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 connectivityk
: we need to remove at leastk
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:
- GNetworkX graph
- flow_funcfunction
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_componentsdict
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 levelk
as values.
- Raises:
- NetworkXNotImplemented
If the input graph is directed.
See also
node_connectivity
all_node_cuts
biconnected_components
special case of this function when k=2
k_edge_components
similar to this function, but uses edge-connectivity instead of node-connectivity
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):Compute node connectivity, k, of the input graph G.
Identify all k-cutsets at the current level of connectivity using Kanevsky’s algorithm.
Generate new graph components based on the removal of these cutsets. Nodes in a cutset belong to both sides of the induced cut.
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.
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. https://arxiv.org/pdf/1503.04476v1
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)