Source code for networkx.algorithms.assortativity.correlation

"""Node assortativity coefficients and correlation measures.
"""
from networkx.algorithms.assortativity.mixing import (
    degree_mixing_matrix,
    attribute_mixing_matrix,
    numeric_mixing_matrix,
)
from networkx.algorithms.assortativity.pairs import node_degree_xy

__all__ = [
    "degree_pearson_correlation_coefficient",
    "degree_assortativity_coefficient",
    "attribute_assortativity_coefficient",
    "numeric_assortativity_coefficient",
]


[docs]def degree_assortativity_coefficient(G, x="out", y="in", weight=None, nodes=None): """Compute degree assortativity of graph. Assortativity measures the similarity of connections in the graph with respect to the node degree. Parameters ---------- G : NetworkX graph x: string ('in','out') The degree type for source node (directed graphs only). y: string ('in','out') The degree type for target node (directed graphs only). weight: string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. nodes: list or iterable (optional) Compute degree assortativity only for nodes in container. The default is all nodes. Returns ------- r : float Assortativity of graph by degree. Examples -------- >>> G = nx.path_graph(4) >>> r = nx.degree_assortativity_coefficient(G) >>> print(f"{r:3.1f}") -0.5 See Also -------- attribute_assortativity_coefficient numeric_assortativity_coefficient degree_mixing_dict degree_mixing_matrix Notes ----- This computes Eq. (21) in Ref. [1]_ , where e is the joint probability distribution (mixing matrix) of the degrees. If G is directed than the matrix e is the joint probability of the user-specified degree type for the source and target. References ---------- .. [1] M. E. J. Newman, Mixing patterns in networks, Physical Review E, 67 026126, 2003 .. [2] Foster, J.G., Foster, D.V., Grassberger, P. & Paczuski, M. Edge direction and the structure of networks, PNAS 107, 10815-20 (2010). """ if nodes is None: nodes = G.nodes degrees = set([d for n, d in G.degree(nodes, weight=weight)]) mapping = {d: i for i, d, in enumerate(degrees)} M = degree_mixing_matrix(G, x=x, y=y, nodes=nodes, weight=weight, mapping=mapping) return numeric_ac(M, mapping=mapping)
[docs]def degree_pearson_correlation_coefficient(G, x="out", y="in", weight=None, nodes=None): """Compute degree assortativity of graph. Assortativity measures the similarity of connections in the graph with respect to the node degree. This is the same as degree_assortativity_coefficient but uses the potentially faster scipy.stats.pearsonr function. Parameters ---------- G : NetworkX graph x: string ('in','out') The degree type for source node (directed graphs only). y: string ('in','out') The degree type for target node (directed graphs only). weight: string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. nodes: list or iterable (optional) Compute pearson correlation of degrees only for specified nodes. The default is all nodes. Returns ------- r : float Assortativity of graph by degree. Examples -------- >>> G = nx.path_graph(4) >>> r = nx.degree_pearson_correlation_coefficient(G) >>> print(f"{r:3.1f}") -0.5 Notes ----- This calls scipy.stats.pearsonr. References ---------- .. [1] M. E. J. Newman, Mixing patterns in networks Physical Review E, 67 026126, 2003 .. [2] Foster, J.G., Foster, D.V., Grassberger, P. & Paczuski, M. Edge direction and the structure of networks, PNAS 107, 10815-20 (2010). """ import scipy as sp import scipy.stats # call as sp.stats xy = node_degree_xy(G, x=x, y=y, nodes=nodes, weight=weight) x, y = zip(*xy) return sp.stats.pearsonr(x, y)[0]
[docs]def attribute_assortativity_coefficient(G, attribute, nodes=None): """Compute assortativity for node attributes. Assortativity measures the similarity of connections in the graph with respect to the given attribute. Parameters ---------- G : NetworkX graph attribute : string Node attribute key nodes: list or iterable (optional) Compute attribute assortativity for nodes in container. The default is all nodes. Returns ------- r: float Assortativity of graph for given attribute Examples -------- >>> G = nx.Graph() >>> G.add_nodes_from([0, 1], color="red") >>> G.add_nodes_from([2, 3], color="blue") >>> G.add_edges_from([(0, 1), (2, 3)]) >>> print(nx.attribute_assortativity_coefficient(G, "color")) 1.0 Notes ----- This computes Eq. (2) in Ref. [1]_ , (trace(M)-sum(M^2))/(1-sum(M^2)), where M is the joint probability distribution (mixing matrix) of the specified attribute. References ---------- .. [1] M. E. J. Newman, Mixing patterns in networks, Physical Review E, 67 026126, 2003 """ M = attribute_mixing_matrix(G, attribute, nodes) return attribute_ac(M)
[docs]def numeric_assortativity_coefficient(G, attribute, nodes=None): """Compute assortativity for numerical node attributes. Assortativity measures the similarity of connections in the graph with respect to the given numeric attribute. Parameters ---------- G : NetworkX graph attribute : string Node attribute key. nodes: list or iterable (optional) Compute numeric assortativity only for attributes of nodes in container. The default is all nodes. Returns ------- r: float Assortativity of graph for given attribute Examples -------- >>> G = nx.Graph() >>> G.add_nodes_from([0, 1], size=2) >>> G.add_nodes_from([2, 3], size=3) >>> G.add_edges_from([(0, 1), (2, 3)]) >>> print(nx.numeric_assortativity_coefficient(G, "size")) 1.0 Notes ----- This computes Eq. (21) in Ref. [1]_ , for the mixing matrix of the specified attribute. References ---------- .. [1] M. E. J. Newman, Mixing patterns in networks Physical Review E, 67 026126, 2003 """ if nodes is None: nodes = G.nodes vals = set(G.nodes[n][attribute] for n in nodes) mapping = {d: i for i, d, in enumerate(vals)} M = attribute_mixing_matrix(G, attribute, nodes, mapping) return numeric_ac(M, mapping)
def attribute_ac(M): """Compute assortativity for attribute matrix M. Parameters ---------- M : numpy.ndarray 2D ndarray representing the attribute mixing matrix. Notes ----- This computes Eq. (2) in Ref. [1]_ , (trace(e)-sum(e^2))/(1-sum(e^2)), where e is the joint probability distribution (mixing matrix) of the specified attribute. References ---------- .. [1] M. E. J. Newman, Mixing patterns in networks, Physical Review E, 67 026126, 2003 """ if M.sum() != 1.0: M = M / M.sum() s = (M @ M).sum() t = M.trace() r = (t - s) / (1 - s) return r def numeric_ac(M, mapping): # M is a numpy matrix or array # numeric assortativity coefficient, pearsonr import numpy as np if M.sum() != 1.0: M = M / float(M.sum()) nx, ny = M.shape # nx=ny x = np.array(list(mapping.keys())) y = x # x and y have the same support idx = list(mapping.values()) a = M.sum(axis=0) b = M.sum(axis=1) vara = (a[idx] * x ** 2).sum() - ((a[idx] * x).sum()) ** 2 varb = (b[idx] * y ** 2).sum() - ((b[idx] * y).sum()) ** 2 xy = np.outer(x, y) ab = np.outer(a[idx], b[idx]) return (xy * (M - ab)).sum() / np.sqrt(vara * varb)