return type

core_values

Example

>>> from sknetwork.data import karate_club
>>> adjacency = karate_club()
>>> core_values = get_core_decomposition(adjacency)
>>> len(core_values)
34

Triangles

class sknetwork.topology.count_triangles

Count the number of triangles in a graph.

Parameters
  • adjacency – Adjacency matrix of the graph.

  • parallelize – If True, use a parallel range while listing the triangles.

Returns

n_triangles – Number of triangles.

Return type

int

Example

>>> from sknetwork.data import karate_club
>>> adjacency = karate_club()
>>> count_triangles(adjacency)
45
class sknetwork.topology.get_clustering_coefficient

Get the clustering coefficient of a graph.

Parameters
  • adjacency – Adjacency matrix of the graph.

  • parallelize – If True, use a parallel range while listing the triangles.

Returns

coefficient – Clustering coefficient.

Return type

float

Example

>>> from sknetwork.data import karate_club
>>> adjacency = karate_club()
>>> np.round(get_clustering_coefficient(adjacency), 2)
0.26

Cliques

class sknetwork.topology.count_cliques

Count the number of cliques of some size.

Parameters
  • adjacency – Adjacency matrix of the graph.

  • clique_size (int) – Clique size (default = 3, corresponding to triangles.

Returns

n_cliques – Number of cliques.

Return type

int

Example

>>> from sknetwork.data import karate_club
>>> adjacency = karate_club()
>>> count_cliques(adjacency, 3)
45

References

Danisch, M., Balalau, O., & Sozio, M. (2018, April). Listing k-cliques in sparse real-world graphs. In Proceedings of the 2018 World Wide Web Conference (pp. 589-598).

Isomorphism

class sknetwork.topology.color_weisfeiler_lehman(adjacency: Union[scipy.sparse._csr.csr_matrix, numpy.ndarray], max_iter: int = - 1)[source]

Color nodes using Weisfeiler-Lehman algorithm.

Parameters
  • adjacency (sparse.csr_matrix) – Adjacency matrix of the graph

  • max_iter (int) – Maximum number of iterations. Negative value means no limit (until convergence).

Returns

labels – Label of each node.

Return type

np.ndarray

Example

>>> from sknetwork.data import house
>>> adjacency = house()
>>> labels = color_weisfeiler_lehman(adjacency)
>>> print(labels)
[0 2 1 1 2]

References

sknetwork.topology.are_isomorphic(adjacency1: scipy.sparse._csr.csr_matrix, adjacency2: scipy.sparse._csr.csr_matrix, max_iter: int = - 1) bool[source]

Weisfeiler-Lehman isomorphism test. If the test is False, the graphs cannot be isomorphic.

Parameters
  • adjacency1 – First adjacency matrix.

  • adjacency2 – Second adjacency matrix.

  • max_iter (int) – Maximum number of iterations. Negative value means no limit (until convergence).

Returns

test_result

Return type

bool

Example

>>> from sknetwork.data import house, bow_tie
>>> are_isomorphic(house(), bow_tie())
False

References