# number_of_walks#

number_of_walks(G, walk_length)[source]#

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:
GNetworkX graph
walk_lengthint

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
```