rich_club_coefficient

rich_club_coefficient(G, normalized=True, Q=100, seed=None)

Returns the rich-club coefficient of the graph G.

For each degree k, the rich-club 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, and E_k is the number of edges among those nodes.

Parameters

GNetworkX graph

Undirected graph with neither parallel edges nor self-loops.

normalizedbool (optional)

Normalize using randomized network as in [1]

Qfloat (optional, default=100)

If normalized is True, perform Q * m double-edge swaps, where m is the number of edges in G, to use as a null-model for normalization.

seedinteger, random_state, or None (default)

Indicator of random number generation state. See Randomness.

Returns

rcdictionary

A dictionary, keyed by degree, with rich-club coefficient values.

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 rich-club 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/cond-mat/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, seed=42)
>>> rc[0]
0.4