Mercurial > repos > shellac > sam_consensus_v3
view env/lib/python3.9/site-packages/networkx/algorithms/community/quality.py @ 0:4f3585e2f14b draft default tip
"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
| author | shellac |
|---|---|
| date | Mon, 22 Mar 2021 18:12:50 +0000 |
| parents | |
| children |
line wrap: on
line source
"""Functions for measuring the quality of a partition (into communities). """ from functools import wraps from itertools import product import networkx as nx from networkx import NetworkXError from networkx.utils import not_implemented_for from networkx.algorithms.community.community_utils import is_partition __all__ = ["coverage", "modularity", "performance"] class NotAPartition(NetworkXError): """Raised if a given collection is not a partition. """ def __init__(self, G, collection): msg = f"{G} is not a valid partition of the graph {collection}" super().__init__(msg) def require_partition(func): """Decorator 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 """ @wraps(func) def new_func(*args, **kw): # Here we assume that the first two arguments are (G, partition). if not is_partition(*args[:2]): raise nx.NetworkXError( "`partition` is not a valid partition of" " the nodes of G" ) return func(*args, **kw) return new_func def intra_community_edges(G, partition): """Returns 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. """ return sum(G.subgraph(block).size() for block in partition) def inter_community_edges(G, partition): """Returns 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`. """ # Alternate implementation that does not require constructing a new # graph object (but does require constructing an affiliation # dictionary): # # aff = dict(chain.from_iterable(((v, block) for v in block) # for block in partition)) # return sum(1 for u, v in G.edges() if aff[u] != aff[v]) # MG = nx.MultiDiGraph if G.is_directed() else nx.MultiGraph return nx.quotient_graph(G, partition, create_using=MG).size() def inter_community_non_edges(G, partition): """Returns 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`. """ # Alternate implementation that does not require constructing two # new graph objects (but does require constructing an affiliation # dictionary): # # aff = dict(chain.from_iterable(((v, block) for v in block) # for block in partition)) # return sum(1 for u, v in nx.non_edges(G) if aff[u] != aff[v]) # return inter_community_edges(nx.complement(G), partition) @not_implemented_for("multigraph") @require_partition def performance(G, partition): """Returns 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 <https://arxiv.org/abs/0906.0612> """ # Compute the number of intra-community edges and inter-community # edges. intra_edges = intra_community_edges(G, partition) inter_edges = inter_community_non_edges(G, partition) # Compute the number of edges in the complete graph (directed or # undirected, as it depends on `G`) on `n` nodes. # # (If `G` is an undirected graph, we divide by two since we have # double-counted each potential edge. We use integer division since # `total_pairs` is guaranteed to be even.) n = len(G) total_pairs = n * (n - 1) if not G.is_directed(): total_pairs //= 2 return (intra_edges + inter_edges) / total_pairs @require_partition def coverage(G, partition): """Returns 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 <https://arxiv.org/abs/0906.0612> """ intra_edges = intra_community_edges(G, partition) total_edges = G.number_of_edges() return intra_edges / total_edges def modularity(G, communities, weight="weight"): r"""Returns 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). <https://arxiv.org/abs/cond-mat/0408187> """ if not isinstance(communities, list): communities = list(communities) if not is_partition(G, communities): raise NotAPartition(G, communities) directed = G.is_directed() if directed: out_degree = dict(G.out_degree(weight=weight)) in_degree = dict(G.in_degree(weight=weight)) m = sum(out_degree.values()) norm = 1 / m ** 2 else: out_degree = in_degree = dict(G.degree(weight=weight)) deg_sum = sum(out_degree.values()) m = deg_sum / 2 norm = 1 / deg_sum ** 2 def community_contribution(community): comm = set(community) L_c = sum(wt for u, v, wt in G.edges(comm, data=weight, default=1) if v in comm) out_degree_sum = sum(out_degree[u] for u in comm) in_degree_sum = sum(in_degree[u] for u in comm) if directed else out_degree_sum return L_c / m - out_degree_sum * in_degree_sum * norm return sum(map(community_contribution, communities))