VhGDdZddlZgdZejj dej dZddZejj dej dddd Z ejj dej dddd Z ejj dej dddd Z ejj dej dddd Z ejj d ejj dej dddZ ejj dejj d ej dZy)u Find the k-cores of a graph. The k-core is found by recursively pruning nodes with degrees less than k. See the following references for details: An O(m) Algorithm for Cores Decomposition of Networks Vladimir Batagelj and Matjaz Zaversnik, 2003. https://arxiv.org/abs/cs.DS/0310049 Generalized Cores Vladimir Batagelj and Matjaz Zaversnik, 2002. https://arxiv.org/pdf/cs/0202039 For directed graphs a more general notion is that of D-cores which looks at (k, l) restrictions on (in, out) degree. The (k, k) D-core is the k-core. D-cores: Measuring Collaboration of Directed Graphs Based on Degeneracy Christos Giatsidis, Dimitrios M. Thilikos, Michalis Vazirgiannis, ICDM 2011. http://www.graphdegeneracy.org/dcores_ICDM_2011.pdf Multi-scale structure and topological anomaly detection via a new network statistic: The onion decomposition L. Hébert-Dufresne, J. A. Grochow, and A. Allard Scientific Reports 6, 31708 (2016) http://doi.org/10.1038/srep31708 N) core_numberk_corek_shellk_crustk_coronak_truss onion_layers multigraphc tj|dkDrd}tj|t|j }t ||j }dg}d}t|D].\}}|||kDs|j|g|||z z||}0t|Dcic]\}}|| } }}|} |Dcic]"}|ttj||$} }|D]t}| |D]j} | | | |kDs| | j|| | }|| | } | | | <|| || <|||| c|| <||<|| | xxdz cc<| | xxdzcc<lv| Scc}}wcc}w)aReturns the core number for each node. A k-core is a maximal subgraph that contains nodes of degree k or more. The core number of a node is the largest value k of a k-core containing that node. Parameters ---------- G : NetworkX graph An undirected or directed graph Returns ------- core_number : dictionary A dictionary keyed by node to the core number. Raises ------ NetworkXNotImplemented If `G` is a multigraph or contains self loops. Notes ----- For directed graphs the node degree is defined to be the in-degree + out-degree. Examples -------- >>> degrees = [0, 1, 2, 2, 2, 2, 3] >>> H = nx.havel_hakimi_graph(degrees) >>> nx.core_number(H) {0: 1, 1: 2, 2: 2, 3: 2, 4: 1, 5: 2, 6: 0} >>> G = nx.DiGraph() >>> G.add_edges_from([(1, 2), (2, 1), (2, 3), (2, 4), (3, 4), (4, 3)]) >>> nx.core_number(G) {1: 2, 2: 2, 3: 2, 4: 2} References ---------- .. [1] An O(m) Algorithm for Cores Decomposition of Networks Vladimir Batagelj and Matjaz Zaversnik, 2003. https://arxiv.org/abs/cs.DS/0310049 rlInput graph has self loops which is not permitted; Consider using G.remove_edges_from(nx.selfloop_edges(G)).key) nxnumber_of_selfloopsNetworkXNotImplementeddictdegreesortedget enumerateextendlist all_neighborsremove)Gmsgdegreesnodesbin_boundaries curr_degreeivposnode_poscorenbrsu bin_starts H/home/dcms/DCMS/lib/python3.12/site-packages/networkx/algorithms/core.pyrr-s^ a 1$ H '',,188:G 7 ,ESNK% %1 1: #  ! !1#k)A"B C!!*K%&/u%566336H6 D56 7AtB$$Q*+ + 7D 7  a AAwa Qq!qk*473 ' -0y)*/4Sz5;K,i %*tAw'1,'Q1    K7 8s / E)'E/c t|tjfdD}|j|j S)aReturns the subgraph induced by nodes passing filter `k_filter`. Parameters ---------- G : NetworkX graph The graph or directed graph to process k_filter : filter function This function filters the nodes chosen. It takes three inputs: A node of G, the filter's cutoff, and the core dict of the graph. The function should return a Boolean value. k : int, optional The order of the core. If not specified use the max core number. This value is used as the cutoff for the filter. core : dict, optional Precomputed core numbers keyed by node for the graph `G`. If not specified, the core numbers will be computed from `G`. c38K|]}|s|ywN).0r#r&kk_filters r* z!_core_subgraph..s 51At 4Q 5s)rmaxvaluessubgraphcopy)rr1r0r&rs ``` r*_core_subgraphr7}sJ& |1~y   5 5E ::e  ! ! ##T)preserve_all_attrs returns_graphc$d}t||||S)a Returns the k-core of G. A k-core is a maximal subgraph that contains nodes of degree `k` or more. Parameters ---------- G : NetworkX graph A graph or directed graph k : int, optional The order of the core. If not specified return the main core. core_number : dictionary, optional Precomputed core numbers for the graph G. Returns ------- G : NetworkX graph The k-core subgraph Raises ------ NetworkXNotImplemented The k-core is not defined for multigraphs or graphs with self loops. Notes ----- The main core is the core with `k` as the largest core_number. For directed graphs the node degree is defined to be the in-degree + out-degree. Graph, node, and edge attributes are copied to the subgraph. Examples -------- >>> degrees = [0, 1, 2, 2, 2, 2, 3] >>> H = nx.havel_hakimi_graph(degrees) >>> H.degree DegreeView({0: 1, 1: 2, 2: 2, 3: 2, 4: 2, 5: 3, 6: 0}) >>> nx.k_core(H).nodes NodeView((1, 2, 3, 5)) See Also -------- core_number References ---------- .. [1] An O(m) Algorithm for Cores Decomposition of Networks Vladimir Batagelj and Matjaz Zaversnik, 2003. https://arxiv.org/abs/cs.DS/0310049 c|||k\Sr-r.r#r0cs r*r1zk_core..k_filtertqyr8r7rr0rr1s r*rrsn !Xq+ 66r8c$d}t||||S)aReturns the k-shell of G. The k-shell is the subgraph induced by nodes with core number k. That is, nodes in the k-core that are not in the (k+1)-core. Parameters ---------- G : NetworkX graph A graph or directed graph. k : int, optional The order of the shell. If not specified return the outer shell. core_number : dictionary, optional Precomputed core numbers for the graph G. Returns ------- G : NetworkX graph The k-shell subgraph Raises ------ NetworkXNotImplemented The k-shell is not implemented for multigraphs or graphs with self loops. Notes ----- This is similar to k_corona but in that case only neighbors in the k-core are considered. For directed graphs the node degree is defined to be the in-degree + out-degree. Graph, node, and edge attributes are copied to the subgraph. Examples -------- >>> degrees = [0, 1, 2, 2, 2, 2, 3] >>> H = nx.havel_hakimi_graph(degrees) >>> H.degree DegreeView({0: 1, 1: 2, 2: 2, 3: 2, 4: 2, 5: 3, 6: 0}) >>> nx.k_shell(H, k=1).nodes NodeView((0, 4)) See Also -------- core_number k_corona References ---------- .. [1] A model of Internet topology using k-shell decomposition Shai Carmi, Shlomo Havlin, Scott Kirkpatrick, Yuval Shavitt, and Eran Shir, PNAS July 3, 2007 vol. 104 no. 27 11150-11154 http://www.pnas.org/content/104/27/11150.full c|||k(Sr-r.r=s r*r1zk_shell..k_filterr?r8r@rAs r*rrsz !Xq+ 66r8ctj|tjdz fdD}|j |j S)aReturns the k-crust of G. The k-crust is the graph G with the edges of the k-core removed and isolated nodes found after the removal of edges are also removed. Parameters ---------- G : NetworkX graph A graph or directed graph. k : int, optional The order of the shell. If not specified return the main crust. core_number : dictionary, optional Precomputed core numbers for the graph G. Returns ------- G : NetworkX graph The k-crust subgraph Raises ------ NetworkXNotImplemented The k-crust is not implemented for multigraphs or graphs with self loops. Notes ----- This definition of k-crust is different than the definition in [1]_. The k-crust in [1]_ is equivalent to the k+1 crust of this algorithm. For directed graphs the node degree is defined to be the in-degree + out-degree. Graph, node, and edge attributes are copied to the subgraph. Examples -------- >>> degrees = [0, 1, 2, 2, 2, 2, 3] >>> H = nx.havel_hakimi_graph(degrees) >>> H.degree DegreeView({0: 1, 1: 2, 2: 2, 3: 2, 4: 2, 5: 3, 6: 0}) >>> nx.k_crust(H, k=1).nodes NodeView((0, 4, 6)) See Also -------- core_number References ---------- .. [1] A model of Internet topology using k-shell decomposition Shai Carmi, Shlomo Havlin, Scott Kirkpatrick, Yuval Shavitt, and Eran Shir, PNAS July 3, 2007 vol. 104 no. 27 11150-11154 http://www.pnas.org/content/104/27/11150.full rc34K|]}|ks |ywr-r.)r/r#rr0s r*r2zk_crust..Ws ;1{1~':Q ; )rrr3r4r5r6)rr0rrs `` r*rrsYvnnQ' y  ""$ % ) ; ;E ::e  ! ! ##r8c*fd}t|||S)a3Returns the k-corona of G. The k-corona is the subgraph of nodes in the k-core which have exactly k neighbors in the k-core. Parameters ---------- G : NetworkX graph A graph or directed graph k : int The order of the corona. core_number : dictionary, optional Precomputed core numbers for the graph G. Returns ------- G : NetworkX graph The k-corona subgraph Raises ------ NetworkXNotImplemented The k-corona is not defined for multigraphs or graphs with self loops. Notes ----- For directed graphs the node degree is defined to be the in-degree + out-degree. Graph, node, and edge attributes are copied to the subgraph. Examples -------- >>> degrees = [0, 1, 2, 2, 2, 2, 3] >>> H = nx.havel_hakimi_graph(degrees) >>> H.degree DegreeView({0: 1, 1: 2, 2: 2, 3: 2, 4: 2, 5: 3, 6: 0}) >>> nx.k_corona(H, k=2).nodes NodeView((1, 2, 3, 5)) See Also -------- core_number References ---------- .. [1] k -core (bootstrap) percolation on complex networks: Critical phenomena and nonlocal effects, A. V. Goltsev, S. N. Dorogovtsev, and J. F. F. Mendes, Phys. Rev. E 73, 056101 (2006) http://link.aps.org/doi/10.1103/PhysRevE.73.056101 cR|k(xrtfd|Dk(S)Nc34K|]}|k\s dyw)rNr.)r/wr>r0s r*r2z)k_corona..func..s%CA1a%CrF)sum)r#r0r>rs ``r*funczk_corona..funcs*tqyCQ#%C1%C"CCCr8r@)rr0rrLs` r*rr[spD !T1k 22r8directedc @tj|dkDrd}tj||j}d}|dkDrd}g}t }|D]p}t ||}|j ||D cgc] } | |vs|  } } | D]6} t |t || z|dz ks$|j|| f8r|j|t |}|jttj||dkDr|Scc} w)aReturns the k-truss of `G`. The k-truss is the maximal induced subgraph of `G` which contains at least three vertices where every edge is incident to at least `k-2` triangles. Parameters ---------- G : NetworkX graph An undirected graph k : int The order of the truss Returns ------- H : NetworkX graph The k-truss subgraph Raises ------ NetworkXNotImplemented If `G` is a multigraph or directed graph or if it contains self loops. Notes ----- A k-clique is a (k-2)-truss and a k-truss is a (k+1)-core. Graph, node, and edge attributes are copied to the subgraph. K-trusses were originally defined in [2] which states that the k-truss is the maximal induced subgraph where each edge belongs to at least `k-2` triangles. A more recent paper, [1], uses a slightly different definition requiring that each edge belong to at least `k` triangles. This implementation uses the original definition of `k-2` triangles. Examples -------- >>> degrees = [0, 1, 2, 2, 2, 2, 3] >>> H = nx.havel_hakimi_graph(degrees) >>> H.degree DegreeView({0: 1, 1: 2, 2: 2, 3: 2, 4: 2, 5: 3, 6: 0}) >>> nx.k_truss(H, k=2).nodes NodeView((0, 1, 2, 3, 4, 5)) References ---------- .. [1] Bounds and Algorithms for k-truss. Paul Burkhardt, Vance Faber, David G. Harris, 2018. https://arxiv.org/abs/1806.05523v2 .. [2] Trusses: Cohesive Subgraphs for Social Network Analysis. Jonathan Cohen, 2005. rr r) rrrr6setaddlenappendremove_edges_fromremove_nodes_fromrisolates) rr0rH n_droppedto_dropseenr(nbrs_ur#new_nbrss r*rrs$l a 1$ H '',, AI a- u +A1YF HHQK#);aQd];H; +vAaD )*a!e4NNAq6* +  + G$L  DQ01 a- H>> degrees = [0, 1, 2, 2, 2, 2, 3] >>> H = nx.havel_hakimi_graph(degrees) >>> H.degree DegreeView({0: 1, 1: 2, 2: 2, 3: 2, 4: 2, 5: 3, 6: 0}) >>> nx.onion_layers(H) {6: 1, 0: 2, 4: 3, 1: 4, 2: 4, 3: 4, 5: 4} See Also -------- core_number References ---------- .. [1] Multi-scale structure and topological anomaly detection via a new network statistic: The onion decomposition L. Hébert-Dufresne, J. A. Grochow, and A. Allard Scientific Reports 6, 31708 (2016) http://doi.org/10.1038/srep31708 .. [2] Percolation and the effective structure of complex networks A. Allard and L. Hébert-Dufresne Physical Review X 9, 011023 (2019) http://doi.org/10.1103/PhysRevX.9.011023 rzqInput graph contains self loops which is not permitted; Consider using G.remove_edges_from(nx.selfloop_edges(G)).rrOr )rrrrrrrrVrRpoprrrSr) rr od_layersr# neighborsr current_core current_layerisolated_nodesr min_degree this_layerns r*r r sf a 1$ H '',,I:;Q A(IaL KKN  g, wGKK0U1X&  $%L  !AqzL(   a  !  A(IaLq\ ,! ##A&$QZ!^  , KKN  &) + g, . K=s'E;)NNr-)__doc__networkxr__all__utilsnot_implemented_for _dispatchablerr7rrrrrr r.r8r*rms> l+K,K\$6l+T>87?,87vl+T>>7?,>7Bl+T>>$?,>$Bl+T>93?,93xj)l+T>L ?,*L ^l+j)^*,^r8