245 lines
8.4 KiB
Python
245 lines
8.4 KiB
Python
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# -*- coding: utf-8 -*-
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# Copyright (C) 2004-2019 by
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# Aric Hagberg <hagberg@lanl.gov>
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# Dan Schult <dschult@colgate.edu>
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# Pieter Swart <swart@lanl.gov>
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# All rights reserved.
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# BSD license.
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#
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# Authors:
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# Aric Hagberg <aric.hagberg@gmail.com>
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# Pieter Swart <swart@lanl.gov>
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# Sasha Gutfraind <ag362@cornell.edu>
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"""Functions for computing eigenvector centrality."""
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from math import sqrt
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import networkx as nx
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from networkx.utils import not_implemented_for
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__all__ = ['eigenvector_centrality', 'eigenvector_centrality_numpy']
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@not_implemented_for('multigraph')
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def eigenvector_centrality(G, max_iter=100, tol=1.0e-6, nstart=None,
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weight=None):
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r"""Compute the eigenvector centrality for the graph `G`.
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Eigenvector centrality computes the centrality for a node based on the
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centrality of its neighbors. The eigenvector centrality for node $i$ is
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the $i$-th element of the vector $x$ defined by the equation
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.. math::
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Ax = \lambda x
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where $A$ is the adjacency matrix of the graph `G` with eigenvalue
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$\lambda$. By virtue of the Perron–Frobenius theorem, there is a unique
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solution $x$, all of whose entries are positive, if $\lambda$ is the
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largest eigenvalue of the adjacency matrix $A$ ([2]_).
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Parameters
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----------
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G : graph
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A networkx graph
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max_iter : integer, optional (default=100)
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Maximum number of iterations in power method.
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tol : float, optional (default=1.0e-6)
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Error tolerance used to check convergence in power method iteration.
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nstart : dictionary, optional (default=None)
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Starting value of eigenvector iteration for each node.
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weight : None or string, optional (default=None)
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If None, all edge weights are considered equal.
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Otherwise holds the name of the edge attribute used as weight.
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Returns
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-------
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nodes : dictionary
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Dictionary of nodes with eigenvector centrality as the value.
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Examples
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--------
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>>> G = nx.path_graph(4)
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>>> centrality = nx.eigenvector_centrality(G)
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>>> sorted((v, '{:0.2f}'.format(c)) for v, c in centrality.items())
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[(0, '0.37'), (1, '0.60'), (2, '0.60'), (3, '0.37')]
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Raises
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------
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NetworkXPointlessConcept
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If the graph `G` is the null graph.
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NetworkXError
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If each value in `nstart` is zero.
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PowerIterationFailedConvergence
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If the algorithm fails to converge to the specified tolerance
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within the specified number of iterations of the power iteration
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method.
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See Also
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--------
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eigenvector_centrality_numpy
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pagerank
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hits
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Notes
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-----
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The measure was introduced by [1]_ and is discussed in [2]_.
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The power iteration method is used to compute the eigenvector and
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convergence is **not** guaranteed. Our method stops after ``max_iter``
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iterations or when the change in the computed vector between two
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iterations is smaller than an error tolerance of
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``G.number_of_nodes() * tol``. This implementation uses ($A + I$)
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rather than the adjacency matrix $A$ because it shifts the spectrum
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to enable discerning the correct eigenvector even for networks with
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multiple dominant eigenvalues.
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For directed graphs this is "left" eigenvector centrality which corresponds
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to the in-edges in the graph. For out-edges eigenvector centrality
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first reverse the graph with ``G.reverse()``.
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References
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----------
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.. [1] Phillip Bonacich.
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"Power and Centrality: A Family of Measures."
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*American Journal of Sociology* 92(5):1170–1182, 1986
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<http://www.leonidzhukov.net/hse/2014/socialnetworks/papers/Bonacich-Centrality.pdf>
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.. [2] Mark E. J. Newman.
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*Networks: An Introduction.*
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Oxford University Press, USA, 2010, pp. 169.
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"""
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if len(G) == 0:
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raise nx.NetworkXPointlessConcept('cannot compute centrality for the'
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' null graph')
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# If no initial vector is provided, start with the all-ones vector.
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if nstart is None:
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nstart = {v: 1 for v in G}
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if all(v == 0 for v in nstart.values()):
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raise nx.NetworkXError('initial vector cannot have all zero values')
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# Normalize the initial vector so that each entry is in [0, 1]. This is
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# guaranteed to never have a divide-by-zero error by the previous line.
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nstart_sum = sum(nstart.values())
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x = {k: v / nstart_sum for k, v in nstart.items()}
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nnodes = G.number_of_nodes()
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# make up to max_iter iterations
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for i in range(max_iter):
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xlast = x
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x = xlast.copy() # Start with xlast times I to iterate with (A+I)
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# do the multiplication y^T = x^T A (left eigenvector)
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for n in x:
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for nbr in G[n]:
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w = G[n][nbr].get(weight, 1) if weight else 1
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x[nbr] += xlast[n] * w
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# Normalize the vector. The normalization denominator `norm`
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# should never be zero by the Perron--Frobenius
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# theorem. However, in case it is due to numerical error, we
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# assume the norm to be one instead.
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norm = sqrt(sum(z ** 2 for z in x.values())) or 1
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x = {k: v / norm for k, v in x.items()}
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# Check for convergence (in the L_1 norm).
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if sum(abs(x[n] - xlast[n]) for n in x) < nnodes * tol:
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return x
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raise nx.PowerIterationFailedConvergence(max_iter)
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def eigenvector_centrality_numpy(G, weight=None, max_iter=50, tol=0):
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r"""Compute the eigenvector centrality for the graph G.
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Eigenvector centrality computes the centrality for a node based on the
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centrality of its neighbors. The eigenvector centrality for node $i$ is
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.. math::
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Ax = \lambda x
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where $A$ is the adjacency matrix of the graph G with eigenvalue $\lambda$.
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By virtue of the Perron–Frobenius theorem, there is a unique and positive
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solution if $\lambda$ is the largest eigenvalue associated with the
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eigenvector of the adjacency matrix $A$ ([2]_).
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Parameters
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----------
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G : graph
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A networkx graph
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weight : None or string, optional (default=None)
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The name of the edge attribute used as weight.
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If None, all edge weights are considered equal.
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max_iter : integer, optional (default=100)
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Maximum number of iterations in power method.
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tol : float, optional (default=1.0e-6)
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Relative accuracy for eigenvalues (stopping criterion).
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The default value of 0 implies machine precision.
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Returns
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-------
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nodes : dictionary
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Dictionary of nodes with eigenvector centrality as the value.
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Examples
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--------
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>>> G = nx.path_graph(4)
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>>> centrality = nx.eigenvector_centrality_numpy(G)
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>>> print(['{} {:0.2f}'.format(node, centrality[node]) for node in centrality])
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['0 0.37', '1 0.60', '2 0.60', '3 0.37']
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See Also
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--------
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eigenvector_centrality
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pagerank
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hits
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Notes
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-----
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The measure was introduced by [1]_.
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This algorithm uses the SciPy sparse eigenvalue solver (ARPACK) to
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find the largest eigenvalue/eigenvector pair.
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For directed graphs this is "left" eigenvector centrality which corresponds
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to the in-edges in the graph. For out-edges eigenvector centrality
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first reverse the graph with ``G.reverse()``.
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Raises
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------
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NetworkXPointlessConcept
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If the graph ``G`` is the null graph.
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References
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----------
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.. [1] Phillip Bonacich:
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Power and Centrality: A Family of Measures.
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American Journal of Sociology 92(5):1170–1182, 1986
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http://www.leonidzhukov.net/hse/2014/socialnetworks/papers/Bonacich-Centrality.pdf
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.. [2] Mark E. J. Newman:
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Networks: An Introduction.
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Oxford University Press, USA, 2010, pp. 169.
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"""
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import scipy as sp
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from scipy.sparse import linalg
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if len(G) == 0:
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raise nx.NetworkXPointlessConcept('cannot compute centrality for the'
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' null graph')
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M = nx.to_scipy_sparse_matrix(G, nodelist=list(G), weight=weight,
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dtype=float)
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eigenvalue, eigenvector = linalg.eigs(M.T, k=1, which='LR',
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maxiter=max_iter, tol=tol)
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largest = eigenvector.flatten().real
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norm = sp.sign(largest.sum()) * sp.linalg.norm(largest)
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return dict(zip(G, largest / norm))
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# fixture for pytest
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def setup_module(module):
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import pytest
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scipy = pytest.importorskip('scipy')
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