all_pairs_node_connectivity#

all_pairs_node_connectivity(G, nbunch=None, cutoff=None)[source]#

Compute node connectivity between all pairs of nodes.

Pairwise or local node connectivity between two distinct and nonadjacent nodes is the minimum number of nodes that must be removed (minimum separating cutset) to disconnect them. By Menger’s theorem, this is equal to the number of node independent paths (paths that share no nodes other than source and target). Which is what we compute in this function.

This algorithm is a fast approximation that gives an strict lower bound on the actual number of node independent paths between two nodes [1]. It works for both directed and undirected graphs.

Parameters:

GNetworkX graph
nbunch: container: Container of nodes. If provided node connectivity will be computed only over pairs of nodes in nbunch.
cutoffinteger: Maximum node connectivity to consider. If None, the minimum degree of source or target is used as a cutoff in each pair of nodes. Default value None.

Returns:

Kdictionary: Dictionary, keyed by source and target, of pairwise node connectivity

See also

local_node_connectivity
node_connectivity

References

[1]

White, Douglas R., and Mark Newman. 2001 A Fast Algorithm for Node-Independent Paths. Santa Fe Institute Working Paper #01-07-035 http://eclectic.ss.uci.edu/~drwhite/working.pdf

Examples

A 3 node cycle with one extra node attached has connectivity 2 between all nodes in the cycle and connectivity 1 between the extra node and the rest:

>>> G = nx.cycle_graph(3)
>>> G.add_edge(2, 3)
>>> import pprint  # for nice dictionary formatting
>>> pprint.pprint(nx.all_pairs_node_connectivity(G))
{0: {1: 2, 2: 2, 3: 1},
 1: {0: 2, 2: 2, 3: 1},
 2: {0: 2, 1: 2, 3: 1},
 3: {0: 1, 1: 1, 2: 1}}