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rich_club_coefficient¶

rich_club_coefficient
(G, normalized=True, Q=100)[source]¶ Return the richclub coefficient of the graph G.
The richclub coefficient is the ratio, for every degree k, of the number of actual to the number of potential edges for nodes with degree greater than k:
\[\phi(k) = \frac{2 Ek}{Nk(Nk1)}\]where Nk is the number of nodes with degree larger than k, and Ek be the number of edges among those nodes.
Parameters: G : NetworkX graph
normalized : bool (optional)
Normalize using randomized network (see [R275])
Q : float (optional, default=100)
If normalized=True build a random network by performing Q*M doubleedge swaps, where M is the number of edges in G, to use as a nullmodel for normalization.
Returns: rc : dictionary
A dictionary, keyed by degree, with rich club coefficient values.
Notes
The rich club definition and algorithm are found in [R275]. 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 [R276].
References
[R275] (1, 2, 3) 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. http://arxiv.org/abs/physics/0701290 [R276] (1, 2) R. Milo, N. Kashtan, S. Itzkovitz, M. E. J. Newman, U. Alon, “Uniform generation of random graphs with arbitrary degree sequences”, 2006. http://arxiv.org/abs/condmat/0312028 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