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k_components

k_components(G, flow_func=None)[source]

Returns the k-component structure of a graph G.

A \(k\)-component is a maximal subgraph of a graph G that has, at least, node connectivity \(k\): we need to remove at least \(k\) nodes to break it into more components. \(k\)-components have an inherent hierarchical structure because they are nested in terms of connectivity: a connected graph can contain several 2-components, each of which can contain one or more 3-components, and so forth.

Parameters:
  • G (NetworkX graph) –
  • flow_func (function) – Function to perform the underlying flow computations. Default value edmonds_karp(). This function performs better in sparse graphs with right tailed degree distributions. shortest_augmenting_path() will perform better in denser graphs.
Returns:

k_components – Dictionary with all connectivity levels \(k\) in the input Graph as keys and a list of sets of nodes that form a k-component of level \(k\) as values.
Return type:

dict
Raises:

NetworkXNotImplemented: – If the input graph is directed.

Examples

>>> # Petersen graph has 10 nodes and it is triconnected, thus all
>>> # nodes are in a single component on all three connectivity levels
>>> G = nx.petersen_graph()
>>> k_components = nx.k_components(G)

Notes

Moody and White [1] (appendix A) provide an algorithm for identifying k-components in a graph, which is based on Kanevsky’s algorithm [2] for finding all minimum-size node cut-sets of a graph (implemented in all_node_cuts() function):

  1. Compute node connectivity, k, of the input graph G.
  2. Identify all k-cutsets at the current level of connectivity using Kanevsky’s algorithm.
  3. Generate new graph components based on the removal of these cutsets. Nodes in a cutset belong to both sides of the induced cut.
  4. If the graph is neither complete nor trivial, return to 1; else end.

This implementation also uses some heuristics (see [3] for details) to speed up the computation.

See also

node_connectivity(), all_node_cuts()

References

[1]Moody, J. and D. White (2003). Social cohesion and embeddedness: A hierarchical conception of social groups. American Sociological Review 68(1), 103–28. http://www2.asanet.org/journals/ASRFeb03MoodyWhite.pdf
[2]Kanevsky, A. (1993). Finding all minimum-size separating vertex sets in a graph. Networks 23(6), 533–541. http://onlinelibrary.wiley.com/doi/10.1002/net.3230230604/abstract
[3]Torrents, J. and F. Ferraro (2015). Structural Cohesion: Visualization and Heuristics for Fast Computation. http://arxiv.org/pdf/1503.04476v1