"""Functions for computing eigenvector centrality."""
import math
import networkx as nx
from networkx.utils import not_implemented_for
__all__ = ["eigenvector_centrality", "eigenvector_centrality_numpy"]
[docs]
@not_implemented_for("multigraph")
@nx._dispatchable(edge_attrs="weight")
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 by adding
the centrality of its predecessors. The centrality for node $i$ is the
$i$-th element of a left eigenvector associated with the eigenvalue $\lambda$
of maximum modulus that is positive. Such an eigenvector $x$ is
defined up to a multiplicative constant by the equation
.. math::
\lambda x^T = x^T A,
where $A$ is the adjacency matrix of the graph G. By definition of
row-column product, the equation above is equivalent to
.. math::
\lambda x_i = \sum_{j\to i}x_j.
That is, adding the eigenvector centralities of the predecessors of
$i$ one obtains the eigenvector centrality of $i$ multiplied by
$\lambda$. In the case of undirected graphs, $x$ also solves the familiar
right-eigenvector equation $Ax = \lambda x$.
By virtue of the Perron–Frobenius theorem [1]_, if G is strongly
connected there is a unique eigenvector $x$, and all its entries
are strictly positive.
If G is not strongly connected there might be several left
eigenvectors associated with $\lambda$, and some of their elements
might be zero.
Parameters
----------
G : graph
A networkx graph.
max_iter : integer, optional (default=100)
Maximum number of power iterations.
tol : float, optional (default=1.0e-6)
Error tolerance (in Euclidean norm) used to check convergence in
power iteration.
nstart : dictionary, optional (default=None)
Starting value of power iteration for each node. Must have a nonzero
projection on the desired eigenvector for the power method to converge.
If None, this implementation uses an all-ones vector, which is a safe
choice.
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. In this measure the
weight is interpreted as the connection strength.
Returns
-------
nodes : dictionary
Dictionary of nodes with eigenvector centrality as the value. The
associated vector has unit Euclidean norm and the values are
nonegative.
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
:func:`~networkx.algorithms.link_analysis.pagerank_alg.pagerank`
:func:`~networkx.algorithms.link_analysis.hits_alg.hits`
Notes
-----
Eigenvector centrality was introduced by Landau [2]_ for chess
tournaments. It was later rediscovered by Wei [3]_ and then
popularized by Kendall [4]_ in the context of sport ranking. Berge
introduced a general definition for graphs based on social connections
[5]_. Bonacich [6]_ reintroduced again eigenvector centrality and made
it popular in link analysis.
This function computes the left dominant eigenvector, which corresponds
to adding the centrality of predecessors: this is the usual approach.
To add the centrality of successors first reverse the graph with
``G.reverse()``.
The implementation uses power iteration [7]_ to compute a dominant
eigenvector starting from the provided vector `nstart`. Convergence is
guaranteed as long as `nstart` has a nonzero projection on a dominant
eigenvector, which certainly happens using the default value.
The method stops when the change in the computed vector between two
iterations is smaller than an error tolerance of ``G.number_of_nodes()
* tol`` or after ``max_iter`` iterations, but in the second case it
raises an exception.
This implementation uses $(A + I)$ rather than the adjacency matrix
$A$ because the change preserves eigenvectors, but it shifts the
spectrum, thus guaranteeing convergence even for networks with
negative eigenvalues of maximum modulus.
References
----------
.. [1] Abraham Berman and Robert J. Plemmons.
"Nonnegative Matrices in the Mathematical Sciences."
Classics in Applied Mathematics. SIAM, 1994.
.. [2] Edmund Landau.
"Zur relativen Wertbemessung der Turnierresultate."
Deutsches Wochenschach, 11:366–369, 1895.
.. [3] Teh-Hsing Wei.
"The Algebraic Foundations of Ranking Theory."
PhD thesis, University of Cambridge, 1952.
.. [4] Maurice G. Kendall.
"Further contributions to the theory of paired comparisons."
Biometrics, 11(1):43–62, 1955.
https://www.jstor.org/stable/3001479
.. [5] Claude Berge
"Théorie des graphes et ses applications."
Dunod, Paris, France, 1958.
.. [6] Phillip Bonacich.
"Technique for analyzing overlapping memberships."
Sociological Methodology, 4:176–185, 1972.
https://www.jstor.org/stable/270732
.. [7] Power iteration:: https://en.wikipedia.org/wiki/Power_iteration
"""
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 _ 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 = math.hypot(*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]
@nx._dispatchable(edge_attrs="weight")
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 by adding
the centrality of its predecessors. The centrality for node $i$ is the
$i$-th element of a left eigenvector associated with the eigenvalue $\lambda$
of maximum modulus that is positive. Such an eigenvector $x$ is
defined up to a multiplicative constant by the equation
.. math::
\lambda x^T = x^T A,
where $A$ is the adjacency matrix of the graph `G`. By definition of
row-column product, the equation above is equivalent to
.. math::
\lambda x_i = \sum_{j\to i}x_j.
That is, adding the eigenvector centralities of the predecessors of
$i$ one obtains the eigenvector centrality of $i$ multiplied by
$\lambda$. In the case of undirected graphs, $x$ also solves the familiar
right-eigenvector equation $Ax = \lambda x$.
By virtue of the Perron--Frobenius theorem [1]_, if `G` is (strongly)
connected, there is a unique eigenvector $x$, and all its entries
are strictly positive.
However, if `G` is not (strongly) connected, there might be several left
eigenvectors associated with $\lambda$, and some of their elements
might be zero.
Depending on the method used to choose eigenvectors, round-off error can affect
which of the infinitely many eigenvectors is reported.
This can lead to inconsistent results for the same graph,
which the underlying implementation is not robust to.
For this reason, only (strongly) connected graphs are accepted.
Parameters
----------
G : graph
A connected NetworkX graph.
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. In this measure the
weight is interpreted as the connection strength.
max_iter : integer, optional (default=50)
Maximum number of Arnoldi update iterations allowed.
tol : float, optional (default=0)
Relative accuracy for eigenvalues (stopping criterion).
The default value of 0 implies machine precision.
Returns
-------
nodes : dict of nodes
Dictionary of nodes with eigenvector centrality as the value. The
associated vector has unit Euclidean norm and the values are
nonnegative.
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']
Raises
------
NetworkXPointlessConcept
If the graph `G` is the null graph.
ArpackNoConvergence
When the requested convergence is not obtained. The currently
converged eigenvalues and eigenvectors can be found as
eigenvalues and eigenvectors attributes of the exception object.
AmbiguousSolution
If `G` is not connected.
See Also
--------
:func:`scipy.sparse.linalg.eigs`
eigenvector_centrality
:func:`~networkx.algorithms.link_analysis.pagerank_alg.pagerank`
:func:`~networkx.algorithms.link_analysis.hits_alg.hits`
Notes
-----
Eigenvector centrality was introduced by Landau [2]_ for chess
tournaments. It was later rediscovered by Wei [3]_ and then
popularized by Kendall [4]_ in the context of sport ranking. Berge
introduced a general definition for graphs based on social connections
[5]_. Bonacich [6]_ reintroduced again eigenvector centrality and made
it popular in link analysis.
This function computes the left dominant eigenvector, which corresponds
to adding the centrality of predecessors: this is the usual approach.
To add the centrality of successors first reverse the graph with
``G.reverse()``.
This implementation uses the
:func:`SciPy sparse eigenvalue solver<scipy.sparse.linalg.eigs>` (ARPACK)
to find the largest eigenvalue/eigenvector pair using Arnoldi iterations
[7]_.
References
----------
.. [1] Abraham Berman and Robert J. Plemmons.
"Nonnegative Matrices in the Mathematical Sciences".
Classics in Applied Mathematics. SIAM, 1994.
.. [2] Edmund Landau.
"Zur relativen Wertbemessung der Turnierresultate".
Deutsches Wochenschach, 11:366--369, 1895.
.. [3] Teh-Hsing Wei.
"The Algebraic Foundations of Ranking Theory".
PhD thesis, University of Cambridge, 1952.
.. [4] Maurice G. Kendall.
"Further contributions to the theory of paired comparisons".
Biometrics, 11(1):43--62, 1955.
https://www.jstor.org/stable/3001479
.. [5] Claude Berge.
"Théorie des graphes et ses applications".
Dunod, Paris, France, 1958.
.. [6] Phillip Bonacich.
"Technique for analyzing overlapping memberships".
Sociological Methodology, 4:176--185, 1972.
https://www.jstor.org/stable/270732
.. [7] Arnoldi, W. E. (1951).
"The principle of minimized iterations in the solution of the matrix eigenvalue problem".
Quarterly of Applied Mathematics. 9 (1): 17--29.
https://doi.org/10.1090/qam/42792
"""
import numpy as np
import scipy as sp
if len(G) == 0:
raise nx.NetworkXPointlessConcept(
"cannot compute centrality for the null graph"
)
connected = nx.is_strongly_connected(G) if G.is_directed() else nx.is_connected(G)
if not connected: # See gh-6888.
raise nx.AmbiguousSolution(
"`eigenvector_centrality_numpy` does not give consistent results for disconnected graphs"
)
M = nx.to_scipy_sparse_array(G, nodelist=list(G), weight=weight, dtype=float)
_, eigenvector = sp.sparse.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).tolist()))
