networkx.algorithms.richclub.rich_club_coefficient¶

rich_club_coefficient
(G, normalized=True, Q=100, seed=None)[source]¶ Returns the richclub coefficient of the graph
G
.For each degree k, the richclub coefficient is the ratio of the number of actual to the number of potential edges for nodes with degree greater than k:
\[\phi(k) = \frac{2 E_k}{N_k (N_k  1)}\]where
N_k
is the number of nodes with degree larger than k, andE_k
is the number of edges among those nodes. Parameters
G (NetworkX graph) – Undirected graph with neither parallel edges nor selfloops.
normalized (bool (optional)) – Normalize using randomized network as in 1
Q (float (optional, default=100)) – If
normalized
is True, performQ * m
doubleedge swaps, wherem
is the number of edges inG
, to use as a nullmodel for normalization.seed (integer, random_state, or None (default)) – Indicator of random number generation state. See Randomness.
 Returns
rc – A dictionary, keyed by degree, with richclub coefficient values.
 Return type
dictionary
Examples
>>> G = nx.Graph([(0, 1), (0, 2), (1, 2), (1, 3), (1, 4), (4, 5)]) >>> rc = nx.rich_club_coefficient(G, normalized=False) >>> rc[0] 0.4
Notes
The rich club definition and algorithm are found in 1. This algorithm ignores any edge weights and is not defined for directed graphs or graphs with parallel edges or self loops.
Estimates for appropriate values of
Q
are found in 2.References
 1(1,2)
Julian J. McAuley, Luciano da Fontoura Costa, and Tibério S. Caetano, “The richclub phenomenon across complex network hierarchies”, Applied Physics Letters Vol 91 Issue 8, August 2007. https://arxiv.org/abs/physics/0701290
 2
R. Milo, N. Kashtan, S. Itzkovitz, M. E. J. Newman, U. Alon, “Uniform generation of random graphs with arbitrary degree sequences”, 2006. https://arxiv.org/abs/condmat/0312028