U 9v f8@sPdZddlmZddlZddlmZdgZededd d dZd d Z dS)z/Functions for computing rich-club coefficients.) accumulateN)not_implemented_forrich_club_coefficientZdirectedZ multigraphTdcsrt|dkrtdt|}|rn|}|}tj|||||d|dt|fdd|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. )Z max_triesseedcsi|]\}}|||qSr).0kvZrcranr@/tmp/pip-unpacked-wheel-_lngutwb/networkx/algorithms/richclub.py Tsz)rich_club_coefficient..)nxZnumber_of_selfloops Exception _compute_rccopynumber_of_edgesZdouble_edge_swapitems)G normalizedQrrcRErr r r s=c st}t|fddt|D}tfddDdd}}|\}}i}t|D]P\}} ||krt |dkrd}q|\}}|d8}qnd|| | d||<qf|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. c3s"|]}|dkr|VqdS)Nr)r cs)totalrr fs z_compute_rc..c3s|]}ttj|VqdS)N)sortedmapZdegree)r e)rrr rlsT)reverserr) rZdegree_histogramsumrredgesrpop enumeratelen) rZdeghistZnksZ edge_degreesZekZk1Zk2rdZnkr)rrr rXs     r)TrN) __doc__ itertoolsrZnetworkxrZnetworkx.utilsr__all__rrrrrr s  K