# 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 (weighted) 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
weightstring or None, optional (default: None)

If None, every edge has weight 1. If a string, use this edge attribute as the edge weight. Any edge attribute not present defaults to 1. The edge weights are used to computing shortest-path distances.

Returns:
number

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.

References

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
```