ó ,µ[c@s#dZddlZddlmZddlmZddddgZed ƒed ƒed ƒd edd „ƒƒƒZ edƒed ƒed ƒd dedd„ƒƒƒZ ed ƒed ƒed ƒdddd„ƒƒƒZ ed ƒed ƒed ƒdddd„ƒƒƒZ dS(sGFunctions for estimating the small-world-ness of graphs. A small world network is characterized by a small average shortest path length, and a large clustering coefficient. Small-worldness is commonly measured with the coefficient sigma or omega. Both coefficients compare the average clustering coefficient and shortest path length of a given graph against the same quantities for an equivalent random or lattice graph. For more information, see the Wikipedia article on small-world network [1]_. .. [1] Small-world network:: https://en.wikipedia.org/wiki/Small-world_network iÿÿÿÿN(tnot_implemented_for(tpy_random_statetrandom_referencetlattice_referencetsigmatomegaitdirectedt multigraphicCsÆ|jƒr$d}tj|ƒ‚nt|ƒdkrHtjdƒ‚nddlm}m}tjj}|j ƒ}t |j ƒŒ\}} tj j| ƒ} t|ƒ} tj |ƒ} || }t| | | | ddƒ} d}xÔt|ƒD]Æ}d}x·|| kr½|dd | d |ƒ\}}||krCqn||}||}|jt|j|ƒƒƒ}|jt|j|ƒƒƒ}|j|ƒ}|j|ƒ}||||gks||||gkráqn|||kr°|||kr°|j||ƒ|j||ƒ|j||ƒ|j||ƒ|r¢||||ƒdkr¢|j||ƒ|j||ƒ|j||ƒ|j||ƒq°|d7}Pn|d7}qWqøW|S( s‚Compute a random graph by swapping edges of a given graph. Parameters ---------- G : graph An undirected graph with 4 or more nodes. niter : integer (optional, default=1) An edge is rewired approximately `niter` times. connectivity : boolean (optional, default=True) When True, ensure connectivity for the randomized graph. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness`. Returns ------- G : graph The randomized graph. Notes ----- The implementation is adapted from the algorithm by Maslov and Sneppen (2002) [1]_. References ---------- .. [1] Maslov, Sergei, and Kim Sneppen. "Specificity and stability in topology of protein networks." Science 296.5569 (2002): 910-913. s3random_reference() not defined for directed graphs.isGraph has less than four nodes.iÿÿÿÿ(tcumulative_distributiontdiscrete_sequenceiiit cdistributiontseed(t is_directedtnxt NetworkXErrortlentnetworkx.utilsRR t connectivitytlocal_edge_connectivitytcopytziptdegreetutilstnumber_of_edgestinttrangetchoicetlistt neighborstindextadd_edget remove_edge(tGtniterRR tmsgRR t local_conntkeystdegreestcdftnnodestnedgestntriest swapcounttitntaitcitatctbtdtbitdi((s=lib/python2.7/site-packages/networkx/algorithms/smallworld.pyRsV%         *  icCs ddl}ddlm}m}tjj}|jƒrRd} tj| ƒ‚nt |ƒdkrvtjdƒ‚n|j ƒ}t |j ƒŒ\} } || ƒ} t |ƒ} tj |ƒ}|dkrÖ|j| | fƒ}|jd| ƒ}|j| dddƒ}|jd |j||k||ƒƒ}xŸtt|j| d ƒƒƒD]{}|j||d||d ƒ|| |ddd…f<|| |ddd…fddd…||dd…f`. Returns ------- G : graph The latticized graph. Notes ----- The implementation is adapted from the algorithm by Sporns et al. [1]_. which is inspired from the original work by Maslov and Sneppen(2002) [2]_. References ---------- .. [1] Sporns, Olaf, and Jonathan D. Zwi. "The small world of the cerebral cortex." Neuroinformatics 2.2 (2004): 145-162. .. [2] Maslov, Sergei, and Kim Sneppen. "Specificity and stability in topology of protein networks." Science 296.5569 (2002): 910-913. iÿÿÿÿN(RR s4lattice_reference() not defined for directed graphs.isGraph has less than four nodes.iiiR R (i(tnumpyRRR R RRR RRRRRRtNonetzerostarangetappendtwhereRRtceilRRRRRR(R R!tDRR tnpRR R#R"R$R%R&R'R(tuntumtutvR)R*R+R,R-R.R/R0R1R2R3R4((s=lib/python2.7/site-packages/networkx/algorithms/smallworld.pyR|sj+       '&:B     * < idi c CsÝddl}igd6gd6}x`t|ƒD]R}t|d|d|ƒ}|djtj|ƒƒ|djtj|ƒƒq-Wtj|ƒ}tj|ƒ} |j|dƒ} |j|dƒ} || | | } | S(s‚Return the small-world coefficient (sigma) of the given graph. The small-world coefficient is defined as: sigma = C/Cr / L/Lr where C and L are respectively the average clustering coefficient and average shortest path length of G. Cr and Lr are respectively the average clustering coefficient and average shortest path length of an equivalent random graph. A graph is commonly classified as small-world if sigma>1. Parameters ---------- G : NetworkX graph An undirected graph. niter : integer (optional, default=100) Approximate number of rewiring per edge to compute the equivalent random graph. nrand : integer (optional, default=10) Number of random graphs generated to compute the average clustering coefficient (Cr) and average shortest path length (Lr). seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness`. Returns ------- sigma : float The small-world coefficient of G. Notes ----- The implementation is adapted from Humphries et al. [1]_ [2]_. References ---------- .. [1] The brainstem reticular formation is a small-world, not scale-free, network M. D. Humphries, K. Gurney and T. J. Prescott, Proc. Roy. Soc. B 2006 273, 503-511, doi:10.1098/rspb.2005.3354. .. [2] Humphries and Gurney (2008). "Network 'Small-World-Ness': A Quantitative Method for Determining Canonical Network Equivalence". PLoS One. 3 (4). PMID 18446219. doi:10.1371/journal.pone.0002051. iÿÿÿÿNtCtLR!R (R5RRR9R t transitivitytaverage_shortest_path_lengthtmean( R R!tnrandR R=t randMetricsR+tGrRBRCtCrtLrR((s=lib/python2.7/site-packages/networkx/algorithms/smallworld.pyRòs0 cCsõddl}igd6gd6}xxt|ƒD]j}t|d|d|ƒ}t|d|d|ƒ}|djtj|ƒƒ|djtj|ƒƒq-Wtj|ƒ} tj|ƒ} |j|dƒ} |j|dƒ} | | | | } | S(s|Return the small-world coefficient (omega) of a graph The small-world coefficient of a graph G is: omega = Lr/L - C/Cl where C and L are respectively the average clustering coefficient and average shortest path length of G. Lr is the average shortest path length of an equivalent random graph and Cl is the average clustering coefficient of an equivalent lattice graph. The small-world coefficient (omega) ranges between -1 and 1. Values close to 0 means the G features small-world characteristics. Values close to -1 means G has a lattice shape whereas values close to 1 means G is a random graph. Parameters ---------- G : NetworkX graph An undirected graph. niter: integer (optional, default=100) Approximate number of rewiring per edge to compute the equivalent random graph. nrand: integer (optional, default=10) Number of random graphs generated to compute the average clustering coefficient (Cr) and average shortest path length (Lr). seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness`. Returns ------- omega : float The small-work coefficient (omega) Notes ----- The implementation is adapted from the algorithm by Telesford et al. [1]_. References ---------- .. [1] Telesford, Joyce, Hayasaka, Burdette, and Laurienti (2011). "The Ubiquity of Small-World Networks". Brain Connectivity. 1 (0038): 367-75. PMC 3604768. PMID 22432451. doi:10.1089/brain.2011.0038. iÿÿÿÿNRBRCR!R ( R5RRRR9R RDRERF(R R!RGR R=RHR+RItGlRBRCtClRKR((s=lib/python2.7/site-packages/networkx/algorithms/smallworld.pyR6s6 ( t__doc__tnetworkxR RRRt__all__tTrueR6RRRR(((s=lib/python2.7/site-packages/networkx/algorithms/smallworld.pyts(    [   s   A