btj(dZddlmZddlZgdZejdddZejdddZejddd Z ejddd Z ejddd Z ejddd Z ejd Z ejdZdS)z5Functions for finding and evaluating cuts in a graph.)chainN)boundary_expansion conductancecut_sizeedge_expansionmixing_expansionnode_expansionnormalized_cut_sizevolumeweight) edge_attrsc tj||||d}|r't|tj||||d}t d|DS)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"K|] \}}}|V dSN).0uvr s V/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/networkx/algorithms/cuts.py zcut_size..Rs(00,!Qv000000)nx edge_boundary is_directedrsum)GSTr edgess rrrssr  Q161 = = =E}}PeR-aAFANNNOO 00%000 0 00rc|r|jn|j}td|||DS)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 K|] \}}|V dSrr)rrds rrzvolume..}s&66TQq666666rr )r out_degreedegreer)rr r r(s rr r UsKN]]__ :Q\\!(F 66VVAf555666 6 66rc|t|t|z }t||||}t|||}t|||}|d|z d|z zzS)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*. Nr!r r&r)setrr rr r!r num_cut_edgesvolume_Svolume_Ts rr r suZ y FFSVVOQQv666Ma6***Ha6***H Q\a(l; < Nr&)r+rr minr,s rrrsrP y FFSVVOQ1V444Ma6***Ha6***H 3x22 22rc|t|t|z }t||||}|tt|t|z S)aReturns 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 Nr*)r+rr1len)rr r!r r-s rrrsVR y FFSVVOQQv666M 3s1vvs1vv.. ..rc`t||||}|}|d|zz S)usReturns 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)rr r!r r-num_total_edgess rrrs<RQQv666M''))O A/ 00rcttjfd|D}t|t|z S)uReturns 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. c3BK|]}|VdSr) neighbors)rrrs rrz!node_expansion..fs-*E*Ea1;;q>>*E*E*E*E*E*Er)r+r from_iterabler3)rr neighborhoods` rr r DsLDu**E*E*E*E1*E*E*EEEFFL |  s1vv %%rcfttj||t|z S)uReturns 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. 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