user2983638 - 1 year ago 77
Python Question

# Efficient way for finding all the complete subgraphs of a given graph (Python)?

Is there an efficient way for finding all the fully-connected components (i.e. the complete subgraphs) of a given (undirected) graph with networkx? For example, I have the following adjacency matrix (without self-loops):

``````    |0 1 1 0 0|
|1 0 1 0 0|
G = |1 1 0 1 0|
|0 0 1 0 1|
|0 0 0 1 0|
``````

which corresponds to the following graph
The code should return the following tuples of nodes:

``````(0,1), (1,2), (0,2), (3,4), (2,3), (0,1,2)
``````

I know networkx has routines for finding cycles, strongly-connected components, etc, but I cannot find anything about fully-connected components. If it's not possible with networkx, it would also be fine with Numpy + Scipy. Many thanks in advance!

EDIT

This is what I did:

``````import networkx as nx
import itertools

def findsubsets(S, m):
return set(itertools.combinations(S, m))

A = np.array([[0, 1, 1, 0, 0],
[1, 0, 1, 0, 0],
[1, 1, 0, 1, 0],
[0, 0, 1, 0, 1],
[0, 0, 0, 1, 0]])

G = nx.from_numpy_matrix(A)

M = np.sqrt(np.size(A))

for m in range(2, M+1):

for a in findsubsets(range(0, M), m):

if(nx.number_of_edges(G.subgraph(a)) == (m**2 - m)/2.):

print nx.nodes(G.subgraph(a))
``````

which basically finds all the possible mXm subgraphs of the given one, and then checks if they have the maximum (i.e. (m**2 - m)/2) number of connections. But I was wondering if there is a more efficient way to do that, because the performance of the function
`itertools.combinations`
is not very good for large graphs.

Ok, I found it. It's simply `list(nx.find_cliques(G))`, just because I didn't know that in graph theory a clique is a fully connected subgraph.

EDIT

More precisely, `list(nx.find_cliques(G))` finds the maximal cliques, therefore it's not what I need. I found a similar post at this link.

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