# wiener_index#

wiener_index(G, weight=None)[source]#

Returns the Wiener index of the given graph.

The Wiener index of a graph is the sum of the shortest-path distances between each pair of reachable nodes. For pairs of nodes in undirected graphs, only one orientation of the pair is counted.

Parameters:
GNetworkX graph
weightobject

The edge attribute to use as distance when computing shortest-path distances. This is passed directly to the `networkx.shortest_path_length()` function.

Returns:
float

The Wiener index of the graph `G`.

Raises:
NetworkXError

If the graph `G` is not connected.

Notes

If a pair of nodes is not reachable, the distance is assumed to be infinity. This means that for graphs that are not strongly-connected, this function returns `inf`.

The Wiener index is not usually defined for directed graphs, however this function uses the natural generalization of the Wiener index to directed graphs.

Examples

The Wiener index of the (unweighted) complete graph on n nodes equals the number of pairs of the n nodes, since each pair of nodes is at distance one:

```>>> n = 10
>>> G = nx.complete_graph(n)
>>> nx.wiener_index(G) == n * (n - 1) / 2
True
```

Graphs that are not strongly-connected have infinite Wiener index:

```>>> G = nx.empty_graph(2)
>>> nx.wiener_index(G)
inf
```