Gd8dZddlmZddlZddlmZdgZededd d Zd Z dS) z/Functions for computing rich-club coefficients.) accumulateN)not_implemented_forrich_club_coefficientdirected multigraphTdchtj|dkrtdt|}|rx|}|}tj|||z||zdz|t|fd|D}|S)u 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*: .. math:: \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 ---------- G : NetworkX graph Undirected graph with neither parallel edges nor self-loops. normalized : bool (optional) Normalize using randomized network as in [1]_ Q : float (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. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness`. Returns ------- rc : dictionary A dictionary, keyed by degree, with rich-club coefficient values. 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 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] 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 rzDrich_club_coefficient is not implemented for graphs with self loops. ) max_triesseedc.i|]\}}|||z Sr).0kvrcrans z)rich_club_coefficient..Ts' 5 5 5$!QaU1X 5 5 5)nxnumber_of_selfloops Exception _compute_rccopynumber_of_edgesdouble_edge_swapitems)G normalizedQr rcRErs @rrr sz a  1$$ U    QB6 FFHH     Aq1uA FFFFA 5 5 5 5"((** 5 5 5 Irc tj}t| fdt|D}t fdDd}}|\}}i}t|D]T\}} ||kr8t|dkrd}n"|\}}|dz}||k8d|z| | dz zz ||<U|S)zReturns the rich-club coefficient for each degree in the graph `G`. `G` is an undirected graph without multiedges. Returns a dictionary mapping degree to rich-club coefficient for that degree. c34K|]}|z dk |z VdS)Nr)rcstotals r z_compute_rc..fs/ F F"urzA~~52:~~~~ F Frc3\K|]&}ttj|V'dS)N)sortedmapdegree)rers rr)z_compute_rc..ls7GG6#ah"2"233GGGGGGrT)reverserr&) rdegree_histogramsumrr+edgesrpop enumeratelen) rdeghistnks edge_degreesekk1k2r!dnkr(s ` @rrrXs+!!$$G LLE G F F F 7 3 3 F F FC GGGGQWWYYGGGQUVVVL    B     FB B3))2Agg<  A%%!%%''FB !GB Agg B"Q-(1 Ir)TrN) __doc__ itertoolsrnetworkxrnetworkx.utilsr__all__rrrrrrDs55 ...... " #Z  \""HHH#"! HV     r