#
#
# multired.py
#
#
# Copyright (C) 2015 Vincenzo (Enzo) Nicosia <katolaz@yahoo.it>
#
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# long with this program. If not, see <http://www.gnu.org/licenses/>.
#
#
#
# This module provides the class multiplex_red, which implements the
# algorithm for the structural reduction of multi-layer networks
# based on the Von Neumann entropy and Quantum Jensen-Shannon
# divergence of graphs.
#
# If you use this code please cite the paper:
#
# M. De Domenico, V. Nicosia, A. Arenas, V. Latora,
# "Structural reducibility of multilayer networks"
# Nat. Commun. 6, 6864 (2015) doi:10.1038/ncomms7864
#
# --------------------------------------------
#
# -- 2015/04/23 -- release 0.1
# -- 2015/05/11 -- release 0.1.1 -- removed the last full matrices
#
import sys
import math
import numpy as np
from scipy.sparse import csr_matrix, eye
from scipy.linalg import eigh, eig
import copy
from scipy.cluster.hierarchy import linkage, dendrogram
has_matplotlib = False
try:
import matplotlib
has_matplotlib = True
except ImportError:
has_matplotlib = False
class XLogx_fit:
def __init__(self, degree, npoints= 100, xmax=1):
if xmax > 1:
xmax = 1
self.degree = degree
x = np.linspace(0, xmax, npoints)
y = [i * math.log(i) for i in x[1:]]
y.insert(0, 0)
self.fit = np.polyfit(x, y, degree)
def __getitem__ (self, index):
if index <= self.degree:
return self.fit[index]
else:
print "Error!!! Index %d is larger than the degree of the fitting polynomial (%d)" \
% (index, degree)
sys.exit(-1)
class layer:
def __init__ (self, layerfile= None, matrix=None):
self.N = 0
self.num_layer = -1
self.fname = layerfile
self.adj_matr = None
self.laplacian = None
self.resc_laplacian = None
self.entropy = None
self.entropy_approx = None
self._ii = []
self._jj = []
self._ww = []
self._matrix_called = False
if layerfile != None:
try:
min_N = 10e10
with open(layerfile, "r") as lines:
for l in lines:
if l[0] == '#':
continue
elems = l.strip(" \n").split(" ")
s = int(elems[0])
d = int(elems[1])
self._ii.append(s)
self._jj.append(d)
if s > self.N:
self.N = s
if d > self.N:
self.N = d
if s < min_N:
min_N = s
if d < min_N:
min_N = d
if len(elems) >2 : ## A weight is specified
val = [float(x) if "e" in x or "." in x else int(x) for x in [elems[2]]][0]
self._ww.append(float(val))
else:
self._ww.append(int(1))
except (IOError):
print "Unable to find/open file %s -- Exiting!!!" % layerfile
exit(-2)
elif matrix != None:
self.adj_matr = copy.copy(matrix)
self.N, _x = matrix.shape
#K = np.multiply(self.adj_matr.sum(0), np.ones((self.N,self.N)))
#D = np.diag(np.diag(K))
K = self.adj_matr.sum(0)
D = csr_matrix((self.N, self.N))
D.setdiag(eye(self.N) * K.transpose())
self.laplacian = csr_matrix(D - self.adj_matr)
K = self.laplacian.diagonal().sum()
self.resc_laplacian = csr_matrix(self.laplacian / K)
self._matrix_called = True
else:
print "The given matrix is BLANK"
def make_matrices(self, N):
self.N = N
self.adj_matr = csr_matrix((self._ww, (self._ii, self._jj)), shape=(self.N, self.N))
self.adj_matr = self.adj_matr + self.adj_matr.transpose()
#K = np.multiply(self.adj_matr.sum(0), np.ones((self.N,self.N)))
#D = np.diag(np.diag(K))
K = self.adj_matr.sum(0)
D = csr_matrix((self.N, self.N))
D.setdiag(eye(self.N) * K.transpose())
self.laplacian = csr_matrix(D - self.adj_matr)
K = self.laplacian.diagonal().sum()
self.resc_laplacian = csr_matrix(self.laplacian / K)
self._matrix_called = True
def dump_info(self):
N, M = self.adj_matr.shape
K = self.adj_matr.nnz
sys.stderr.write("Layer File: %s\nNodes: %d Edges: %d\nEntropy: %g Approx. Entropy: %g\n" % \
(self.fname, N, K, self.entropy, self.entropy_approx) )
def compute_VN_entropy(self):
eigvals = eigh(self.resc_laplacian.todense())
self.entropy = 0
for l_i in eigvals[0]:
if (l_i > 10e-20):
self.entropy -= l_i * math.log (l_i)
def compute_VN_entropy_approx(self, poly):
p = poly.degree
h = - poly[p] * self.N
M = csr_matrix(np.eye(self.N))
for i in range(p-1, -1, -1):
M = M * self.resc_laplacian
h += - poly[i] * sum(M.diagonal())
self.entropy_approx = h
def aggregate(self, other_layer):
if self.adj_matr != None:
self.adj_matr = self.adj_matr + other_layer.adj_matr
else:
self.adj_matr = copy.copy(other_layer.adj_matr)
#K = np.multiply(self.adj_matr.sum(0), np.ones((self.N,self.N)))
#D = np.diag(np.diag(K))
K = self.adj_matr.sum(0)
D = csr_matrix((self.N, self.N))
D.setdiag(eye(self.N) * K. transpose())
self.laplacian = csr_matrix(D - self.adj_matr)
K = self.laplacian.diagonal().sum()
self.resc_laplacian = csr_matrix(self.laplacian / K)
self._matrix_called = True
def dump_laplacian(self):
print self.laplacian
class multiplex_red:
def __init__ (self, multiplexfile, directed = None, fit_degree=10, verbose=False):
self.layers = []
self.N = 0
self.M = 0
self.entropy = 0
self.entropy_approx = 0
self.JSD = None
self.JSD_approx = None
self.Z = None
self.Z_approx = None
self.aggr = None
self.q_vals = None
self.q_vals_approx = None
self.fit_degree = fit_degree
self.poly = XLogx_fit(self.fit_degree)
self.verb = verbose
self.cuts = None
self.cuts_approx = None
try:
with open(multiplexfile, "r") as lines:
for l in lines:
if (self.verb):
sys.stderr.write("Loading layer %d from file %s" % (len(self.layers), l))
A = layer(l.strip(" \n"))
if A.N > self.N:
self.N = A.N+1
self.layers.append(A)
n = 0
for l in self.layers:
l.make_matrices(self.N)
l.num_layer = n
n += 1
self.M = len(self.layers)
except ( IOError):
print "Unable to find/open file %s -- Exiting!!!" % layer_file
exit(-2)
def dump_info(self):
i = 0
for l in self.layers:
sys.stderr.write("--------\nLayer: %d\n" % i)
l.dump_info()
i += 1
def compute_aggregated(self):
self.aggr = copy.copy(self.layers[0])
self.aggr.entropy = 0
self.aggr.entropy_approx = 0
for l in self.layers[1:]:
self.aggr.aggregate(l)
def compute_layer_entropies(self):
for l in self.layers:
l.compute_VN_entropy()
def compute_layer_entropies_approx(self):
for l in self.layers:
l.compute_VN_entropy_approx(self.poly)
def compute_multiplex_entropy(self, force_compute=False):
### The entropy of a multiplex is defined as the sum of the entropies of its layers
for l in self.layers:
if l.entropy == None:
l.compute_VN_entropy()
self.entropy += l.entropy
def compute_multiplex_entropy_approx(self, force_compute=False):
### The entropy of a multiplex is defined as the sum of the entropies of its layers
for l in self.layers:
if l.entropy_approx == None:
l.compute_VN_entropy_approx(self.poly)
self.entropy_approx += l.entropy_approx
def compute_JSD_matrix(self):
if (self.verb):
sys.stderr.write("Computing JSD matrix\n")
self.JSD = np.zeros((self.M, self.M))
for i in range(len(self.layers)):
for j in range(i+1, len(self.layers)):
li = self.layers[i]
lj = self.layers[j]
if not li.entropy:
li.compute_VN_entropy()
if not lj.entropy:
lj.compute_VN_entropy()
# m_sigma = (li.resc_laplacian + lj.resc_laplacian)/2.0
# m_sigma_entropy = mr.compute_VN_entropy_LR(m_sigma)
m_sigma_matr = (li.adj_matr + lj.adj_matr)/2.0
m_sigma = layer(matrix=m_sigma_matr)
m_sigma.compute_VN_entropy()
d = m_sigma.entropy - 0.5 * (li.entropy + lj.entropy)
d = math.sqrt(d)
self.JSD[i][j] = d
self.JSD[j][i] = d
pass
def compute_JSD_matrix_approx(self):
if (self.verb):
sys.stderr.write("Computing JSD matrix (approx)\n")
self.JSD_approx = np.zeros((self.M, self.M))
for i in range(len(self.layers)):
for j in range(i+1, len(self.layers)):
li = self.layers[i]
lj = self.layers[j]
if not li.entropy_approx:
li.compute_VN_entropy_approx(self.poly)
if not lj.entropy_approx:
lj.compute_VN_entropy_approx(self.poly)
m_sigma_matr = (li.adj_matr + lj.adj_matr)/2.0
m_sigma = layer(matrix=m_sigma_matr)
m_sigma.compute_VN_entropy_approx(self.poly)
d = m_sigma.entropy_approx - 0.5 * (li.entropy_approx + lj.entropy_approx)
d = math.sqrt(d)
self.JSD_approx[i][j] = d
self.JSD_approx[j][i] = d
def dump_JSD(self, force_compute=False):
if self.JSD == None:
if force_compute:
self.compute_JSD_matrix()
else:
print "Error!!! call to dump_JSD but JSD matrix has not been computed!!!"
sys.exit(1)
idx = 0
for i in range(self.len):
for j in range(i+1, self.len):
print i, j, self.JSD[idx]
idx += 1
def dump_JSD_approx(self, force_compute=False):
if self.JSD_approx == None:
if force_compute:
self.compute_JSD_matrix_approx()
else:
print "Error!!! call to dump_JSD_approx but JSD approximate matrix has not been computed!!!"
sys.exit(1)
idx = 0
for i in range(self.M):
for j in range(i+1, self.M):
print i, j, self.JSD_approx[idx]
idx += 1
def reduce(self, method="ward"):
if (self.verb):
sys.stderr.write("Performing '%s' reduction\n" % method)
if self.JSD == None:
self.compute_JSD_matrix()
self.Z = linkage(self.JSD, method=method)
return self.Z
def reduce_approx(self, method="ward"):
if (self.verb):
sys.stderr.write("Performing '%s' reduction (approx)\n" % method)
if self.JSD_approx == None:
self.compute_JSD_matrix_approx()
self.Z_approx = linkage(self.JSD_approx, method=method)
return self.Z_approx
def get_linkage(self):
return self.Z
def get_linkage_approx(self):
return self.Z_approx
def __compute_q(self, layers):
H_avg = 0
if not self.aggr:
self.compute_aggregated()
self.aggr.compute_VN_entropy()
for l in layers:
if not l.entropy:
l.compute_VN_entropy()
H_avg += l.entropy
H_avg /= len(layers)
q = 1.0 - H_avg / self.aggr.entropy
return q
def get_q_profile(self):
mylayers = copy.copy(self.layers)
rem_layers = copy.copy(self.layers)
q_vals = []
if self.Z == None:
self.reduce()
q = self.__compute_q(rem_layers)
q_vals.append(q)
n = len(self.layers)
for l1, l2, _d, _x in self.Z:
l_new = layer(matrix=mylayers[int(l1)].adj_matr)
l_new.num_layer = n
n += 1
l_new.aggregate(mylayers[int(l2)])
rem_layers.remove(mylayers[int(l1)])
rem_layers.remove(mylayers[int(l2)])
rem_layers.append(l_new)
mylayers.append(l_new)
q = self.__compute_q(rem_layers)
q_vals.append(q)
self.q_vals = q_vals
return q_vals
pass
def __compute_q_approx(self, layers):
H_avg = 0
if not self.aggr:
self.compute_aggregated()
self.aggr.compute_VN_entropy_approx(self.poly)
for l in layers:
if not l.entropy_approx:
l.compute_VN_entropy_approx(self.poly)
H_avg += l.entropy_approx
H_avg /= len(layers)
q = 1.0 - H_avg / self.aggr.entropy_approx
return q
def get_q_profile_approx(self):
mylayers = copy.copy(self.layers)
rem_layers = copy.copy(self.layers)
q_vals = []
if self.Z_approx == None:
self.reduce_approx()
q = self.__compute_q_approx(rem_layers)
q_vals.append(q)
n = len(self.layers)
for l1, l2, _d, _x in self.Z_approx:
l_new = layer(matrix=mylayers[int(l1)].adj_matr)
l_new.num_layer = n
n += 1
l_new.aggregate(mylayers[int(l2)])
rem_layers.remove(mylayers[int(l1)])
rem_layers.remove(mylayers[int(l2)])
rem_layers.append(l_new)
mylayers.append(l_new)
q = self.__compute_q_approx(rem_layers)
q_vals.append(q)
self.q_vals_approx = q_vals
return q_vals
def compute_partitions(self):
if (self.verb):
sys.stderr.write("Getting partitions...\n")
if self.Z == None:
self.reduce()
if self.q_vals == None:
self.get_q_profile()
sets = {}
M = len(self.layers)
for i in range(len(self.layers)):
sets[i] = [i]
best_pos = self.q_vals.index(max(self.q_vals))
j = 0
cur_part = sets.values()
self.cuts = [copy.deepcopy(cur_part)]
while j < M-1:
l1, l2, _x, _y = self.Z[j]
l1 = int(l1)
l2 = int(l2)
val = sets[l1]
val.extend(sets[l2])
sets[M+j] = val
r1 = cur_part.index(sets[l1])
cur_part.pop(r1)
r2 = cur_part.index(sets[l2])
cur_part.pop(r2)
cur_part.append(val)
j += 1
self.cuts.append(copy.deepcopy(cur_part))
self.cuts.append(copy.deepcopy(cur_part))
return zip(self.q_vals, self.cuts)
def compute_partitions_approx(self):
if (self.verb):
sys.stderr.write("Getting partitions (approx)...\n")
if self.Z_approx == None:
self.reduce_approx()
if self.q_vals_approx == None:
self.get_q_profile_approx()
sets = {}
M = len(self.layers)
for i in range(len(self.layers)):
sets[i] = [i]
best_pos = self.q_vals_approx.index(max(self.q_vals_approx))
j = 0
cur_part = sets.values()
self.cuts_approx = [copy.deepcopy(cur_part)]
while j < M-1:
l1, l2, _x, _y = self.Z_approx[j]
l1 = int(l1)
l2 = int(l2)
val = sets[l1]
val.extend(sets[l2])
sets[M+j] = val
r1 = cur_part.index(sets[l1])
cur_part.pop(r1)
r2 = cur_part.index(sets[l2])
cur_part.pop(r2)
cur_part.append(val)
j += 1
self.cuts_approx.append(copy.deepcopy(cur_part))
self.cuts_approx.append(copy.deepcopy(cur_part))
return zip(self.q_vals_approx, self.cuts_approx)
def draw_dendrogram(self, force = False):
if not has_matplotlib:
sys.stderr.write("No matplotlib module found in draw_dendrogram...Exiting!!!\n")
sys.exit(3)
if self.Z == None:
if not force:
sys.stderr.write("Please call reduce() first or specify 'force=True'")
else:
self.reduce()
dendrogram(self.Z, no_plot=False)
matplotlib.pyplot.draw()
matplotlib.pyplot.show()
def draw_dendrogram_approx(self, force = False):
if not has_matplotlib:
sys.stderr.write("No matplotlib module found in draw_dendrogram_approx...Exiting!!!\n")
sys.exit(3)
if self.Z_approx == None:
if not force:
sys.stderr.write("Please call reduce_approx() first or specify 'force=True'")
else:
self.reduce_approx()
dendrogram(self.Z_approx, no_plot=False)
matplotlib.pyplot.draw()
matplotlib.pyplot.show()
def dump_partitions(self):
part = zip(self.q_vals, self.cuts)
for q, p in part:
print q, "->", p
def dump_partitions_approx(self):
part = zip(self.q_vals_approx, self.cuts_approx)
for q, p in part:
print q, "->", p