+ j\vRt^RIHt^RIt^RIHtHtHt^RI H t .ROt ]P!RR7RRl4t Rt]P!RR7RR l4tR tR t]P!RR7RR l4t]PR 4t] !R4]PR44t] !R4]PR44tR#)zGroup centrality measures.)deepcopyN)_accumulate_endpoints"_single_source_dijkstra_path_basic"_single_source_shortest_path_basic)not_implemented_forweight) edge_attrsc aaa*.pRp\;QJdV3RlV4F 'gK RM RM!V3RlV44'dV.pRpVUUu0uF qwFqkK K p ppV SP, 'd,\P!RV SP, R24hSP 4p V 'd\P !S4M\P !S4p \SW4wro*VEFLo\S4o^p\V 4p\V 4pSEF<pVVV,V,, pS*V,pVV,pVV,pSEFpS*V,pVV,pVV,pVV,pVV,pVV9pVV9pSVP4,EFpS*V,pVV,pVV,p VV9p!VV9p"V!'drVV,VV,VV,,8XdOV'dGVV8wd@VV;;,VV,V,VV,V,,V, ,uu&V'ddVV,VV,VV,,8XdAV 'd9VV8wd2VV;;,VV,V,V,V , ,uu&V'gEKV"'gEK#VV,VV,VV,,8XgEKIV'gEKTVV ,p#VV;;,^V#V, , ,uu&VV;;,V#,uu&VV8XgEKVV#,pEK EK EK? \S4p$V'dT$p%M\S4p&V$V&, ;p'p%V 'dV&V$^, V',,p(MKV 'd#V*3RlS4p)\V3RlV)44p(M!V*3RlS4p)\V3R lV)44p(VV(,pV'dVV%V%^, ,,pMV 'g V^,pVPV4EKO V'dV#V^,#uuppi) uCompute the group betweenness centrality for a group of nodes. Group betweenness centrality of a group of nodes $C$ is the sum of the fraction of all-pairs shortest paths that pass through any vertex in $C$ .. math:: c_B(v) =\sum_{s,t \in V} \frac{\sigma(s, t|v)}{\sigma(s, t)} where $V$ is the set of nodes, $\sigma(s, t)$ is the number of shortest $(s, t)$-paths, and $\sigma(s, t|C)$ is the number of those paths passing through some node in group $C$. Note that $(s, t)$ are not members of the group ($V-C$ is the set of nodes in $V$ that are not in $C$). Parameters ---------- G : graph A NetworkX graph. C : list or set or list of lists or list of sets A group or a list of groups containing nodes which belong to G, for which group betweenness centrality is to be calculated. normalized : bool, optional (default=True) If True, group betweenness is normalized by $1/(N_{out}(N_{out}-1))$ where $N_{out}$ is the number of nodes in `G` that are not in `C`. This ensures the reported value is between 0 and 1. If `endpoints` is True, the normalization uses all nodes in `G`. The reported value is then between $2N_{in}/(N-1)$ and 1 where $N_{in}$ is the number of nodes in `C` and $N$ the number of nodes in `G`. weight : None or string, optional (default=None) If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight. The weight of an edge is treated as the length or distance between the two sides. endpoints : bool, optional (default=False) By default, only node-pairs that are both not in `C` are counted for group betweenness centrality. The count is how many non-`C` node-pairs have nodes from `C` "between" them on a shortest path. When ``endpoints=True``, we also count node-pairs with one or both nodes in `C` while considering endpoint nodes as being between the node-pairs. So we count paths that start in `C` whether or not they pass through any other nodes in `C`. This adds centrality to large groups without any reference to the connectivity of the group. The minimum normalized score is $N_{in}(N_{in}-1)/(N(N-1))$ instead of 0. For that reason, this feature is rarely used. We don't currently support considering node-pairs with nodes in `C` without also counting their endpoints. Nor do we support counting endpoints while only considering node-pairs that are both not in `C`. This keyword indicates both coutning endpoints of paths and allowing node-pairs in C. Raises ------ NodeNotFound If node(s) in `C` are not present in `G`. Returns ------- betweenness : list of floats or float If `C` is a single group then return a float. If `C` is a list with several groups then return a list of group betweenness centralities. See Also -------- betweenness_centrality Notes ----- Group betweenness centrality is defined in [1]_ and discussed in [3]_. The algorithm is described in [2]_ and is based on techniques mentioned in [4]_. The number of nodes in the group must be a maximum of ``N - 2`` where ``N`` is the total number of nodes in the graph. For weighted graphs the edge weights must be greater than zero. Zero edge weights can produce an infinite number of equal length paths between pairs of nodes. The total number of paths between source and target is counted differently for directed and undirected graphs. Directed paths between "u" and "v" are counted as two possible paths (one each direction) while undirected paths between "u" and "v" are counted as one path. Said another way, the sum in the expression above is over all ``s != t`` for directed graphs and for ``s < t`` for undirected graphs. References ---------- .. [1] M G Everett and S P Borgatti: The Centrality of Groups and Classes. Journal of Mathematical Sociology. 23(3): 181-201. 1999. http://www.analytictech.com/borgatti/group_centrality.htm .. [2] Ulrik Brandes: On Variants of Shortest-Path Betweenness Centrality and their Generic Computation. Social Networks 30(2):136-145, 2008. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.72.9610&rep=rep1&type=pdf .. [3] Sourav Medya et. al.: Group Centrality Maximization via Network Design. SIAM International Conference on Data Mining, SDM 2018, 126–134. https://sites.cs.ucsb.edu/~arlei/pubs/sdm18.pdf .. [4] Rami Puzis, Yuval Elovici, and Shlomi Dolev. "Fast algorithm for successive computation of group betweenness centrality." https://journals.aps.org/pre/pdf/10.1103/PhysRevE.76.056709 Tc3,<"TF qS9xK R#5iN).0elGs& e/home/luca/screen-checking/.venv/lib/python3.14/site-packages/networkx/algorithms/centrality/group.py /group_betweenness_centrality..s Qr7QsF The node(s)  are in C but not in G.c3Z<"TF pSV,Fq"V8wgK W3xK K" R#5ir r r uvDs& rrrs'Ha1Aafqffa++c3L<"TFwrVS9g VS9gK^xK R#5iNr r rrgroups& rrrs!S*$!eqEzAA*s$ $c3Z<"TF pSV,Fq"V8wgK W3xK K" R#5ir r rs& rrrs'LeQqTT!VfqfTferc3><"TFwrVS9d^M^xK R#5irr rs& rrrsL!u*Q!3s)anynodesnx NodeNotFound is_directedis_strongly_connected is_connected_group_preprocessingsetrkeyslensumappend)+rC normalizedr endpointsGBClist_of_groupsrnodeset_vr& connectedPBsigma GBC_groupsigma_mPB_mrDvPB_vsig_vxDxPB_xsig_xsig_xvsig_vxx_in_Dvv_in_DxyDysig_xysig_vyv_in_Dyy_in_Dvsig_xvyNNscaleN_inN_outextra reachablesrs+f&&&& ` @rgroup_betweenness_centralityrTs` CN s Q sss Q  C 3qeUTTUTqE 3 qwwoo UQWW_,==TUVV--/K/:((+PQ@RI(59LBqE  5/|A a #I1B7DAJEqTAw qqr'r'**A1B"1XF"1XF2gG2gG2a5BqEBqEM#9f6 GtAw'7'!*Q-'G&'PPG2a5BqEBqEM#9f6 GtAw'7&'@6'IIGw77r!u11 /E&&"(6/Q1w'7#77aG+6"g-F1+\ F Fu:DX %EFA .HaH S*SSMeL LLL  I  6A:. .I NI 9ij q6MG 4sQ$c /p/p/p\PV^4pVFpVf\W4wrW7&WW&M\WV4wrW7&WW&\ WhWV,V4wqdV&WG,FIp Wz8wdWG,V ;;,^, uu&VfK*W7,V ,^, W7,V &KK K \PV4p VFp \PVR4W&VFp WV ,9dKVFpWV,9gKWV,9gK!W^,V ,W^,V ,W\,V ,,8XgKXW,V ;;,WN,V ,W>,V ,,W<,V ,,W>,V ,, , uu&K K K WV3#)g)dictfromkeysrrr)rr5rr8deltar betweennesssSPir7 group_node1 group_node2r4s&&& rr)r)s E E A--1%K  >#Ea#K A%(AD#EaF#S A%(AD 5kaqST U 1XAv q !#hqkAo   q B --3/ KK.0D')ktW.D ,7;/!.2MMN 4!K 4#k+67#0=>$k+674!$ a<c^RIp^RIpVek\V4pW0P, 'd+\P !RW0P, R24h\ VPV, 4p M\ VP4p \P!4p RV n\W V4wrp VPPV 4pVe.VF'pVPVR7pVPVR7pK) \\VPV4V 4RR7UUu.uFwpqNK ppp^p.pV P!^VV^.V \#\WPV444R7^V P^,R &\%V4FgpV P^,R ;;,V P^,R ,V P^,R ,V,,, uu&Ki \'WV V^V VW4 wpp p\)V4pV'g^p\P*!V4'd<\P,!V4'dV^V,V, ^, ,pM:\P.!V4'dV^V,V, ^, ,pV^8Xd>VF7pV V,F'pVV8wgK VV9d V^, pKV^, pK) K9 VV,pV'd0^VV, VV, ^, ,, pVV,pMVP+4'g V^,p\1VR 4pVV3#uuppi) uFind the prominent group of size $k$ in graph $G$. The prominence of the group is evaluated by the group betweenness centrality. Group betweenness centrality of a group of nodes $C$ is the sum of the fraction of all-pairs shortest paths that pass through any vertex in $C$ .. math:: c_B(v) =\sum_{s,t \in V} \frac{\sigma(s, t|v)}{\sigma(s, t)} where $V$ is the set of nodes, $\sigma(s, t)$ is the number of shortest $(s, t)$-paths, and $\sigma(s, t|C)$ is the number of those paths passing through some node in group $C$. Note that $(s, t)$ are not members of the group ($V-C$ is the set of nodes in $V$ that are not in $C$). Parameters ---------- G : graph A NetworkX graph. k : int The number of nodes in the group. normalized : bool, optional (default=True) If True, group betweenness is normalized by ``1/((|V|-|C|)(|V|-|C|-1))`` where ``|V|`` is the number of nodes in G and ``|C|`` is the number of nodes in C. weight : None or string, optional (default=None) If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight. The weight of an edge is treated as the length or distance between the two sides. endpoints : bool, optional (default=False) If True include the endpoints in the shortest path counts. C : list or set, optional (default=None) list of nodes which won't be candidates of the prominent group. greedy : bool, optional (default=False) Using a naive greedy algorithm in order to find non-optimal prominent group. For scale free networks the results are negligibly below the optimal results. Raises ------ NodeNotFound If node(s) in C are not present in G. Returns ------- max_GBC : float The group betweenness centrality of the prominent group. max_group : list The list of nodes in the prominent group. See Also -------- betweenness_centrality, group_betweenness_centrality Notes ----- Group betweenness centrality is defined in [1]_ and discussed in [3]_. The algorithm is described in [2]_ and is based on techniques mentioned in [4]_. The number of nodes in the group must be a maximum of ``N - 2`` where ``N`` is the total number of nodes in the graph. For weighted graphs the edge weights must be greater than zero. Zero edge weights can produce an infinite number of equal length paths between pairs of nodes. The total number of paths between source and target is counted differently for directed and undirected graphs. Directed paths between "u" and "v" are counted as two possible paths (one each direction) while undirected paths between "u" and "v" are counted as one path. Said another way, the sum in the expression above is over all ``s != t`` for directed graphs and for ``s < t`` for undirected graphs. References ---------- .. [1] M G Everett and S P Borgatti: The Centrality of Groups and Classes. Journal of Mathematical Sociology. 23(3): 181-201. 1999. http://www.analytictech.com/borgatti/group_centrality.htm .. [2] Rami Puzis, Yuval Elovici, and Shlomi Dolev: "Finding the Most Prominent Group in Complex Networks" AI communications 20(4): 287-296, 2007. https://www.researchgate.net/profile/Rami_Puzis2/publication/220308855 .. [3] Sourav Medya et. al.: Group Centrality Maximization via Network Design. SIAM International Conference on Data Mining, SDM 2018, 126–134. https://sites.cs.ucsb.edu/~arlei/pubs/sdm18.pdf .. [4] Rami Puzis, Yuval Elovici, and Shlomi Dolev. "Fast algorithm for successive computation of group betweenness centrality." https://journals.aps.org/pre/pdf/10.1103/PhysRevE.76.056709 Nrr)index)columnsTreverse)CLrZr2GMr8contheurirgz.2f)numpypandasr*r#r$r%listGraph__networkx_cache__r) DataFrame from_dictdropsortedzipdiagadd_noderWrange_dfbnbr,r&r'r(float)rkrr/r1r0greedynppdr#DF_treer7r8rrZr4_rgmax_GBC max_groupr^rscaler_s&&&&&&& rprominent_grouprsN} F ww;;//LWW =T"UV VQWWq[!QWW hhjG!%G'&9LBq,,((,K}D%***6K%**4*8K%S)=u%EtT UT71d$TB UGI    #eWW[12 3 GMM!U 1X a7==#3F#;GMM!>!  ''**QUQY]+ __Q  QQ'E A:( kNND{*9,!QJE!QJE + ) 5a!eA *+5]]__1 wsm%G I g Vs, M4c F\VPV,R,4V8XdTVPV,R,V8d5VPV,R,W"PV,R,3#\VPV,R,4V8Xg\VPV,R,4V\VPV,R,4, 8:g?VPV,R,VPV,R,,V8:dW2V3#\WW%Wx4wrpV'd\WW#WWgV4 wr2pMVPV ,R,VPV ,R,,VPV ,R,VPV ,R,,8d&\WW#WWgV4 wr2p\WW#WWgV4 wr2pM$\WW#WWgV4 wr2p\WW#WWgV4 wr2pW2V3#)rhr2rgrj)r,r# _heuristicrx) rrzr~rrootrrr#r{node_pnode_ms &&&&&&&&& rrxrxs 7== t $%*w}}T/B5/IG/S}}T"5)7MM$4G4MMM GMM$  %&!+ w}}T"4( )QW]]45H5N1O-O O == u % d(;E(B Bg M**)'eLFG&, 'Fy' #)  fe$w}}V'u)E E F'- 'Fy' #)'- 'Fy' #)'- 'Fy' #)'- 'Fy' #) Y &&rac j^RIpVP4^,pVP4^,pVPV,R,^,p VPV\ VPV,43.4VPV,R,P V 4VPV,R;;,VPV,R,V ,, uu&VPV,p VEFVp VEFKp ^p ^p^pV R,V ,V ,^8XEgV R,V ,V ,^8XEgV R,V ,V ,^8XEgW;,V ,W;,V ,W<,V ,,8XdPV R,V ,V ,V R,V ,V ,,V R,V ,V ,, pW;,V ,W;,V ,W9,V ,,8XdPV R,V ,V ,V R,V ,V ,,V R,V ,V ,, p W9,V ,W9,V ,W;,V ,,8XdPV R,V ,V ,V R,V ,V ,,V R,V ,V ,, pV R,V ,V ,^V , ,VPV,R,V ,V &V R,V ,V ,V R,V ,V ,V ,, VPV,R,P W3&W8wdUVPV,R,P W3;;,V R,V ,V ,V,,uu&W8wgEKVPV,R,P W3;;,V R,V ,V ,V,,uu&EKN EKY \\VPVPV,R,4V4RR 7UUu.uF)wppVVPV,R,9gK'VNK+ uppVPV,R&\\WFPVPV,R,444VPV,R&^VPV,R &\V\VPV,R,4, 4FgpVPV,R ;;,VPV,R,VPV,R,V,,, uu&Ki V'Eg6VPV\ VPV,43.4VPV,R,P^4VPV,R,PV 4^VPV,R &\V\VPV,R,4, 4FgpVPV,R ;;,VPV,R,VPV,R,V,,, uu&Ki MRpWxV3#uuppi) rVNrgrhr2rir8rZTrerj)rknumber_of_nodesr#add_nodes_fromrr.locrsrtrurWrwr,pop)rzrr~rr#r{r|rr added_node root_noder?rGdxvydxyvdvxyrr4r^s&&&&&& rrrs $ $ & *F  $ $ & *Ft$T*1-J VXgmmD.A%BCDE MM&$&&z2 MM&% GMM&$9&$A*$MM  d#I ADDD'"1%a(A-W%a(49W%j1!494 #qtAwj1A'AA!'*1-a0#G,Q/ ;<#G,Q/ ;< 47ad:.q1AAA!'*1-j9#G,Z8;<#G,Q/23 =#q}Q'7!$q''AA!'*:6q9#G,Q/23#G,Z8;< 4=W3Ea3H3KqSWx3XGMM& !' *1 -a 0-(+A.=1I!1LQ1ORV1VV MM& !- 0 4 4QT : f%m488>m,Q/ ;dB> f%m488>m,Z8;dB>IV  f-m<=u Et # GAt w}}V,T2 2  #GMM&$%) E777==0?@A%GMM&&!$%GMM&% 1s7==0677 8 fe$ f(=f(E MM& !$ ' *)  $9 6'--2E)F GHI fd#''* ff%))*5'( fe$q3w}}V4T:;;rs  : X&X'Xv$NX&l'l^)'XU#pX&]']@//d\"*3#*3Z\"*)#*)ra