h8mSrSSKJr SSKrSSKJrJrJr SSK J r /SQr \R"SS9SS j5r S r\R"SS9SS j5rS rS r\R"SS9SSj5r\RS5r\ "S5\RS55r\ "S5\RS55rg)zGroup centrality measures.)deepcopyN)_accumulate_endpoints"_single_source_dijkstra_path_basic"_single_source_shortest_path_basic)not_implemented_for)group_betweenness_centralitygroup_closeness_centralitygroup_degree_centralitygroup_in_degree_centralitygroup_out_degree_centralityprominent_groupweight) edge_attrsc ^/nSn[U4SjU55(aU/nSnUVVs1sH owHoiM M n nnU TR- (a'[R"SU TR- S35e[ TX5upn UGHn[ U5nSn [ U 5n[ U 5n[ U5n[ U5nUGHdnXUU- n UGHGnUGH<nSnSnSnUUUS:XdUUUS:XdUUUS:XdU UUU UUU UU-:XaUUUUUU-UUU- nU UUU UUU UU-:XaUUUUUU-UUU- nU UUU UUU UU-:XaUUUU UU-U UU- nUUUSU- -UUU'UUUUUUU-- UUU'UU:waUUU==UUUU--ss'UU:wdGM#UUU==UUUU--ss'GM? GMJ UUnnUUnnGMg [T5[U5nnU(dSn[R"T5(a*[R"T5(aUSU-U- S- -nO)[R"T5(aUSU-U- S- -nUS:Xa.UH(nU UHnUU:wdM X;aUS- nMUS- nM M* U U-n U(aSUU- UU- S- -- nU U-n OTR5(dU S-n URU 5 GM U(aU$US$s snnf) u[ Compute 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/((|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. 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 described in [1]_ and its importance discussed in [3]_. The initial implementation of the algorithm is mentioned in [2]_. This function uses an improved algorithm presented 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,># UH oT;v M g7fN).0elGs V/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/centrality/group.py /group_betweenness_centrality..xs Qr7QsF The node(s)  are in C but not in G.r) anynodesnx NodeNotFound_group_preprocessingsetrlen is_directedis_strongly_connected is_connectedappend)rC normalizedr endpointsGBClist_of_groupsgroupnodeset_vPBsigmaD GBC_groupsigma_mPB_m sigma_m_vPB_m_vvxydxvydxyvdvxycscale group_node1s` rrrs| CN Q  C 3qeUTTUTqE 3 qwwoo UQWW_,==TUVV(59LBqE  5/|W% $A a #IADDD 1 *gajmq.@GAJqMUVDVQ47ad1g!Q&77#*1:a=71:a=#@71:a=#PDQ47ad1g!Q&77#*1:a=71:a=#@71:a=#PDQ47ad1g!Q&77#*1:a=58A;#>q!#LD&-ajmq4x&@IaLO#'71:Q T0A#AF1IaLAvq ! Q T(99 Avq ! 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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 described in [1]_ and its importance 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 rNrrT)indexinplace)columnsrOreverser)CLrFr,GMr2contheurUrSrz.2f)numpypandasr#rr r!listGraph__networkx_cache__r" DataFrame from_dictdropsortedzipdiagadd_noderCrange_dfbnbr$r%r&r'float)rkrr)r+r*greedynppdrDF_treer1r2r3rFr/_rSmax_GBC max_grouprJr9r@rAs rr r sN} F ww;//LWW =T"UV VQWWq[!QWW hhjG!%G'&9LBq,,((,K}D   4  6   T4  8%S)=u%EtT UT71d$TB UGI    #eWW[12 3 GMM!U 1X a7==#3F#;GMM!>!  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