# Source code for networkx.algorithms.walks

```"""Function for computing walks in a graph.
"""

import networkx as nx

__all__ = ["number_of_walks"]

[docs]
@nx._dispatchable
def number_of_walks(G, walk_length):
"""Returns the number of walks connecting each pair of nodes in `G`

A *walk* is a sequence of nodes in which each adjacent pair of nodes
in the sequence is adjacent in the graph. A walk can repeat the same
edge and go in the opposite direction just as people can walk on a
set of paths, but standing still is not counted as part of the walk.

This function only counts the walks with `walk_length` edges. Note that
the number of nodes in the walk sequence is one more than `walk_length`.
The number of walks can grow very quickly on a larger graph
and with a larger walk length.

Parameters
----------
G : NetworkX graph

walk_length : int
A nonnegative integer representing the length of a walk.

Returns
-------
dict
A dictionary of dictionaries in which outer keys are source
nodes, inner keys are target nodes, and inner values are the
number of walks of length `walk_length` connecting those nodes.

Raises
------
ValueError
If `walk_length` is negative

Examples
--------

>>> G = nx.Graph([(0, 1), (1, 2)])
>>> walks = nx.number_of_walks(G, 2)
>>> walks
{0: {0: 1, 1: 0, 2: 1}, 1: {0: 0, 1: 2, 2: 0}, 2: {0: 1, 1: 0, 2: 1}}
>>> total_walks = sum(sum(tgts.values()) for _, tgts in walks.items())

You can also get the number of walks from a specific source node using the
returned dictionary. For example, number of walks of length 1 from node 0
can be found as follows:

>>> walks = nx.number_of_walks(G, 1)
>>> walks[0]
{0: 0, 1: 1, 2: 0}
>>> sum(walks[0].values())  # walks from 0 of length 1
1

Similarly, a target node can also be specified:

>>> walks[0][1]
1

"""
import numpy as np

if walk_length < 0:
raise ValueError(f"`walk_length` cannot be negative: {walk_length}")

A = nx.adjacency_matrix(G, weight=None)
# TODO: Use matrix_power from scipy.sparse when available
# power = sp.sparse.linalg.matrix_power(A, walk_length)
power = np.linalg.matrix_power(A.toarray(), walk_length)
result = {
u: {v: power.item(u_idx, v_idx) for v_idx, v in enumerate(G)}
for u_idx, u in enumerate(G)
}
return result

```