+ jt{<Rt^RIt^RIt^RIt^RIt^RIHt^RIt^RI H t H t ^RI H t HtRR.t]!R4]P !RR 7RRR R R RRRR RRR/Rl44t]!R4]P !RR 7RRR RR R RRRR/Rl44tRtRtRtR#)zGDetecting communities based on the Leiden Community Detection algorithmN)deque)constant_potts_model modularity)not_implemented_forpy_random_stateleiden_communitiesleiden_partitionsseedweight) edge_attrs resolutiong? max_levelmetriccpmthetag{Gz?c \WW$WVR7pVe)V^8:d \R4h\P!Ws4p\ V^R7pVP 4#)a5Return best node communities of `G`. This function uses the Leiden Community Detection algorithm [1]_ to estimate the community structure based on metric optimization. The metric can be modularity or Constant Potts Model (CPM). Leiden ensures that the communities are well connected whereas Louvain does not. The functions which compute those two metrics are :func:`~networkx.algorithms.community.quality.modularity` and :any:`constant_potts_model` The algorithm works in 3 phases. In the first phase, starting with a partition of all singleton nodes, and a randomly shuffled queue of all nodes, each node in the queue is considered in order and moved to the current community which most increases the metric value. The node can stay in its current community. If it moves, all neighbors of the moved node which are not currently in the queue are added to the end of the queue to be considered for moving in turn. The first phase continues until the queue is empty, resulting in a node partition $P$. The second phase refines $P$ to obtain $P_{refined}$. Each community in $P$ is considered on its own and split into subcommunities using a method closely resembling the first phase. Starting from a subpartition of singleton nodes, and a randomly ordered queue, subcommunities are merged only if both are sufficiently well connected. This is determined randomly using a distribution based on edge weights and controlled by the parameter `theta`. The merging results in a splitting or refinement of the communities in $P$ such that each set in $P_{refined}$ is a subset of one community in $P$. Thus each community in $P$ also represents a community of subcommunities from $P_{refined}$. The third phase involves creating an aggregate network of community connections. The communities in $P_{refined}$ act as nodes in the new network. Edges exist between communities that have underlying nodes connected to nodes in the other community. The weight of each such edge is the sum of the original graph edge weights for edges between nodes in the two partitions. An initial partition of the new network is provided via $P$ by joining the subcommunities of each community in $P$ into an aggregated community of the new network. Each subcommunity from $P_{refined}$ is a subset of one community in $P$ and thus belongs to one community in the new network. The result of the 3 phases is a new network (level of aggregation) which encodes a new partition of the original network. We can reapply the 3 phases to create bigger aggregations with increased metric value. At each level the network is smaller with fewer nodes and edges, so convergence is guaranteed. The three phases are repeatedly applied until no gain is achieved or `max_level` applications have been performed. Parameters ---------- G : NetworkX graph weight : string or None, optional (default="weight") The name of an edge attribute that holds the numerical value used as a weight. If None then each edge has weight 1. metric : {"cpm", "modularity"}, default="cpm" The name of the partition quality metric that the algorithm optimises. Allowed names are "cpm" and "modularity" for constant potts model and modularity respectively. See functions :func:`~networkx.algorithms.community.quality.modularity` and :any:`constant_potts_model` for more info. resolution : float, optional (default=1) Resolution should be a positive number indicating the coarseness of the communities produced. With a lower resolution, larger communities are produced. To see the smaller sub-communities, use a higher resolution. max_level : int or None, optional (default=None) The maximum number of levels (steps of the algorithm) to compute. Must be a positive integer or None indicating the process is reapplied until no increase in metric is obtained. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness`. theta : float (default=0.01) Parameter that determines the degree of randomness in the second phase _refine_partition step of the algorithm, Returns ------- list of sets of nodes A partition of `G` as a list of disjoint sets of nodes. Each set represents one community of nodes and each node of `G` appears in exactly one set. Examples -------- >>> G = nx.barbell_graph(3, 4) >>> cpm_comm = nx.community.leiden_communities(G, resolution=0.2, seed=62537113) >>> len(cpm_comm) 3 >>> sorted(sorted(c) for c in cpm_comm) [[0, 1, 2], [3, 4, 5, 6], [7, 8, 9]] Higher resolution produces smaller sub-communities: >>> cpm_comm = nx.community.leiden_communities(G, resolution=0.4, seed=62537114) >>> len(cpm_comm) 4 >>> sorted(sorted(c) for c in cpm_comm) [[0, 1, 2], [3, 4], [5, 6], [7, 8, 9]] Notes ----- The order in which the nodes are considered can affect the final output. In the algorithm the ordering happens using a random shuffle controlled by `seed` which also controls the randomness involved in selecting communities to merge during the refining (2nd) stage. References ---------- .. [1] Traag, V.A., Waltman, L. & van Eck, N.J. From Louvain to Leiden: guaranteeing well-connected communities. Sci Rep 9, 5233 (2019). https://doi.org/10.1038/s41598-019-41695-z See Also -------- leiden_partitions :any:`louvain_communities` :func:`networkx.algorithms.community.quality.modularity` :any:`constant_potts_model` )r r r rrz5max_level argument must be a positive integer or None)maxlen)r ValueError itertoolsislicerpop) Gr r r r rr partitionsfinal_partitions &$$$$$$ e/home/luca/screen-checking/.venv/lib/python3.14/site-packages/networkx/algorithms/community/leiden.pyrrs\D# Z6J >TU U%%j< Jq1O     c# aa'a(a)a*a+a,a-a."VUu.uFqf0NK pp\P!V4'dVxR#TpVP4p V 'd\P!4M\P!4pVP V4VP 4'dmVPV^R7FTwrjp VPWj4'd%W,V ,R;;,V , uu&KAVPWjV R7KV M"VPVPV^R74VPP4UU U U u/uF8wrlYlP4U U u/uFwrWj8wgK WR,bK up p bK: up p p po)V 'd|VPP4F]wrS)V ,pV P4F;wrmWj8XdK Wn9dW;;,V R,, uu&K0V R,W&K= K_ S)VPR&\P!YUu/uFqV0bK upR4VR8Xd`\ P"!\$SRRR 7p\&P)V^4o,RS,/pV 'd^S,o*V)V*V,3R lpEM0So*V)V*V,3R lpEM$VR 8Xd\ P"!\*SRR 7pV 'dj\'VP-RR74o+\'VP/RR74o-RS+RS-/p\1S-P344pSV, o*V)V*V+V-3RlpEM\'VP5RR74o(RS(/p\1S(P344pSV, o*V(V)V*3RlpEMDVR8XEd$V 'd\P6!R4hRpTpVP9RR7UUu0uFwppV^8XgKVkK pppVP9RR7UUu0uFwppV^8XgKVkK ppp\'VP5RR74o(VUu/uF qfS(V,bK upo.S.P;RV44VUu/uF qfS(V,bK upo'S'P;RV44RS.RS'/pV'V)V.V3Rlp\P6!R4h\P6!RV 24hV!W4pRp\=V4UUu/uF wppVVbK pppV'Ed3\?VVVVR 7pVUu.uFp\AVVVWE4NK p pV U!U"u.uFp!V!Fp"V"NK K p#p!p"V!VV#4p$V$V, R!8pT$p\CVV V4wppVP9RR7U%U&u.uFwp%p&V&PE4NK up&p%xVPR,o)VR8Xd$VR,o,\P!VS,R4KVR 8Xd*V 'dVR,o+VR,o-EKVR,o(EKVR8XgEK&VR,o.VR,o'EK;R#uupiuup p iuup p p piuupiuuppiuuppiuupiuupiuuppiuupiuup"p!iuup&p%i5i)"aVYield partitions at each level of Leiden Community Detection Leiden Community Detection is an algorithm to extract the community structure of a network based on modularity optimization. The partitions across levels (steps of the algorithm) form a dendrogram of communities. A dendrogram is a diagram representing a tree and each level represents a partition of the graph `G`. The top level contains the smallest communities and as you traverse the tree the communities get bigger and the overall partition quality metric increases. Each level is generated by executing the three phases of the `Leiden Community Detection algorithm `. Parameters ---------- G : NetworkX graph weight : string or None, optional (default="weight") The name of an edge attribute that holds the numerical value used as a weight. If None then each edge has weight 1. metric : {"cpm", "modularity"}, default="cpm" The name of the partition quality metric that the algorithm optimises. Allowed names are "cpm" and "modularity". resolution : float, optional (default=1) Resolution should be a positive number indicating the coarseness of the communities produced. With a lower resolution, larger communities are produced. To see the smaller sub-communities, use a higher resolution. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness`. theta : float (default=0.01) Parameter that determines the degree of randomness in the second phase _refine_partition step of the algorithm, Yields ------ list of sets of nodes A partition of `G` as a list of disjoint sets of nodes. Each set represents one community of nodes and each node of `G` appears in exactly one set. The quality metric is nondecreasing across the yielded partitions. References ---------- .. [1] Traag, V.A., Waltman, L. & van Eck, N.J. From Louvain to Leiden: guaranteeing well-connected communities. Sci Rep 9, 5233 (2019). https://doi.org/10.1038/s41598-019-41695-z See Also -------- leiden_communities :any:`louvain_partitions` N)datadefaultr r edge_wtsnodesr node_weight)r r"r cb<aaS'g^#\V\4'df\V 3RlV44p\V 3RlS44p\V4\S48:dTSupoMSVupo\VV 3RlV44pMTpS V,p\S4^8Xd8\ \ S44pS V,pS V,P V^4pM?\V 3RlS44p\V3RlS V,P444pVS V,V,, #)c36<"TFpSV,xK R#5iN.0u node_weightss& r 8leiden_partitions..delta_func..0$Kl\!__lc36<"TFpSV,xK R#5ir&r'r(s& rr,r-1#GYLOOYr/c3z<"TF0pSV,P4Fwr#VS9gKVxK K2 R#5ir&itemsr)r*vwtCr s& rr,r-72U1x{7H7H7JeaaSTfBB7JB%;;c36<"TFpSV,xK R#5ir&r'r(s& rr,r-@'KA Qr/c3><"TFwrVS9gKVxK R#5ir&r'r)r6r7 communitys& rr,r-AV0CuqqI~0C   isinstancesetsumlennextitergetr4 nodes_to_addr? nodes_size comm_sizeAEr*r6r8r gammar+s &f @r delta_func%leiden_partitions..delta_func, lC00!$$Kl$K!KJ ##GY#G GI<(C N:+Y1(,1UUUA$A!-aJ9~* i1$0O $QKOOAq1$''K'K$K V 0A0A0CVV59,z999rcb<aaS'g^#\V\4'df\V 3RlV44p\V 3RlS44p\V4\S48:dTSupoMSVupo\VV 3RlV44pMTpS V,p\S4^8Xd8\ \ S44pS V,pS V,P V^4pM?\V 3RlS44p\V3RlS V,P444pVS V,V,, #)r$c36<"TFpSV,xK R#5ir&r'r(s& rr,r-Qr.r/c36<"TFpSV,xK R#5ir&r'r(s& rr,r-Rr1r/c3z<"TF0pSV,P4Fwr#VS9gKVxK K2 R#5ir&r3r5s& rr,r-Xr9r:c36<"TFpSV,xK R#5ir&r')r)r6r+s& rr,r-ar<r/c3><"TFwrVS9gKVxK R#5ir&r'r>s& rr,r-br@rArBrJs &f @rrQrRMrSrr)r r in_degree out_degreec<aa S'g^#\V\4'd\V4\S48:dTSupo MSVupo \V 3RlV44p\V3RlV44p\V 3RlS 44p\V3RlS 44p\V V 3RlV44pMTpS V,pSV,p\S4^8XdA\ \ S44p S V ,pSV ,pS V,P V ^4pMT\V 3RlS44p\V3RlS44p\V3RlS V,P444pVS W6,WE,,,, #) r$c36<"TFpSV,xK R#5ir&r'r)r* in_degreess& rr,r-8az!}}ar/c36<"TFpSV,xK R#5ir&r'r)r* out_degreess& rr,r-:1 Ar/c36<"TFpSV,xK R#5ir&r'r^s& rr,r-r`r/c36<"TFpSV,xK R#5ir&r'rbs& rr,r-rdr/c3z<"TF0pSV,P4Fwr#VS9gKVxK K2 R#5ir&r3r5s& rr,r-r9r:c36<"TFpSV,xK R#5ir&r'r^s& rr,r-s"D)Q:a==)r/c36<"TFpSV,xK R#5ir&r'rbs& rr,r-s#FIqKNNIr/c3><"TFwrVS9gKVxK R#5ir&r'r>s& rr,r-r@rA)rCrDrFrErGrHrIr4)rKr?rNA_inA_outC_inC_outrOr*r6r8r rPr_rcs&f @rrQrR|s< lC00<(C N:+Y1(,18a88D:::E8a88D:::EUUUA$A%a=D'NE9~* i1)!} +A$QKOOAq1""D)"DD ##FI#F FV 0A0A0CVV 5DL5<$?@@@rdegreecb<aaS'g^#\V\4'df\V 3RlS44p\V 3RlV44p\V4\S48:dTSupoMSVupo\VV 3RlV44pMTpS V,p\S4^8Xd8\ \ S44pS V,pS V,P V^4pM?\V 3RlS44p\V3RlS V,P444pVS V,V,, #)r$c36<"TFpSV,xK R#5ir&r'r)r*degreess& rr,r-s#B 1GAJJ r/c36<"TFpSV,xK R#5ir&r'rrs& rr,r-s$FAWQZZr/c3z<"TF0pSV,P4Fwr#VS9gKVxK K2 R#5ir&r3r5s& rr,r-r9r:c36<"TFpSV,xK R#5ir&r'rrs& rr,r-s'FIq Ir/c3><"TFwrVS9gKVxK R#5ir&r'r>s& rr,r-r@rArB) rKr?rMrLrNrOr*r6r8rsr rPs &f @rrQrRs lC00 ##B #B BI!$$F$F!FJ<(C N:+Y1(,1UUUA$A!(J9~* i1$+AJ $QKOOAq1$''FI'F$F V 0A0A0CVV 5:- 999rbarber_modularityz-barber_modularity not implemented for DiGraphcR#r&r'r'rrrx,leiden_partitions..barber_modularitys r bipartiterc3("TFq^3xK R#5ir$Nr'r)r*s& rr,$leiden_partitions..s6:aq6:c3("TFq^3xK R#5ir~r'rs& rr,rs6IqFIr red_degree blue_degreec<a \V 3RlV44p\V 3RlV44p\V\4'dd\V 3RlV44p\V 3RlV44p\V4\V48:dYupo MYupo \V V 3RlV44pM@S V,pS V,pYupo \V 3RlS V,P 444pVS WC,WR,,,, #)c36<"TFpSV,xK R#5ir&r'r)r* red_degreess& rr,r-s=9a;q>>9r/c36<"TFpSV,xK R#5ir&r'r)r* blue_degreess& rr,r-s?YLOOYr/c36<"TFpSV,xK R#5ir&r'rs& rr,r-sE 1 A r/c36<"TFpSV,xK R#5ir&r'rs& rr,r-s G,Qa,r/c3z<"TF0pSV,P4Fwr#VS9gKVxK K2 R#5ir&r3r5s& rr,r-s1QAq8A;3D3D3F%!!q&3FAr:c3><"TFwrVS9gKVxK R#5ir&r')r)r6r7r8s& rr,r-sF(;uqqAv(;rA)rErCrDrFr4)rKr?comm_red comm_blue nodes_red nodes_bluerNrOr*r8rr rr s&& @rrQrRs=9==H?Y??I,,,E EE  G, GG |$I6'DAq$DAqQAQQ' 5 ),7 #1F (9(9(;FFzY%:Z=R%RSS Srz,barber_modularity not implemented for leidenz:leiden only supports metrics "cpm" and "modularity". Got: T)r gHz>)#nxis_empty is_directedDiGraphGraphadd_nodes_from is_multigraphedgeshas_edgeadd_edgeadd_weighted_edges_from_adjr4_predgraphset_node_attributes functoolspartialrdictfromkeysrrZr[rEvaluesro NetworkXErrorr!update enumerate_move_nodes_fast_merge_node_subset_create_aggregate_graphcopy)/rr rr r rr* partitionorig_Grr6r7nbrsddv_wtsnmetric_functionnode_attributesrQmtwo_mrxc red_nodes blue_nodesQimprovement_madeinodenode2comPr8 P_refinedpcommP_refined_flatQ_new_r!rrsr rPr_r+rcrs/&$$d$$ @@@@@@@@rrrsB  aaI  {{1~~F$$&K# AV  &! 2!AI2  .aOL 8 -.GGG,AB,A5uzz|,ABB77:& U?*=9L  " "1lM B | #,[9 -l; )(3 * *),7K*=9L m !D A,7HHH9:R5S@CsZ  Y=Z AZ *B-Z Y 1 Y  Y  Y  Z #BZ  Y AZ A Z 'CZ +0Z  Y,Y2Z  Y$Y$!"Z Y*Z 4Y/A1Z 7Y4 Z Z %Y:<Z Y??Z Z2AZ %Z /Z  Y :Z caaaVUu.uF p\4NK upoVP4FwrVSV,PV4K \V4pVP V4\ P V4pV'Ed\\V44oVPS4VS,p \\P!VS44p V U u0uF qV ,kK up V 0, p V 'gK\VVV3RlV 44wrSV ,pVS0, pT V'd S!SV4M^8gKWS&VPS4SV,PS4V SV,, pVPRV44EKSUu.uFpV'gK VNK up#uupiuup iuupi)c3H<"TFpS!SSV,4V3xK R#5ir&r')r)rrrQr*s& rr,#_move_nodes_fast..Rs! L8a*Q!"5q!98s"c3("TFqR3xK R#5ir&r')r)rs& rr,r`sB1AA$i1Ar)rDr4addlistshufflerrrGrHrr all_neighborsmaxremover)rrrQr rrr rand_nodes dict_dequeold_comu_nbrsr6nbr_comsbest_addbest_com P_old_com without_unodes_to_revisitrrr*s&&f& @@rrr8s| !1!Ann&  $ D'hJLL z*J * j! "q1+R%%a+,)/0AQKK0G9< ! L8 LLgJ O 9z!Y/! D"QK   Q  hKOOA  &( 3    B1AB B q!AAAq U "(1, sF;G# G2Gc R\V4UUu/uF'wrVYQV0, 'dV!WaV0, 4M^bK) ppp\V4UUu/uFwrVWu,^8gKWebK pppVUu.uFqf0NK p pVP4EF`wrj\W,4^8wdK\\P !W44p .p .p \V 4F[wr^W,'dW8XgWu,^8:dK(V!Wn4pV^8gK9V P V4V P V4K] V 'gK\V 4pV Uu.uF'p\P!VV, V, 4NK) ppVPV VR7^,pW,PV4^Wz&V V,pVPV4W, pV'd V!VV4M^VV&EKc V Uu.uFpV'gK VNK up#uuppiuuppiuupiuupiuupi)r$)weights) rr4rFrDrrappendrmathexpchoicesrr)rr8rQr rrr*TR C_refinedcomm_iunbrs cand_commscand_comm_deltasnew_commQ_delta max_deltaxcand_wts new_comm_i C_removedrs&&&&& rrrfs DMQ<P<41A3wwZs7 #A 5U 's)-H H#H0 H-H6 H$H$cVPRR7p/pVP4p/p^pVUu/uFq/bK p p\V4FwrV Fp WV&VU u/uFq^bK pp \4pV FTpWtV&VP VV,4VP 4F!wppW;;,VV,, uu&K# KV VP WR7VFpW,W,V&K V^, pK K VF pW,W(&K VPRR7FjwpppVV,pVV,pVPVV4'd&VV,V,R;;,V, uu&KVVPVVVR7Kl VPP 4UUUUu/uF=wppTVP 4UUu/uFwppVV8wgKVVR,bK uppbK? pppppVP4'dVPP 4FcwppVV,pVP 4F@wppVV8XdKVV9dVV;;,VR,, uu&K4VR,VV&KB Ke VVPR&WV3#uupiuup iuuppiuuppppi)a8Return a new graph based on P_refined. Each community becomes a node. Note: `node_attributes` is changed in place by this function! P_refined is a list of lists of community sets. The outer list indicates communities in the new graph which comes from P. Each inner list holds the refined sets of nodes each of which becomes a new node. So P_refined contains P as the outer list and its refinement as the inner lists which hold the refined community sets that make up the new graph's nodes. node_attributes is a dict keyed by each attribute name to be aggregated which depends on the metric being used. The value is a dict for that attribute keyed by node to the attribute value. Aggregate means sum within a G community into a new node in H. After this function the incoming dict values are replaced by aggregated dicts. r!r|)r!r rr )r! __class__rrDrr4add_noderrrrrrr)rrrG_original_nodes nodes_G2HH H_node2com H_node_id attr_nameH_node_attributes H_comm_id refined_comms refined_comm attributeagg_valsoriginal_nodesG_node attr_dictr*r6r7H_uH_vrr H_edge_wtsrs&&& rrrs"wwGw,I AJI8GH9BH$-i$8 )L%.y !6EFo1 oHF!UN'$-&!%%&6v&>?,;,A,A,C(Iy'9V+<<'-D ' JJyJ 7, :B:M!,Y7- NI+*%98% %6%A"%GGG*1bll ::c3   cF3K !R ' ! JJsCJ + +vv||~%GAt  @2aOAr(|O @@% }}ww}}GAtqME26:!H8 ,H!(|E!H &'%AGGJ =sI GH As) J( J-7J8  J2$J22J8 2J8 )__doc__rrrrandom collectionsrnetworkxr%networkx.algorithms.community.qualityrrnetworkx.utilsrr__all__ _dispatchablerrrrrr'rrr sM R? !4 5X&I! I! I!  I!  I! I! I!'I!XX&U  U   U  U  U  U 'U p +\6'rPr