# Topology

Functions related to graph topology.

## Connectivity

sknetwork.topology.get_connected_components(input_matrix: csr_matrix, connection: str = 'weak', force_bipartite: bool = False) 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: 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: csr_matrix, connection: str = 'weak', force_bipartite: bool = False, return_index: bool = False) csr_matrix | Tuple[csr_matrix, ndarray][source]

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

Parameters:

• 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: csr_matrix, return_biadjacency: bool = False) bool | Tuple[bool, csr_matrix | None, ndarray | None, ndarray | None][source]

Check whether a graph is bipartite.

Parameters:

• 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
```

## Cycles

sknetwork.topology.is_acyclic(adjacency: csr_matrix, directed: bool | None = None) bool[source]

Check whether a graph has no cycle.

Parameters:

• 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
```
sknetwork.topology.get_cycles(adjacency: csr_matrix, directed: bool | None = None) List[List[int]][source]

Get all possible cycles of a graph.

Parameters:

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

Returns:

cycles – List of cycles, each cycle represented as a list of nodes.

Return type:

list

Example

```>>> from sknetwork.topology import get_cycles
>>> from sknetwork.data import cyclic_digraph
[[0, 1, 2, 3]]
```
sknetwork.topology.break_cycles(adjacency: csr_matrix, root: int | List[int], directed: bool | None = None) csr_matrix[source]

Break cycles of a graph from given roots.

Parameters:

• root – The root node or list of root nodes to break cycles from.

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

Returns:

Return type:

sparse.csr_matrix

Example

```>>> from sknetwork.topology import break_cycles, is_acyclic
>>> from sknetwork.data import cyclic_digraph
>>> dag = break_cycles(adjacency, root=0, directed=True)
>>> is_acyclic(dag, directed=True)
True
```

## Core decomposition

Get the k-core decomposition of a graph.

Parameters:

Returns:

Core value of each node.

Return type:

core_values

Example

```>>> from sknetwork.data import karate_club
>>> len(core_values)
34
```

## Triangles

class sknetwork.topology.count_triangles(adjacency: csr_matrix, parallelize: bool = False)

Count the number of triangles in a graph. The graph is considered undirected.

Parameters:

• parallelize (bool) – 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
45
```
class sknetwork.topology.get_clustering_coefficient(adjacency: csr_matrix, parallelize: bool = False)

Get the clustering coefficient of a graph.

Parameters:

• parallelize (bool) – If `True`, use a parallel range while listing the triangles.

Returns:

coefficient – Clustering coefficient.

Return type:

float

Example

```>>> from sknetwork.data import karate_club
0.26
```

## Cliques

class sknetwork.topology.count_cliques(adjacency: csr_matrix, clique_size: int = 3)

Count the number of cliques of some size.

Parameters:

• 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
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: csr_matrix | ndarray, max_iter: int = -1)[source]

Color nodes using Weisfeiler-Lehman algorithm.

Parameters:

• 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
>>> print(labels)
[0 2 1 1 2]
```

References

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

Parameters:

• 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