"""Functions for computing eigenvector centrality."""
from math import sqrt
import networkx as nx
from networkx.utils import not_implemented_for
__all__ = ["eigenvector_centrality", "eigenvector_centrality_numpy"]
[docs]@not_implemented_for("multigraph")
def eigenvector_centrality(G, max_iter=100, tol=1.0e-6, nstart=None, weight=None):
r"""Compute the eigenvector centrality for the graph `G`.
Eigenvector centrality computes the centrality for a node based on the
centrality of its neighbors. The eigenvector centrality for node $i$ is
the $i$-th element of the vector $x$ defined by the equation
.. math::
Ax = \lambda x
where $A$ is the adjacency matrix of the graph `G` with eigenvalue
$\lambda$. By virtue of the Perron–Frobenius theorem, there is a unique
solution $x$, all of whose entries are positive, if $\lambda$ is the
largest eigenvalue of the adjacency matrix $A$ ([2]_).
Parameters
----------
G : graph
A networkx graph
max_iter : integer, optional (default=100)
Maximum number of iterations in power method.
tol : float, optional (default=1.0e-6)
Error tolerance used to check convergence in power method iteration.
nstart : dictionary, optional (default=None)
Starting value of eigenvector iteration for each node.
weight : None or string, optional (default=None)
If None, all edge weights are considered equal.
Otherwise holds the name of the edge attribute used as weight.
Returns
-------
nodes : dictionary
Dictionary of nodes with eigenvector centrality as the value.
Examples
--------
>>> G = nx.path_graph(4)
>>> centrality = nx.eigenvector_centrality(G)
>>> sorted((v, f"{c:0.2f}") for v, c in centrality.items())
[(0, '0.37'), (1, '0.60'), (2, '0.60'), (3, '0.37')]
Raises
------
NetworkXPointlessConcept
If the graph `G` is the null graph.
NetworkXError
If each value in `nstart` is zero.
PowerIterationFailedConvergence
If the algorithm fails to converge to the specified tolerance
within the specified number of iterations of the power iteration
method.
See Also
--------
eigenvector_centrality_numpy
pagerank
hits
Notes
-----
The measure was introduced by [1]_ and is discussed in [2]_.
The power iteration method is used to compute the eigenvector and
convergence is **not** guaranteed. Our method stops after ``max_iter``
iterations or when the change in the computed vector between two
iterations is smaller than an error tolerance of
``G.number_of_nodes() * tol``. This implementation uses ($A + I$)
rather than the adjacency matrix $A$ because it shifts the spectrum
to enable discerning the correct eigenvector even for networks with
multiple dominant eigenvalues.
For directed graphs this is "left" eigenvector centrality which corresponds
to the in-edges in the graph. For out-edges eigenvector centrality
first reverse the graph with ``G.reverse()``.
References
----------
.. [1] Phillip Bonacich.
"Power and Centrality: A Family of Measures."
*American Journal of Sociology* 92(5):1170–1182, 1986
<http://www.leonidzhukov.net/hse/2014/socialnetworks/papers/Bonacich-Centrality.pdf>
.. [2] Mark E. J. Newman.
*Networks: An Introduction.*
Oxford University Press, USA, 2010, pp. 169.
"""
if len(G) == 0:
raise nx.NetworkXPointlessConcept(
"cannot compute centrality for the null graph"
)
# If no initial vector is provided, start with the all-ones vector.
if nstart is None:
nstart = {v: 1 for v in G}
if all(v == 0 for v in nstart.values()):
raise nx.NetworkXError("initial vector cannot have all zero values")
# Normalize the initial vector so that each entry is in [0, 1]. This is
# guaranteed to never have a divide-by-zero error by the previous line.
nstart_sum = sum(nstart.values())
x = {k: v / nstart_sum for k, v in nstart.items()}
nnodes = G.number_of_nodes()
# make up to max_iter iterations
for i in range(max_iter):
xlast = x
x = xlast.copy() # Start with xlast times I to iterate with (A+I)
# do the multiplication y^T = x^T A (left eigenvector)
for n in x:
for nbr in G[n]:
w = G[n][nbr].get(weight, 1) if weight else 1
x[nbr] += xlast[n] * w
# Normalize the vector. The normalization denominator `norm`
# should never be zero by the Perron--Frobenius
# theorem. However, in case it is due to numerical error, we
# assume the norm to be one instead.
norm = sqrt(sum(z ** 2 for z in x.values())) or 1
x = {k: v / norm for k, v in x.items()}
# Check for convergence (in the L_1 norm).
if sum(abs(x[n] - xlast[n]) for n in x) < nnodes * tol:
return x
raise nx.PowerIterationFailedConvergence(max_iter)
[docs]def eigenvector_centrality_numpy(G, weight=None, max_iter=50, tol=0):
r"""Compute the eigenvector centrality for the graph G.
Eigenvector centrality computes the centrality for a node based on the
centrality of its neighbors. The eigenvector centrality for node $i$ is
.. math::
Ax = \lambda x
where $A$ is the adjacency matrix of the graph G with eigenvalue $\lambda$.
By virtue of the Perron–Frobenius theorem, there is a unique and positive
solution if $\lambda$ is the largest eigenvalue associated with the
eigenvector of the adjacency matrix $A$ ([2]_).
Parameters
----------
G : graph
A networkx graph
weight : None or string, optional (default=None)
The name of the edge attribute used as weight.
If None, all edge weights are considered equal.
max_iter : integer, optional (default=100)
Maximum number of iterations in power method.
tol : float, optional (default=1.0e-6)
Relative accuracy for eigenvalues (stopping criterion).
The default value of 0 implies machine precision.
Returns
-------
nodes : dictionary
Dictionary of nodes with eigenvector centrality as the value.
Examples
--------
>>> G = nx.path_graph(4)
>>> centrality = nx.eigenvector_centrality_numpy(G)
>>> print([f"{node} {centrality[node]:0.2f}" for node in centrality])
['0 0.37', '1 0.60', '2 0.60', '3 0.37']
See Also
--------
eigenvector_centrality
pagerank
hits
Notes
-----
The measure was introduced by [1]_.
This algorithm uses the SciPy sparse eigenvalue solver (ARPACK) to
find the largest eigenvalue/eigenvector pair.
For directed graphs this is "left" eigenvector centrality which corresponds
to the in-edges in the graph. For out-edges eigenvector centrality
first reverse the graph with ``G.reverse()``.
Raises
------
NetworkXPointlessConcept
If the graph ``G`` is the null graph.
References
----------
.. [1] Phillip Bonacich:
Power and Centrality: A Family of Measures.
American Journal of Sociology 92(5):1170–1182, 1986
http://www.leonidzhukov.net/hse/2014/socialnetworks/papers/Bonacich-Centrality.pdf
.. [2] Mark E. J. Newman:
Networks: An Introduction.
Oxford University Press, USA, 2010, pp. 169.
"""
import numpy as np
import scipy as sp
from scipy.sparse import linalg
if len(G) == 0:
raise nx.NetworkXPointlessConcept(
"cannot compute centrality for the null graph"
)
M = nx.to_scipy_sparse_matrix(G, nodelist=list(G), weight=weight, dtype=float)
eigenvalue, eigenvector = linalg.eigs(
M.T, k=1, which="LR", maxiter=max_iter, tol=tol
)
largest = eigenvector.flatten().real
norm = np.sign(largest.sum()) * sp.linalg.norm(largest)
return dict(zip(G, largest / norm))