local_node_connectivity(G, s, t, flow_func=None, auxiliary=None, residual=None, cutoff=None)
Local node connectivity for two non adjacent nodes s and t is the minimum number of nodes that must be removed (along with their incident edges) to disconnect them.
This is a flow based implementation of node connectivity. We compute the maximum flow on an auxiliary digraph build from the original input graph (see below for details).
This is a flow based implementation of node connectivity. We compute the maximum flow using, by default, the edmonds_karp
algorithm (see: maximum_flow
) on an auxiliary digraph build from the original input graph:
For an undirected graph G having n nodes and :None:None:`m` edges we derive a directed graph H with :None:None:`2n` nodes and :None:None:`2m+n` arcs by replacing each original node :None:None:`v` with two nodes :None:None:`v_A`, :None:None:`v_B` linked by an (internal) arc in H. Then for each edge (u, :None:None:`v`) in G we add two arcs (:None:None:`u_B`, :None:None:`v_A`) and (:None:None:`v_B`, :None:None:`u_A`) in H. Finally we set the attribute capacity = 1 for each arc in H .
For a directed graph G having n nodes and :None:None:`m` arcs we derive a directed graph H with :None:None:`2n` nodes and :None:None:`m+n` arcs by replacing each original node :None:None:`v` with two nodes :None:None:`v_A`, :None:None:`v_B` linked by an (internal) arc (:None:None:`v_A`, :None:None:`v_B`) in H. Then for each arc (u, :None:None:`v`) in G we add one arc (:None:None:`u_B`, :None:None:`v_A`) in H. Finally we set the attribute capacity = 1 for each arc in H.
This is equal to the local node connectivity because the value of a maximum s-t-flow is equal to the capacity of a minimum s-t-cut.
Undirected graph
Source node
Target node
A function for computing the maximum flow among a pair of nodes. The function has to accept at least three parameters: a Digraph, a source node, and a target node. And return a residual network that follows NetworkX conventions (see maximum_flow
for details). If flow_func is None, the default maximum flow function ( edmonds_karp
) is used. See below for details. The choice of the default function may change from version to version and should not be relied on. Default value: None.
Auxiliary digraph to compute flow based node connectivity. It has to have a graph attribute called mapping with a dictionary mapping node names in G and in the auxiliary digraph. If provided it will be reused instead of recreated. Default value: None.
Residual network to compute maximum flow. If provided it will be reused instead of recreated. Default value: None.
If specified, the maximum flow algorithm will terminate when the flow value reaches or exceeds the cutoff. This is only for the algorithms that support the cutoff parameter: edmonds_karp
and shortest_augmenting_path
. Other algorithms will ignore this parameter. Default value: None.
local node connectivity for nodes s and t
Computes local node connectivity for nodes s and t.
edmonds_karp
meth
maximum_flow
meth
minimum_node_cut
meth
node_connectivity
meth
preflow_push
meth
This function is not imported in the base NetworkX namespace, so you have to explicitly import it from the connectivity package:
>>> from networkx.algorithms.connectivity import local_node_connectivity
We use in this example the platonic icosahedral graph, which has node connectivity 5.
>>> G = nx.icosahedral_graph()
... local_node_connectivity(G, 0, 6) 5
If you need to compute local connectivity on several pairs of nodes in the same graph, it is recommended that you reuse the data structures that NetworkX uses in the computation: the auxiliary digraph for node connectivity, and the residual network for the underlying maximum flow computation.
Example of how to compute local node connectivity among all pairs of nodes of the platonic icosahedral graph reusing the data structures.
>>> import itertools
... # You also have to explicitly import the function for
... # building the auxiliary digraph from the connectivity package
... from networkx.algorithms.connectivity import build_auxiliary_node_connectivity ...
>>> H = build_auxiliary_node_connectivity(G)
... # And the function for building the residual network from the
... # flow package
... from networkx.algorithms.flow import build_residual_network
... # Note that the auxiliary digraph has an edge attribute named capacity
... R = build_residual_network(H, "capacity")
... result = dict.fromkeys(G, dict())
... # Reuse the auxiliary digraph and the residual network by passing them
... # as parameters
... for u, v in itertools.combinations(G, 2):
... k = local_node_connectivity(G, u, v, auxiliary=H, residual=R)
... result[u][v] = k ...
>>> all(result[u][v] == 5 for u, v in itertools.combinations(G, 2)) True
You can also use alternative flow algorithms for computing node connectivity. For instance, in dense networks the algorithm shortest_augmenting_path
will usually perform better than the default edmonds_karp
which is faster for sparse networks with highly skewed degree distributions. Alternative flow functions have to be explicitly imported from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_pathSee :
... local_node_connectivity(G, 0, 6, flow_func=shortest_augmenting_path) 5
The following pages refer to to this document either explicitly or contain code examples using this.
networkx.algorithms.connectivity.connectivity.edge_connectivity
networkx.algorithms.connectivity.connectivity.local_node_connectivity
networkx.algorithms.connectivity.connectivity.average_node_connectivity
networkx.algorithms.connectivity.connectivity.local_edge_connectivity
networkx.algorithms.connectivity.connectivity.node_connectivity
networkx.algorithms.connectivity.connectivity.all_pairs_node_connectivity
Hover to see nodes names; edges to Self not shown, Caped at 50 nodes.
Using a canvas is more power efficient and can get hundred of nodes ; but does not allow hyperlinks; , arrows or text (beyond on hover)
SVG is more flexible but power hungry; and does not scale well to 50 + nodes.
All aboves nodes referred to, (or are referred from) current nodes; Edges from Self to other have been omitted (or all nodes would be connected to the central node "self" which is not useful). Nodes are colored by the library they belong to, and scaled with the number of references pointing them