from collections import defaultdict
import networkx as nx
__all__ = ["k_clique_communities"]
[docs]def k_clique_communities(G, k, cliques=None):
"""Find k-clique communities in graph using the percolation method.
A k-clique community is the union of all cliques of size k that
can be reached through adjacent (sharing k-1 nodes) k-cliques.
Parameters
----------
G : NetworkX graph
k : int
Size of smallest clique
cliques: list or generator
Precomputed cliques (use networkx.find_cliques(G))
Returns
-------
Yields sets of nodes, one for each k-clique community.
Examples
--------
>>> from networkx.algorithms.community import k_clique_communities
>>> G = nx.complete_graph(5)
>>> K5 = nx.convert_node_labels_to_integers(G, first_label=2)
>>> G.add_edges_from(K5.edges())
>>> c = list(k_clique_communities(G, 4))
>>> sorted(list(c[0]))
[0, 1, 2, 3, 4, 5, 6]
>>> list(k_clique_communities(G, 6))
[]
References
----------
.. [1] Gergely Palla, Imre Derényi, Illés Farkas1, and Tamás Vicsek,
Uncovering the overlapping community structure of complex networks
in nature and society Nature 435, 814-818, 2005,
doi:10.1038/nature03607
"""
if k < 2:
raise nx.NetworkXError(f"k={k}, k must be greater than 1.")
if cliques is None:
cliques = nx.find_cliques(G)
cliques = [frozenset(c) for c in cliques if len(c) >= k]
# First index which nodes are in which cliques
membership_dict = defaultdict(list)
for clique in cliques:
for node in clique:
membership_dict[node].append(clique)
# For each clique, see which adjacent cliques percolate
perc_graph = nx.Graph()
perc_graph.add_nodes_from(cliques)
for clique in cliques:
for adj_clique in _get_adjacent_cliques(clique, membership_dict):
if len(clique.intersection(adj_clique)) >= (k - 1):
perc_graph.add_edge(clique, adj_clique)
# Connected components of clique graph with perc edges
# are the percolated cliques
for component in nx.connected_components(perc_graph):
yield (frozenset.union(*component))
def _get_adjacent_cliques(clique, membership_dict):
adjacent_cliques = set()
for n in clique:
for adj_clique in membership_dict[n]:
if clique != adj_clique:
adjacent_cliques.add(adj_clique)
return adjacent_cliques