~i(SrSSKJr SSKr/SQr\R "SS9SSj5r\R "SS9SSj5r\R "SS9SS j5r \R "SS9SS j5r \R "SS9SS j5r \R "SS9SS j5r \R S 5r \R S5rg)z5Functions for finding and evaluating cuts in a graph.)chainN)boundary_expansion conductancecut_sizeedge_expansionmixing_expansionnode_expansionnormalized_cut_sizevolumeweight) edge_attrsc [R"XX#SS9nUR5(a [U[R"XXSS95n[ SU55$)aReturns the size of the cut between two sets of nodes. A *cut* is a partition of the nodes of a graph into two sets. The *cut size* is the sum of the weights of the edges "between" the two sets of nodes. Parameters ---------- G : NetworkX graph S : collection A collection of nodes in `G`. T : collection A collection of nodes in `G`. If not specified, this is taken to be the set complement of `S`. weight : object Edge attribute key to use as weight. If not specified, edges have weight one. Returns ------- number Total weight of all edges from nodes in set `S` to nodes in set `T` (and, in the case of directed graphs, all edges from nodes in `T` to nodes in `S`). Examples -------- In the graph with two cliques joined by a single edges, the natural bipartition of the graph into two blocks, one for each clique, yields a cut of weight one: >>> G = nx.barbell_graph(3, 0) >>> S = {0, 1, 2} >>> T = {3, 4, 5} >>> nx.cut_size(G, S, T) 1 Each parallel edge in a multigraph is counted when determining the cut size: >>> G = nx.MultiGraph(["ab", "ab"]) >>> S = {"a"} >>> T = {"b"} >>> nx.cut_size(G, S, T) 2 Notes ----- In a multigraph, the cut size is the total weight of edges including multiplicity. )datadefaultc3*# UH upo3v M g7fN).0uvr s U/home/mande/repo/quber/.venv/lib/python3.13/site-packages/networkx/algorithms/cuts.py cut_size..Rs0%,!v%)nx edge_boundary is_directedrsum)GSTr edgess rrrsPr   Q11 =E}}eR--aAANO 0%0 00cUR5(a URO URn[SU"XS955$)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*# UH upUv M g7frr)rrds rrvolume..}s65TQq5rr )r out_degreedegreer)r r!r r+s rr r Us4N]]__Q\\!((F 6VA56 66r$cUc[U5[U5- n[XX#S9n[XUS9n[XUS9nUSU- SU- --$)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*. r"r r)r)setrr r r!r"r num_cut_edgesvolume_Svolume_Ts rr r sWZ y FSVOQQ6Ma6*Ha6*H Q\a(l; < r))r.rr minr/s rrrsOP y FSVOQ14Ma6*Ha6*H 3x2 22r$cUc[U5[U5- n[XX#S9nU[[U5[U55- $)a5Returns the edge expansion between two node sets. The *edge expansion* is the quotient of the cut size and the smaller of the cardinalities of the two sets. [1] Parameters ---------- G : NetworkX graph S : collection A collection of nodes in `G`. T : collection A collection of nodes in `G`. weight : object Edge attribute key to use as weight. If not specified, edges have weight one. Returns ------- number The edge expansion between the two sets `S` and `T`. See also -------- boundary_expansion mixing_expansion node_expansion References ---------- .. [1] Fan Chung. *Spectral Graph Theory*. (CBMS Regional Conference Series in Mathematics, No. 92), American Mathematical Society, 1997, ISBN 0-8218-0315-8 r-)r.rr4len)r r!r"r r0s rrrsAR y FSVOQQ6M 3s1vs1v. ..r$cF[XX#S9nUR5nUSU-- $)uReturns the mixing expansion between two node sets. The *mixing expansion* is the quotient of the cut size and twice the number of edges in the graph. [1] Parameters ---------- G : NetworkX graph S : collection A collection of nodes in `G`. T : collection A collection of nodes in `G`. weight : object Edge attribute key to use as weight. If not specified, edges have weight one. Returns ------- number The mixing expansion between the two sets `S` and `T`. See also -------- boundary_expansion edge_expansion node_expansion References ---------- .. [1] Vadhan, Salil P. "Pseudorandomness." *Foundations and Trends in Theoretical Computer Science* 7.1–3 (2011): 1–336. r-)rnumber_of_edges)r r!r"r r0num_total_edgess rrrs/RQQ6M'')O A/ 00r$c^[[R"U4SjU555n[U5[U5- $)uQReturns the node expansion of the set `S`. The *node expansion* is the quotient of the size of the node boundary of *S* and the cardinality of *S*. [1] Parameters ---------- G : NetworkX graph S : collection A collection of nodes in `G`. Returns ------- number The node expansion of the set `S`. See also -------- boundary_expansion edge_expansion mixing_expansion References ---------- .. [1] Vadhan, Salil P. "Pseudorandomness." *Foundations and Trends in Theoretical Computer Science* 7.1–3 (2011): 1–336. c3F># UHnTRU5v M g7fr) neighbors)rrr s rr!node_expansion..fs*E1a1;;q>>1s!)r.r from_iterabler6)r r! neighborhoods` rr r Ds5Du***E1*EEFL | s1v %%r$cX[[R"X55[U5- $)u[Returns the boundary expansion of the set `S`. The *boundary expansion* of a set `S` is the ratio between the size of its node boundary and the cardinality of the set itself [1]_ . Parameters ---------- G : NetworkX graph The input graph. S : collection A collection of nodes in `G`. Returns ------- number The boundary expansion ratio: size of node boundary / size of `S`. Examples -------- The node boundary is {2, 3} (size 2), divided by ``|S|=2``: >>> G = nx.cycle_graph(4) >>> S = {0, 1} >>> nx.boundary_expansion(G, S) 1.0 For disconnected sets, e.g. here where the node boundary is ``{1, 3, 5}``: >>> G = nx.cycle_graph(6) >>> S = {0, 2, 4} >>> nx.boundary_expansion(G, S) 1.0 See also -------- :func:`~networkx.algorithms.boundary.node_boundary` edge_expansion mixing_expansion node_expansion Notes ----- The node boundary is defined as all nodes not in `S` that are adjacent to nodes in `S`. References ---------- .. [1] Vadhan, Salil P. "Pseudorandomness." *Foundations and Trends in Theoretical Computer Science* 7.1–3 (2011): 1–336. )r6r node_boundary)r r!s rrrjs$l r% &Q //r$)NNr)__doc__ itertoolsrnetworkxr__all__ _dispatchablerr r rrrr rrr$rrHs ; X&;1';1|X&'7''7TX&1='1=hX&,3',3^X&+/'+/\X&*1'*1^"&"&J5050r$