Note

This documents the development version of NetworkX. Documentation for the current release can be found here.

# 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

--------
attribute_assortativity_coefficient
numeric_assortativity_coefficient
neighbor_connectivity
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).
"""
M = degree_mixing_matrix(G, x=x, y=y, nodes=nodes, weight=weight)
return numeric_ac(M)

[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.stats as stats

xy = node_degree_xy(G, x=x, y=y, nodes=nodes, weight=weight)
x, y = zip(*xy)
return 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()
>>> 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.
The numeric attribute must be an integer.

Parameters
----------
G : NetworkX graph

attribute : string
Node attribute key.  The corresponding attribute value must be an
integer.

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()
>>> print(nx.numeric_assortativity_coefficient(G, "size"))
1.0

Notes
-----
This computes Eq. (21) in Ref. [1]_ , for the mixing matrix of
of the specified attribute.

References
----------
.. [1] M. E. J. Newman, Mixing patterns in networks
Physical Review E, 67 026126, 2003
"""
a = numeric_mixing_matrix(G, attribute, nodes)
return numeric_ac(a)

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):
# 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.arange(nx)
y = np.arange(ny)
a = M.sum(axis=0)
b = M.sum(axis=1)
vara = (a * x ** 2).sum() - ((a * x).sum()) ** 2
varb = (b * x ** 2).sum() - ((b * x).sum()) ** 2
xy = np.outer(x, y)
ab = np.outer(a, b)
return (xy * (M - ab)).sum() / np.sqrt(vara * varb)