3 Mdp)@sdZddlmZddlmZddlZddlmZddlm Z ddl m Z dd d gZ Gd d d eZ d dZddZddZddZe dedd ZeddZddd ZdS)zHFunctions for measuring the quality of a partition (into communities). )wraps)productN) NetworkXError)not_implemented_for) is_partitioncoverage modularity performancecs eZdZdZfddZZS) NotAPartitionz6Raised if a given collection is not a partition. cs|d|}tj|dS)Nz' is not a valid partition of the graph )super__init__)selfGZ collectionmsg) __class__W/var/www/html/virt/lib/python3.6/site-packages/networkx/algorithms/community/quality.pyr szNotAPartition.__init__)__name__ __module__ __qualname____doc__r __classcell__rr)rrr sr cstfdd}|S)aDecorator to check that a valid partition is input to a function Raises :exc:`networkx.NetworkXError` if the partition is not valid. This decorator should be used on functions whose first two arguments are a graph and a partition of the nodes of that graph (in that order):: >>> @require_partition ... def foo(G, partition): ... print("partition is valid!") ... >>> G = nx.complete_graph(5) >>> partition = [{0, 1}, {2, 3}, {4}] >>> foo(G, partition) partition is valid! >>> partition = [{0}, {2, 3}, {4}] >>> foo(G, partition) Traceback (most recent call last): ... networkx.exception.NetworkXError: `partition` is not a valid partition of the nodes of G >>> partition = [{0, 1}, {1, 2, 3}, {4}] >>> foo(G, partition) Traceback (most recent call last): ... networkx.exception.NetworkXError: `partition` is not a valid partition of the nodes of G cs$t|ddstjd||S)Nz6`partition` is not a valid partition of the nodes of G)rnxr)argskw)funcrrnew_func9sz#require_partition..new_func)r)rrr)rrrequire_partitions rcstfdd|DS)aRReturns the number of intra-community edges for a partition of `G`. Parameters ---------- G : NetworkX graph. partition : iterable of sets of nodes This must be a partition of the nodes of `G`. The "intra-community edges" are those edges joining a pair of nodes in the same block of the partition. c3s|]}j|jVqdS)N)Zsubgraphsize).0block)rrr Ssz(intra_community_edges..)sum)r partitionr)rrintra_community_edgesEsr%cCs(|jrtjntj}tj|||djS)aReturns the number of inter-community edges for a prtition of `G`. according to the given partition of the nodes of `G`. Parameters ---------- G : NetworkX graph. partition : iterable of sets of nodes This must be a partition of the nodes of `G`. The *inter-community edges* are those edges joining a pair of nodes in different blocks of the partition. Implementation note: this function creates an intermediate graph that may require the same amount of memory as that of `G`. )Z create_using) is_directedrZ MultiDiGraphZ MultiGraphZquotient_graphr)rr$ZMGrrrinter_community_edgesVsr'cCsttj||S)aHReturns the number of inter-community non-edges according to the given partition of the nodes of `G`. `G` must be a NetworkX graph. `partition` must be a partition of the nodes of `G`. A *non-edge* is a pair of nodes (undirected if `G` is undirected) that are not adjacent in `G`. The *inter-community non-edges* are those non-edges on a pair of nodes in different blocks of the partition. Implementation note: this function creates two intermediate graphs, which may require up to twice the amount of memory as required to store `G`. )r'rZ complement)rr$rrrinter_community_non_edgesusr(Z multigraphcCsDt||}t||}t|}||d}|js8|d}|||S)aReturns the performance of a partition. The *performance* of a partition is the ratio of the number of intra-community edges plus inter-community non-edges with the total number of potential edges. Parameters ---------- G : NetworkX graph A simple graph (directed or undirected). partition : sequence Partition of the nodes of `G`, represented as a sequence of sets of nodes. Each block of the partition represents a community. Returns ------- float The performance of the partition, as defined above. Raises ------ NetworkXError If `partition` is not a valid partition of the nodes of `G`. References ---------- .. [1] Santo Fortunato. "Community Detection in Graphs". *Physical Reports*, Volume 486, Issue 3--5 pp. 75--174 r)r%r(lenr&)rr$ intra_edgesZ inter_edgesnZ total_pairsrrrr s'   cCst||}|j}||S)aReturns the coverage of a partition. The *coverage* of a partition is the ratio of the number of intra-community edges to the total number of edges in the graph. Parameters ---------- G : NetworkX graph partition : sequence Partition of the nodes of `G`, represented as a sequence of sets of nodes. Each block of the partition represents a community. Returns ------- float The coverage of the partition, as defined above. Raises ------ NetworkXError If `partition` is not a valid partition of the nodes of `G`. Notes ----- If `G` is a multigraph, the multiplicity of edges is counted. References ---------- .. [1] Santo Fortunato. "Community Detection in Graphs". *Physical Reports*, Volume 486, Issue 3--5 pp. 75--174 )r%Znumber_of_edges)rr$r+Z total_edgesrrrrs& weightcst|tst|}t|s&t|jrltjdtjdtj ddn4tj dtj }|dd|dfdd}tt ||S)aUReturns the modularity of the given partition of the graph. Modularity is defined in [1]_ as .. math:: Q = \frac{1}{2m} \sum_{ij} \left( A_{ij} - \frac{k_ik_j}{2m}\right) \delta(c_i,c_j) where $m$ is the number of edges, $A$ is the adjacency matrix of `G`, $k_i$ is the degree of $i$ and $\delta(c_i, c_j)$ is 1 if $i$ and $j$ are in the same community and 0 otherwise. According to [2]_ (and verified by some algebra) this can be reduced to .. math:: Q = \sum_{c=1}^{n} \left[ \frac{L_c}{m} - \left( \frac{k_c}{2m} \right) ^2 \right] where the sum iterates over all communities $c$, $m$ is the number of edges, $L_c$ is the number of intra-community links for community $c$, $k_c$ is the sum of degrees of the nodes in community $c$. The second formula is the one actually used in calculation of the modularity. Parameters ---------- G : NetworkX Graph communities : list or iterable of set of nodes These node sets must represent a partition of G's nodes. weight : string or None, optional (default="weight") The edge attribute that holds the numerical value used as a weight. If None or an edge does not have that attribute, then that edge has weight 1. Returns ------- Q : float The modularity of the paritition. Raises ------ NotAPartition If `communities` is not a partition of the nodes of `G`. Examples -------- >>> import networkx.algorithms.community as nx_comm >>> G = nx.barbell_graph(3, 0) >>> nx_comm.modularity(G, [{0, 1, 2}, {3, 4, 5}]) 0.35714285714285715 >>> nx_comm.modularity(G, nx_comm.label_propagation_communities(G)) 0.35714285714285715 References ---------- .. [1] M. E. J. Newman *Networks: An Introduction*, page 224. Oxford University Press, 2011. .. [2] Clauset, Aaron, Mark EJ Newman, and Cristopher Moore. "Finding community structure in very large networks." Physical review E 70.6 (2004). )r-r)rcsrt|tfddjddD}tfddD}rZtfddDn|}|||S)Nc3s |]\}}}|kr|VqdS)Nr)r uvwt)commrrr"Gsz=modularity..community_contribution..r))datadefaultc3s|]}|VqdS)Nr)r r.) out_degreerrr"Isc3s|]}|VqdS)Nr)r r.) in_degreerrr"Js)setr#edges)Z communityZL_cZout_degree_sumZ in_degree_sum)rdirectedr5mnormr4r-)r1rcommunity_contributionEs "z*modularity..community_contribution) isinstancelistrr r&dictr4r5r#valuesZdegreemap)rZ communitiesr-Zdeg_sumr;r)rr8r5r9r:r4r-rrs A       )r-)r functoolsr itertoolsrZnetworkxrrZnetworkx.utilsrZ-networkx.algorithms.community.community_utilsr__all__r rr%r'r(r rrrrrrs        *5 +