# Source code for nngt.analysis.nx_functions

#-*- coding:utf-8 -*-
#
# analysis/nx_functions.py
#
# This file is part of the NNGT project, a graph-library for standardized and
# and reproducible graph analysis: generate and analyze networks with your
# favorite graph library (graph-tool/igraph/networkx) on any platform, without
# any change to your code.
# Copyright (C) 2015-2021 Tanguy Fardet
#
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# (at your option) any later version.
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# GNU General Public License for more details.
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""" Tools to analyze graphs with the networkx backend """

import numpy as np
import scipy.sparse as ssp

from ..lib.test_functions import nonstring_container, is_integer
from ..lib.graph_helpers import _get_nx_weights, _get_nx_graph

import networkx as nx

[docs]def global_clustering_binary_undirected(g): ''' Returns the undirected global clustering coefficient. This corresponds to the ratio of undirected triangles to the number of undirected triads. Parameters ---------- g : :class:~nngt.Graph Graph to analyze. References ---------- .. [nx-global-clustering] :nxdoc:algorithms.cluster.transitivity ''' return nx.transitivity(g.graph.to_undirected(as_view=True))
[docs]def local_clustering_binary_undirected(g, nodes=None): ''' Returns the undirected local clustering coefficient of some nodes. If g is directed, then it is converted to a simple undirected graph (no parallel edges). Parameters ---------- g : :class:~nngt.Graph Graph to analyze. nodes : list, optional (default: all nodes) The list of nodes for which the clustering will be returned Returns ------- lc : :class:numpy.ndarray The list of clustering coefficients, on per node. References ---------- .. [nx-local-clustering] :nxdoc:algorithms.cluster.clustering ''' num_nodes = g.node_nb() if nonstring_container(nodes): num_nodes = len(nodes) elif nodes is not None: num_nodes = 1 lc = nx.clustering(g.graph.to_undirected(as_view=True), nodes=nodes, weight=None) if num_nodes == 1: return lc if nodes is None: nodes = list(range(num_nodes)) return np.array([lc[n] for n in nodes], dtype=float)
[docs]def assortativity(g, degree, weights=None): ''' Returns the assortativity of the graph. This tells whether nodes are preferentially connected together depending on their degree. Parameters ---------- g : :class:~nngt.Graph Graph to analyze. degree : str The type of degree that should be considered. weights : bool or str, optional (default: binary edges) Whether edge weights should be considered; if None or False then use binary edges; if True, uses the 'weight' edge attribute, otherwise uses any valid edge attribute required. References ---------- .. [nx-assortativity] :nxdoc:algorithms.assortativity.degree_assortativity_coefficient ''' w = _get_nx_weights(g, weights) return nx.degree_pearson_correlation_coefficient( g.graph, x=degree, y=degree, weight=w)
[docs]def reciprocity(g): ''' Calculate the edge reciprocity of the graph. The reciprocity is defined as the number of edges that have a reciprocal edge (an edge between the same nodes but in the opposite direction) divided by the total number of edges. This is also the probability for any given edge, that its reciprocal edge exists. By definition, the reciprocity of undirected graphs is 1. @todo: check whether we can get this for single nodes for all libraries. Parameters ---------- g : :class:~nngt.Graph Graph to analyze. References ---------- .. [nx-reciprocity] :nxdoc:algorithms.reciprocity.overall_reciprocity ''' if not g.is_directed(): return 1. return nx.overall_reciprocity(g.graph)
[docs]def closeness(g, weights=None, nodes=None, mode="out", harmonic=True, default=np.NaN): r''' Returns the closeness centrality of some nodes. Closeness centrality of a node u is defined, for the harmonic version, as the sum of the reciprocal of the shortest path distance :math:d_{uv} from u to the N - 1 other nodes in the graph (if mode is "out", reciprocally :math:d_{vu}, the distance to u from another node v, if mode is "in"): .. math:: C(u) = \frac{1}{N - 1} \sum_{v \neq u} \frac{1}{d_{uv}}, or, using the arithmetic definition, as the reciprocal of the average shortest path distance to/from u over to all other nodes: .. math:: C(u) = \frac{n - 1}{\sum_{v \neq u} d_{uv}}, where d_{uv} is the shortest-path distance from u to v, and n is the number of nodes in the component. By definition, the distance is infinite when nodes are not connected by a path in the harmonic case (such that :math:\frac{1}{d(v, u)} = 0), while the distance itself is taken as zero for unconnected nodes in the first equation. Parameters ---------- g : :class:~nngt.Graph Graph to analyze. weights : bool or str, optional (default: binary edges) Whether edge weights should be considered; if None or False then use binary edges; if True, uses the 'weight' edge attribute, otherwise uses any valid edge attribute required. nodes : list, optional (default: all nodes) The list of nodes for which the clutering will be returned mode : str, optional (default: "out") For directed graphs, whether the distances are computed from ("out") or to ("in") each of the nodes. harmonic : bool, optional (default: True) Whether the arithmetic or the harmonic (recommended) version of the closeness should be used. Returns ------- c : :class:numpy.ndarray The list of closeness centralities, on per node. References ---------- .. [nx-harmonic] :nxdoc:algorithms.centrality.harmonic_centrality .. [nx-closeness] :nxdoc:algorithms.centrality.closeness_centrality ''' w = _get_nx_weights(g, weights) graph = g.graph if graph.is_directed() and mode == "out": graph = g.graph.reverse(copy=False) c = None if harmonic: c = nx.harmonic_centrality(graph, distance=w) else: c = nx.closeness_centrality(graph, distance=w, wf_improved=False) c = np.array([v for _, v in c.items()]) # normalize if harmonic: c *= 1 / (len(graph) - 1) elif default != 0: c[c == 0.] = default if nodes is None: return c return c[nodes]
[docs]def betweenness(g, btype="both", weights=None): ''' Returns the normalized betweenness centrality of the nodes and edges. Parameters ---------- g : :class:~nngt.Graph Graph to analyze. btype : str, optional (default 'both') The centrality that should be returned (either 'node', 'edge', or 'both'). By default, both betweenness centralities are computed. weights : bool or str, optional (default: binary edges) Whether edge weights should be considered; if None or False then use binary edges; if True, uses the 'weight' edge attribute, otherwise uses any valid edge attribute required. Returns ------- nb : :class:numpy.ndarray The nodes' betweenness if btype is 'node' or 'both' eb : :class:numpy.ndarray The edges' betweenness if btype is 'edge' or 'both' References ---------- .. [nx-ebetw] :nxdoc:algorithms.centrality.edge_betweenness_centrality .. [nx-nbetw] :nxdoc:networkx.algorithms.centrality.betweenness_centrality ''' w = _get_nx_weights(g, weights) nb, eb = None, None if btype in ("both", "node"): di_nb = nx.betweenness_centrality(g.graph, weight=w) nb = np.array([di_nb[i] for i in g.get_nodes()]) if btype in ("both", "edge"): di_eb = nx.edge_betweenness_centrality(g.graph, weight=w) eb = np.array([di_eb[tuple(e)] for e in g.edges_array]) if btype == "node": return nb elif btype == "edge": return eb return nb, eb
[docs]def connected_components(g, ctype=None): ''' Returns the connected component to which each node belongs. Parameters ---------- g : :class:~nngt.Graph Graph to analyze. ctype : str, optional (default 'scc') Type of component that will be searched: either strongly connected ('scc', by default) or weakly connected ('wcc'). Returns ------- cc, hist : :class:numpy.ndarray The component associated to each node (cc) and the number of nodes in each of the component (hist). References ---------- .. [nx-ucc] :nxdoc:algorithms.components.connected_components .. [nx-scc] :nxdoc:algorithms.components.strongly_connected_components .. [nx-wcc] :nxdoc:algorithms.components.weakly_connected_components ''' if ctype is None: ctype = "scc" if g.is_directed() else "wcc" res = None if not g.is_directed(): res = nx.connected_components(g.graph) elif ctype == "scc": res = nx.strongly_connected_components(g.graph) elif ctype == "wcc": res = nx.weakly_connected_components(g.graph) else: raise ValueError("Invalid ctype, only 'scc' and 'wcc' are allowed.") cc = np.zeros(g.node_nb(), dtype=int) hist = [] for i, nodes in enumerate(res): cc[list(nodes)] = i hist.append(len(nodes)) return cc, np.array(hist, dtype=int)
[docs]def shortest_path(g, source, target, directed=None, weights=None, combine_weights="mean"): ''' Returns a shortest path between sourceand target. The algorithms returns an empty list if there is no path between the nodes. Parameters ---------- g : :class:~nngt.Graph Graph to analyze. source : int Node from which the path starts. target : int Node where the path ends. directed : bool, optional (default: g.is_directed()) Whether the edges should be considered as directed or not (automatically set to False if g is undirected). weights : str or array, optional (default: binary) Whether to use weighted edges to compute the distances. By default, all edges are considered to have distance 1. combine_weights : str, optional (default: 'mean') How to combine the weights of reciprocal edges if the graph is directed but directed is set to False. It can be: * "sum": the sum of the edge attribute values will be used for the new edge. * "mean": the mean of the edge attribute values will be used for the new edge. * "min": the minimum of the edge attribute values will be used for the new edge. * "max": the maximum of the edge attribute values will be used for the new edge. Returns ------- path : list of ints Order of the nodes making up the path from source to target. References ---------- .. [nx-sp] :nxdoc:algorithms.shortest_paths.generic.shortest_path ''' g, graph, w = _get_nx_graph(g, directed, weights, combine_weights, return_all=True) w = _get_nx_weights(g, w) try: return nx.shortest_path(graph, source, target, weight=w) except nx.NetworkXNoPath: return []
[docs]def all_shortest_paths(g, source, target, directed=None, weights=None, combine_weights="mean"): ''' Yields all shortest paths from source to target. The algorithms returns an empty generator if there is no path between the nodes. Parameters ---------- g : :class:~nngt.Graph Graph to analyze. source : int Node from which the paths starts. target : int, optional (default: all nodes) Node where the paths ends. directed : bool, optional (default: g.is_directed()) Whether the edges should be considered as directed or not (automatically set to False if g is undirected). weights : str or array, optional (default: binary) Whether to use weighted edges to compute the distances. By default, all edges are considered to have distance 1. combine_weights : str, optional (default: 'mean') How to combine the weights of reciprocal edges if the graph is directed but directed is set to False. It can be: * "sum": the sum of the edge attribute values will be used for the new edge. * "mean": the mean of the edge attribute values will be used for the new edge. * "min": the minimum of the edge attribute values will be used for the new edge. * "max": the maximum of the edge attribute values will be used for the new edge. Returns ------- all_paths : generator Generator yielding paths as lists of ints. References ---------- .. [nx-sp] :nxdoc:algorithms.shortest_paths.generic.all_shortest_paths ''' g, graph, w = _get_nx_graph(g, directed, weights, combine_weights, return_all=True) w = _get_nx_weights(g, w) try: return nx.all_shortest_paths(graph, source, target, weight=w) except nx.NetworkXNoPath: return (_ for _ in [])
[docs]def shortest_distance(g, sources=None, targets=None, directed=None, weights=None, combine_weights="mean"): ''' Returns the length of the shortest paths between sourcesand targets. The algorithms return infinity if there are no paths between nodes. Parameters ---------- g : :class:~nngt.Graph Graph to analyze. sources : list of nodes, optional (default: all) Nodes from which the paths must be computed. targets : list of nodes, optional (default: all) Nodes to which the paths must be computed. directed : bool, optional (default: g.is_directed()) Whether the edges should be considered as directed or not (automatically set to False if g is undirected). weights : str or array, optional (default: binary) Whether to use weighted edges to compute the distances. By default, all edges are considered to have distance 1. combine_weights : str, optional (default: 'mean') How to combine the weights of reciprocal edges if the graph is directed but directed is set to False. It can be: * "sum": the sum of the edge attribute values will be used for the new edge. * "mean": the mean of the edge attribute values will be used for the new edge. * "min": the minimum of the edge attribute values will be used for the new edge. * "max": the maximum of the edge attribute values will be used for the new edge. Returns ------- distance : float, or 1d/2d numpy array of floats Distance (if single source and single target) or distance array. For multiple sources and targets, the shape of the matrix is (S, T), with S the number of sources and T the number of targets; for a single source or target, return a 1d-array of length T or S. References ---------- .. [nx-sp] :nxdoc:algorithms.shortest_paths.weighted.multi_source_dijkstra ''' num_nodes = g.node_nb() # check consistency for weights and directed g, graph, w = _get_nx_graph(g, directed, weights, combine_weights, return_all=True) w = _get_nx_weights(g, w) # check for single source/target case and convert sources and targets if is_integer(sources): if is_integer(targets): try: return nx.shortest_path_length(graph, sources, targets, weight=w) except Exception as e: return np.inf sources = [sources] elif sources is None: sources = range(num_nodes) if is_integer(targets): targets = [targets] # compute distances data, ii, jj = [], [], [] def _nx_sp(nx_graph, s, weight): if weight is None: return nx.single_source_shortest_path_length(nx_graph, s) dist, _ = nx.multi_source_dijkstra(graph, [s], weight=weight) return dist for s in sources: dist = _nx_sp(graph, s, w) if targets is None: data.extend(dist.values()) ii.extend((s for _ in range(len(dist)))) jj.extend(dist.keys()) else: for t in targets: if t in dist: data.append(dist[t]) ii.append(s) jj.append(t) num_sources = num_nodes if sources is None else len(sources) num_targets = num_nodes if targets is None else len(targets) mat_dist = np.full((num_sources, num_targets), np.inf) mat_dist[ii, jj] = data if num_sources == 1: return mat_dist[0] if num_targets == 1: return mat_dist.T[0] return mat_dist
[docs]def average_path_length(g, sources=None, targets=None, directed=None, weights=None, combine_weights="mean", unconnected=False): r''' Returns the average shortest path length between sources and targets. The algorithms raises an error if all nodes are not connected unless unconnected is set to True. The average path length is defined as .. math:: L = \frac{1}{N_p} \sum_{u,v} d(u, v), where :math:N_p is the number of paths between sources and targets, and :math:d(u, v) is the shortest path distance from u to v. If sources and targets are both None, then the total number of paths is :math:N_p = N(N - 1), with :math:N the number of nodes in the graph. Parameters ---------- g : :class:~nngt.Graph Graph to analyze. sources : list of nodes, optional (default: all) Nodes from which the paths must be computed. targets : list of nodes, optional (default: all) Nodes to which the paths must be computed. directed : bool, optional (default: g.is_directed()) Whether the edges should be considered as directed or not (automatically set to False if g is undirected). weights : str or array, optional (default: binary) Whether to use weighted edges to compute the distances. By default, all edges are considered to have distance 1. combine_weights : str, optional (default: 'mean') How to combine the weights of reciprocal edges if the graph is directed but directed is set to False. It can be: * "sum": the sum of the edge attribute values will be used for the new edge. * "mean": the mean of the edge attribute values will be used for the new edge. * "min": the minimum of the edge attribute values will be used for the new edge. * "max": the maximum of the edge attribute values will be used for the new edge. unconnected : bool, optional (default: False) If set to true, ignores unconnected nodes and returns the average path length of the existing paths. References ---------- .. [nx-sp] :nxdoc:algorithms.shortest_paths.generic.average_shortest_path_length ''' directed = g.is_directed() if directed is None else directed if sources is None and targets is None and not unconnected: g, graph, w = _get_nx_graph(g, directed, weights, combine_weights, return_all=True) w = _get_nx_weights(g, w) return nx.average_shortest_path_length(graph, weight=w) mat_dist = shortest_distance(g, sources=sources, targets=targets, directed=directed, weights=weights) if not unconnected and np.any(np.isinf(mat_dist)): raise nx.NetworkXNoPath("sources and target do not belong to the " "same connected component.") # compute the number of path num_paths = np.sum(mat_dist != 0) # compute average path length if unconnected: num_paths -= np.sum(np.isinf(mat_dist)) return np.nansum(mat_dist) / num_paths return np.sum(mat_dist) / num_paths
[docs]def diameter(g, directed=None, weights=None, combine_weights="mean", is_connected=False): ''' Returns the diameter of the graph. .. versionchanged:: 2.3 Added combine_weights argument. .. versionchanged:: 2.0 Added directed and is_connected arguments. It returns infinity if the graph is not connected (strongly connected for directed graphs) unless is_connected is True, in which case it returns the longest existing shortest distance. Parameters ---------- g : :class:~nngt.Graph Graph to analyze. directed : bool, optional (default: g.is_directed()) Whether to compute the directed diameter if the graph is directed. If False, then the graph is treated as undirected. The option switches to False automatically if g is undirected. weights : bool or str, optional (default: binary edges) Whether edge weights should be considered; if None or False then use binary edges; if True, uses the 'weight' edge attribute, otherwise uses any valid edge attribute required. combine_weights : str, optional (default: 'mean') How to combine the weights of reciprocal edges if the graph is directed but directed is set to False. It can be: * "sum": the sum of the edge attribute values will be used for the new edge. * "mean": the mean of the edge attribute values will be used for the new edge. * "min": the minimum of the edge attribute values will be used for the new edge. * "max": the maximum of the edge attribute values will be used for the new edge. is_connected : bool, optional (default: False) If False, check whether the graph is connected or not and return infinite diameter if graph is unconnected. If True, the graph is assumed to be connected. See also -------- :func:nngt.analysis.shortest_distance References ---------- .. [nx-diameter] :nxdoc:algorithms.distance_measures.diameter .. [nx-dijkstra] :nxdoc:algorithms.shortest_paths.weighted.all_pairs_dijkstra ''' w = _get_nx_weights(g, weights) # weighted or "connected" cases if w is not None or is_connected: dist = shortest_distance(g, directed=directed, weights=weights, combine_weights=combine_weights) if is_connected: return np.max(dist[~np.isinf(dist)]) return np.max(dist) # unweighted case graph = _get_nx_graph(g, directed, w, combine_weights) try: return nx.diameter(graph) except nx.exception.NetworkXError: return np.inf
def adj_mat(g, weights=None, mformat="csr"): r''' Returns the adjacency matrix :math:A of the graph. With edge :math:i \leftarrow j corresponding to entry :math:A_{ij}. Parameters ---------- g : :class:~nngt.Graph Graph to analyze. weights : bool or str, optional (default: binary edges) Whether edge weights should be considered; if None or False then returns the binary adjacency matrix; if True, returns the weighted matrix, otherwise fills the matrix with any valid edge attribute values. mformat : str, optional (default: "csr") Type of :mod:scipy.sparse matrix that will be returned, by default :class:scipy.sparse.csr_matrix. Returns ------- The adjacency matrix as a :class:scipy.sparse.csr_matrix. References ---------- .. [nx-adjacency] :nxdoc:.convert_matrix.to_scipy_sparse_matrix ''' w = _get_nx_weights(g, weights) return nx.to_scipy_sparse_matrix(g.graph, nodelist=range(g.node_nb()), weight=w, format=mformat) def get_edges(g): ''' Returns the edges in the graph by order of creation. ''' return g.edges_array