https://github.com/kit-parco/networkit
Tip revision: 0d7ba4f0d7f0358437ec203f49c3c1515a16243d authored by cls on 07 September 2015, 10:35:15 UTC
close branch 'profiling' after merge -> Dev
close branch 'profiling' after merge -> Dev
Tip revision: 0d7ba4f
centrality.py
""" This module contains algorithms for the calculation of centrality, i.e. ranking nodes by their structural importance
to the network """
__author__ = "Christian Staudt"
__credits__ = ["Christian Staudt", "Elisabetta Bergamini", "Henning Meyerhenke", "Marc Nemes"]
# extension imports
# TODO: (+) ApproxCloseness
from _NetworKit import Betweenness, PageRank, EigenvectorCentrality, DegreeCentrality, ApproxBetweenness, ApproxBetweenness2, DynApproxBetweenness, Closeness, KPathCentrality, CoreDecomposition, KatzCentrality, LocalClusteringCoefficient, ApproxCloseness
# local imports
from networkit.algebraic import adjacencyEigenvector, PageRankMatrix, symmetricEigenvectors
# external imports
import math
def ranking(G, algorithm=Betweenness, normalized=False):
""" Return a ranking of nodes by the specified centrality type"""
# FIXME: some centrality algorithms take more parameters
centrality = algorithm(G, normalized)
centrality.run()
return centrality.ranking()
def scores(G, algorithm=Betweenness, normalized=False):
""" Return the centrality scores of nodes using the specified centrality type"""
centrality = algorithm(G, normalized)
centrality.run()
return centrality.scores()
def centralization(G, centralityMeasure):
"""
Compute the centralization of a network with respect to some centrality measure.
The centralization of any network is a measure of how central its most central
node is in relation to how central all the other nodes are.
Centralization measures then (a) calculate the sum in differences
in centrality between the most central node in a network and all other nodes;
and (b) divide this quantity by the theoretically largest such sum of
differences in any network of the same size.
Parameters
----------
G : graph
The graph of which to compute the centrality
centralityMeasure : instance of Centrality (sub)class
initialized algorithm object will be run
Returns
-------
double
centralization score
"""
centralityMeasure.run()
ranking = centralityMeasure.ranking()
(center, centerScore) = ranking[0]
maxScore = centralityMeasure.maximum()
diff1 = sum([(centerScore - c) for (u, c) in ranking])
diff2 = sum([(maxScore - c) for (u, c) in ranking])
return diff1 / diff2
def rankPerNode(ranking):
"""
Parameters
----------
ranking: ordered list of tuples (node, score)
Returns
_______
for each node (sorted by node ID), the ranking of the node
"""
n_nodes = len(ranking)
ranking_id = [0]*n_nodes
for index, pair in enumerate(ranking):
ranking_id[pair[0]] = index
#we assign to all nodes the ranking of the first node with the same score
for index, pair in enumerate(ranking):
if index == 0:
continue
if pair[1] == ranking[index-1][1]:
prev_node = ranking[index-1][0]
ranking_id[pair[0]] = ranking_id[prev_node]
return ranking_id
def relativeRankErrors(rx, ry):
"""
Let $r_x(u)$ be the rank of node $u$ in ranking $x$.
The relative rank error of node $u$ is defined as
$$r_x(u) / r_y(u)$$
Parameters
----------
rx : list
ranking - ordered list of tuples (node, score)
ry: list
ranking - ordered list of tuples (node, score)
Returns
_______
list of rank errors ordered by node ID
"""
diff = []
n = len(rx)
if not(n == len(ry)):
return diff
rnode_x = rankPerNode(rx)
rnode_y = rankPerNode(ry)
for i in range(n):
diff.append((rnode_x[i]+1)/(rnode_y[i]+1))
return diff
class SpectralCentrality:
"""
Abstract class to compute the spectral centrality of a graph. This class needs to be supplied with methods
to generate the correct matrices and do the correct normalization.
"""
def __init__(self, G, normalized=False):
"""
Constructor.
Parameters
----------
G : graph
The graph of which to compute the centrality
normalized : boolean
Whether to normalize the results or not
"""
super(SpectralCentrality, self).__init__()
self.graph = G
self.normalized = normalized
self.scoreList = None
self.rankList = None
self.evz = {}
def prepareSpectrum(self):
""" Method that must be implemented to set the following values:
self.eigenvectors = list of eigenvectors desired for centrality measure
self.eigenvalues = list of corresponding eigenvalues
"""
raise NotImplemented
def normFactor(self):
""" Method that must be implemented to return a correct normalization factor"""
raise NotImplemented
def run(self):
self.prepareSpectrum()
self.scoreList = None
self.rankList = None
self.evz = {}
if self.normalized:
normFactor = self.normFactor()
else:
normFactor = 1
for v in self.graph.nodes():
self.evz[v] = self.eigenvector[v] * normFactor
return self
def scores(self):
if self.scoreList is None:
self.scoreList = [v for k,v in self.evz.items()]
return self.scoreList
def ranking(self):
if self.rankList is None:
self.rankList = sorted(self.evz.items(),key=lambda x: float(x[1]), reverse=True)
return self.rankList
class SciPyEVZ(SpectralCentrality):
"""
Compute Eigenvector centrality using algebraic meh
Parameters
----------
G : graph
The graph of which to compute the centrality
normalized : boolean
Whether to normalize the results or not
"""
def __init__(self, G, normalized=False):
if G.isDirected():
raise NotImplementedError("Not implemented for directed graphs; use centrality.EigenvectorCentrality instead")
super(SciPyEVZ, self).__init__(G, normalized=normalized)
def _length(self, vector):
square = sum([val * val for val in vector])
return math.sqrt(square)
def normFactor(self):
return 1 / self._length(self.eigenvector)
def prepareSpectrum(self):
spectrum = adjacencyEigenvector(self.graph, order=0)
self.eigenvector = spectrum[1]
self.eigenvalue = spectrum[0]
class SciPyPageRank(SpectralCentrality):
# TODO: docstring
def __init__(self, G, damp=0.95, normalized=False):
super(SciPyPageRank, self).__init__(G, normalized=normalized)
self.damp = damp
def _length(self, vector):
return sum(vector)
def normFactor(self):
return 1 / self._length(self.eigenvector)
def prepareSpectrum(self):
prMatrix = PageRankMatrix(self.graph, self.damp)
spectrum = symmetricEigenvectors(prMatrix, cutoff=0, reverse=False)
self.eigenvector = spectrum[1][0]
self.eigenvalue = spectrum[0][0]