+ )i(Rt^RIHt^RIt.R Ot]P !RR7RRl4t]P !RR7RRl4t]P !RR7RRl4t ]P !RR7RRl4t ]P !RR7RR l4t ]P !RR7RR l4t ]P R 4t ]P R 4tR#)z5Functions for finding and evaluating cuts in a graph.)chainNweight) edge_attrsc \P!WW#^R7pVP4'd$\V\P!WW^R74p\ RV44#)aReturns the size of the cut between two sets of nodes. A *cut* is a partition of the nodes of a graph into two sets. The *cut size* is the sum of the weights of the edges "between" the two sets of nodes. Parameters ---------- G : NetworkX graph S : collection A collection of nodes in `G`. T : collection A collection of nodes in `G`. If not specified, this is taken to be the set complement of `S`. weight : object Edge attribute key to use as weight. If not specified, edges have weight one. Returns ------- number Total weight of all edges from nodes in set `S` to nodes in set `T` (and, in the case of directed graphs, all edges from nodes in `T` to nodes in `S`). Examples -------- In the graph with two cliques joined by a single edges, the natural bipartition of the graph into two blocks, one for each clique, yields a cut of weight one: >>> G = nx.barbell_graph(3, 0) >>> S = {0, 1, 2} >>> T = {3, 4, 5} >>> nx.cut_size(G, S, T) 1 Each parallel edge in a multigraph is counted when determining the cut size: >>> G = nx.MultiGraph(["ab", "ab"]) >>> S = {"a"} >>> T = {"b"} >>> nx.cut_size(G, S, T) 2 Notes ----- In a multigraph, the cut size is the total weight of edges including multiplicity. )datadefaultc3*"TF wrq3xK R#5iN).0uvrs& V/var/www/html/photoedit/myenv/lib/python3.14/site-packages/networkx/algorithms/cuts.py cut_size..Rs0%,!v%)nx edge_boundary is_directedrsum)GSTredgess&&&& rcut_sizersPr   Q11 =E}}eR--aAANO 0%0 00cVP4'd VPM VPp\RV!WR744#)a Returns the volume of a set of nodes. The *volume* of a set *S* is the sum of the (out-)degrees of nodes in *S* (taking into account parallel edges in multigraphs). [1] Parameters ---------- G : NetworkX graph S : collection A collection of nodes in `G`. weight : object Edge attribute key to use as weight. If not specified, edges have weight one. Returns ------- number The volume of the set of nodes represented by `S` in the graph `G`. See also -------- conductance cut_size edge_expansion edge_boundary normalized_cut_size References ---------- .. [1] David Gleich. *Hierarchical Directed Spectral Graph Partitioning*. c3*"TF wrVxK R#5ir r )r r ds& rrvolume..}s65TQq5rr)r out_degreedegreer)rrrr"s&&& rvolumer#Us4N]]__Q\\!((F 6VA56 66rcVf\V4\V4, p\WW#R7p\WVR7p\WVR7pV^V, ^V, ,,#)aReturns the normalized size of the cut between two sets of nodes. The *normalized cut size* is the cut size times the sum of the reciprocal sizes of the volumes of the two sets. [1] Parameters ---------- G : NetworkX graph S : collection A collection of nodes in `G`. T : collection A collection of nodes in `G`. weight : object Edge attribute key to use as weight. If not specified, edges have weight one. Returns ------- number The normalized cut size between the two sets `S` and `T`. Notes ----- In a multigraph, the cut size is the total weight of edges including multiplicity. See also -------- conductance cut_size edge_expansion volume References ---------- .. [1] David Gleich. *Hierarchical Directed Spectral Graph Partitioning*. rrr )setrr#rrrr num_cut_edgesvolume_Svolume_Ts&&&& rnormalized_cut_sizer+sWZ y FSVOQQ6Ma6*Ha6*H Q\a(l; < r )r&rr#minr's&&&& r conductancer.sOP y FSVOQ14Ma6*Ha6*H 3x2 22rcVf\V4\V4, p\WW#R7pV\\V4\V44, #)a5Returns the edge expansion between two node sets. The *edge expansion* is the quotient of the cut size and the smaller of the cardinalities of the two sets. [1] Parameters ---------- G : NetworkX graph S : collection A collection of nodes in `G`. T : collection A collection of nodes in `G`. weight : object Edge attribute key to use as weight. If not specified, edges have weight one. Returns ------- number The edge expansion between the two sets `S` and `T`. See also -------- boundary_expansion mixing_expansion node_expansion References ---------- .. [1] Fan Chung. *Spectral Graph Theory*. (CBMS Regional Conference Series in Mathematics, No. 92), American Mathematical Society, 1997, ISBN 0-8218-0315-8 r%)r&rr-len)rrrrr(s&&&& redge_expansionr1sAR y FSVOQQ6M 3s1vs1v. ..rc\\WW#R7pVP4pV^V,, #)uReturns the mixing expansion between two node sets. The *mixing expansion* is the quotient of the cut size and twice the number of edges in the graph. [1] Parameters ---------- G : NetworkX graph S : collection A collection of nodes in `G`. T : collection A collection of nodes in `G`. weight : object Edge attribute key to use as weight. If not specified, edges have weight one. Returns ------- number The mixing expansion between the two sets `S` and `T`. See also -------- boundary_expansion edge_expansion node_expansion References ---------- .. [1] Vadhan, Salil P. "Pseudorandomness." *Foundations and Trends in Theoretical Computer Science* 7.1–3 (2011): 1–336. r%)rnumber_of_edges)rrrrr(num_total_edgess&&&& rmixing_expansionr5s/RQQ6M'')O A/ 00rca\\P!V3RlV444p\V4\V4, #)uQReturns the node expansion of the set `S`. The *node expansion* is the quotient of the size of the node boundary of *S* and the cardinality of *S*. [1] Parameters ---------- G : NetworkX graph S : collection A collection of nodes in `G`. Returns ------- number The node expansion of the set `S`. See also -------- boundary_expansion edge_expansion mixing_expansion References ---------- .. [1] Vadhan, Salil P. "Pseudorandomness." *Foundations and Trends in Theoretical Computer Science* 7.1–3 (2011): 1–336. c3F<"TFpSPV4xK R#5ir ) neighbors)r r rs& rr!node_expansion..fs*E1a1;;q>>1s!)r&r from_iterabler0)rr neighborhoodsf& rnode_expansionr<Ds5Du***E1*EEFL | s1v %%rc`\\P!W44\V4, #)u[Returns the boundary expansion of the set `S`. The *boundary expansion* of a set `S` is the ratio between the size of its node boundary and the cardinality of the set itself [1]_ . Parameters ---------- G : NetworkX graph The input graph. S : collection A collection of nodes in `G`. Returns ------- number The boundary expansion ratio: size of node boundary / size of `S`. Examples -------- The node boundary is {2, 3} (size 2), divided by ``|S|=2``: >>> G = nx.cycle_graph(4) >>> S = {0, 1} >>> nx.boundary_expansion(G, S) 1.0 For disconnected sets, e.g. here where the node boundary is ``{1, 3, 5}``: >>> G = nx.cycle_graph(6) >>> S = {0, 2, 4} >>> nx.boundary_expansion(G, S) 1.0 See also -------- :func:`~networkx.algorithms.boundary.node_boundary` edge_expansion mixing_expansion node_expansion Notes ----- The node boundary is defined as all nodes not in `S` that are adjacent to nodes in `S`. References ---------- .. [1] Vadhan, Salil P. "Pseudorandomness." *Foundations and Trends in Theoretical Computer Science* 7.1–3 (2011): 1–336. )r0r node_boundary)rrs&&rboundary_expansionr?js$l r% &Q //r)r?r.rr1r5r<r+r#)NNr )__doc__ itertoolsrnetworkxr__all__ _dispatchablerr#r+r.r1r5r<r?r rrrEs ; X&;1';1|X&'7''7TX&1='1=hX&,3',3^X&+/'+/\X&*1'*1^"&"&J5050r