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# multired
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/**
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*
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* Copyright (C) 2015 Vincenzo (Enzo) Nicosia <katolaz@yahoo.it>
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*
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*
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* This program is free software: you can redistribute it and/or modify
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* it under the terms of the GNU General Public License as published by
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* the Free Software Foundation, either version 3 of the License, or
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* (at your option) any later version.
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*
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* This program is distributed in the hope that it will be useful, but
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* WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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* General Public License for more details.
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*
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* You should have received a copy of the GNU General Public License
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* long with this program. If not, see <http://www.gnu.org/licenses/>.
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*
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*/
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This is multired-0.1.
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This is a Python implmementation of the algorithm for structural
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reduction of multi-layer networks based on the Von Neumann and on the
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Quantum Jensen-Shannon divergence of graphs, as explained in:
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M. De Domenico. V. Nicosia, A. Arenas, V. Latora
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"Structural reducibility of multilayer networks",
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Nat. Commun. 6, 6864 (2015) doi:10.1038/ncomms7864
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If you happen to find any use of this code please do not forget to
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cite that paper ;-)
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--------------------
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INFO
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--------------------
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The module "multired.py" provides the class "multiplex_red", which
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implements the algorithm to reduce a multilayer network described in
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the paper cited above.
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In order to use it, you just need to
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import multired as mr
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in your python script and create a multiplex_red object. Please make
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sure that "multired.py" is in PYTHONPATH. The constructor requires as
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its first argument the path of a file which in turn contains a list of
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files (one for each line) where the graph of each layer is to be
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found.
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The class provides one set of methods which perform the exact
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evaluation of the Von Neumann entropy, and another set of methods
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(those whose name end with the suffix "_approx") which rely on a
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polynomial approximation of the Von Neumann entropy. By default the
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approximation is based on a 10th order polynomial fit of x log(x) in
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[0,1], but the order of the polynomial can be set through the
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parameter "fit_degree" of the constructor.
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Several sample scripts can be found in the "test/" directory.
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--------------------
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DEPENDENCIES
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--------------------
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The only strict dependencies are a recent version of Python, Numpy and
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SciPy. The methods "draw_dendrogram" and "draw_dendrogram_approx" will
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work only if matplotlib is installed.
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The module has been tested on a Debian GNU/Linux system, using:
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- Python 2.7.8,
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- SciPy 0.13.3
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- Numpy 1.8.2
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- matplotlib 1.3.1
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but it will almost surely work on other platforms and/or with other
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versions of those packages. If you would like to report a working
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configuration, just email me (the address is at the beginning of this
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file).
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@ -0,0 +1,487 @@
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import sys
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import math
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import numpy as np
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from scipy.sparse import csr_matrix, eye
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from scipy.linalg import eigh, eig
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import copy
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from scipy.cluster.hierarchy import linkage, dendrogram
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has_matplotlib = False
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try:
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import matplotlib
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has_matplotlib = True
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except ImportError:
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has_matplotlib = False
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class XLogx_fit:
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def __init__(self, degree, npoints= 100, xmax=1):
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if xmax > 1:
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xmax = 1
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self.degree = degree
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x = np.linspace(0, xmax, npoints)
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y = [i * math.log(i) for i in x[1:]]
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y.insert(0, 0)
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self.fit = np.polyfit(x, y, degree)
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def __getitem__ (self, index):
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if index <= self.degree:
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return self.fit[index]
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else:
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print "Error!!! Index %d is larger than the degree of the fitting polynomial (%d)" \
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% (index, degree)
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sys.exit(-1)
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class layer:
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def __init__ (self, layerfile= None, matrix=None):
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self.N = 0
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self.num_layer = -1
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self.fname = layerfile
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self.adj_matr = None
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self.laplacian = None
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self.resc_laplacian = None
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self.entropy = None
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self.entropy_approx = None
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self._ii = []
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self._jj = []
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self._ww = []
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self._matrix_called = False
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if layerfile != None:
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try:
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min_N = 10e10
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with open(layerfile, "r") as lines:
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for l in lines:
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if l[0] == '#':
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continue
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elems = l.strip(" \n").split(" ")
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s = int(elems[0])
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d = int(elems[1])
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self._ii.append(s)
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self._jj.append(d)
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if s > self.N:
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self.N = s
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if d > self.N:
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self.N = d
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if s < min_N:
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min_N = s
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if d < min_N:
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min_N = d
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if len(elems) >2 : ## A weight is specified
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val = [float(x) if "e" in x or "." in x else int(x) for x in [elems[2]]][0]
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self._ww.append(float(val))
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else:
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self._ww.append(int(1))
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except (IOError):
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print "Unable to find/open file %s -- Exiting!!!" % layerfile
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exit(-2)
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elif matrix != None:
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self.adj_matr = copy.copy(matrix)
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self.N, _x = matrix.shape
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K = np.multiply(self.adj_matr.sum(0), np.ones((self.N,self.N)))
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D = np.diag(np.diag(K))
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self.laplacian = csr_matrix(D - self.adj_matr)
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K = self.laplacian.diagonal().sum()
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self.resc_laplacian = csr_matrix(self.laplacian / K)
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self._matrix_called = True
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else:
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print "The given matrix is BLANK"
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def make_matrices(self, N):
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self.N = N
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self.adj_matr = csr_matrix((self._ww, (self._ii, self._jj)), shape=(self.N, self.N))
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self.adj_matr = self.adj_matr + self.adj_matr.transpose()
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K = np.multiply(self.adj_matr.sum(0), np.ones((self.N,self.N)))
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D = np.diag(np.diag(K))
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self.laplacian = csr_matrix(D - self.adj_matr)
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K = self.laplacian.diagonal().sum()
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self.resc_laplacian = csr_matrix(self.laplacian / K)
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self._matrix_called = True
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def dump_info(self):
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N, M = self.adj_matr.shape
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K = self.adj_matr.nnz
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sys.stderr.write("Layer File: %s\nNodes: %d Edges: %d\nEntropy: %g Approx. Entropy: %g\n" % \
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(self.fname, N, K, self.entropy, self.entropy_approx) )
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def compute_VN_entropy(self):
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eigvals = eigh(self.resc_laplacian.todense())
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self.entropy = 0
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for l_i in eigvals[0]:
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if (l_i > 10e-20):
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self.entropy -= l_i * math.log (l_i)
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def compute_VN_entropy_approx(self, poly):
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p = poly.degree
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h = - poly[p] * self.N
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M = csr_matrix(np.eye(self.N))
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for i in range(p-1, -1, -1):
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M = M * self.resc_laplacian
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h += - poly[i] * sum(M.diagonal())
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self.entropy_approx = h
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def aggregate(self, other_layer):
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if self.adj_matr != None:
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self.adj_matr = self.adj_matr + other_layer.adj_matr
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else:
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self.adj_matr = copy.copy(other_layer.adj_matr)
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K = np.multiply(self.adj_matr.sum(0), np.ones((self.N,self.N)))
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D = np.diag(np.diag(K))
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self.laplacian = csr_matrix(D - self.adj_matr)
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K = self.laplacian.diagonal().sum()
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self.resc_laplacian = csr_matrix(self.laplacian / K)
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self._matrix_called = True
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class multiplex_red:
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def __init__ (self, multiplexfile, directed = None, fit_degree=10, verbose=False):
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self.layers = []
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self.N = 0
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self.M = 0
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self.entropy = 0
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self.entropy_approx = 0
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self.JSD = None
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self.JSD_approx = None
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self.Z = None
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self.Z_approx = None
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self.aggr = None
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self.q_vals = None
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self.q_vals_approx = None
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self.fit_degree = fit_degree
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self.poly = XLogx_fit(self.fit_degree)
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self.verb = verbose
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self.cuts = None
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self.cuts_approx = None
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try:
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with open(multiplexfile, "r") as lines:
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for l in lines:
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if (self.verb):
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sys.stderr.write("Loading layer %d from file %s" % (len(self.layers), l))
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A = layer(l.strip(" \n"))
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if A.N > self.N:
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self.N = A.N+1
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self.layers.append(A)
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n = 0
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for l in self.layers:
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l.make_matrices(self.N)
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l.num_layer = n
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n += 1
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self.M = len(self.layers)
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except ( IOError):
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print "Unable to find/open file %s -- Exiting!!!" % layer_file
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exit(-2)
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def dump_info(self):
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i = 0
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for l in self.layers:
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sys.stderr.write("--------\nLayer: %d\n" % i)
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l.dump_info()
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i += 1
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def compute_aggregated(self):
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self.aggr = copy.copy(self.layers[0])
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self.aggr.entropy = 0
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self.aggr.entropy_approx = 0
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for l in self.layers[1:]:
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self.aggr.aggregate(l)
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def compute_layer_entropies(self):
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for l in self.layers:
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l.compute_VN_entropy()
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def compute_layer_entropies_approx(self):
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for l in self.layers:
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l.compute_VN_entropy_approx(self.poly)
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def compute_multiplex_entropy(self, force_compute=False):
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### The entropy of a multiplex is defined as the sum of the entropies of its layers
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for l in self.layers:
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if l.entropy == None:
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l.compute_VN_entropy()
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self.entropy += l.entropy
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def compute_multiplex_entropy_approx(self, force_compute=False):
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### The entropy of a multiplex is defined as the sum of the entropies of its layers
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for l in self.layers:
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if l.entropy_approx == None:
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l.compute_VN_entropy_approx(self.poly)
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self.entropy_approx += l.entropy_approx
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def compute_JSD_matrix(self):
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if (self.verb):
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sys.stderr.write("Computing JSD matrix\n")
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self.JSD = np.zeros((self.M, self.M))
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for i in range(len(self.layers)):
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for j in range(i+1, len(self.layers)):
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li = self.layers[i]
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lj = self.layers[j]
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if not li.entropy:
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li.compute_VN_entropy()
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if not lj.entropy:
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lj.compute_VN_entropy()
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# m_sigma = (li.resc_laplacian + lj.resc_laplacian)/2.0
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# m_sigma_entropy = mr.compute_VN_entropy_LR(m_sigma)
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m_sigma_matr = (li.adj_matr + lj.adj_matr)/2.0
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m_sigma = layer(matrix=m_sigma_matr)
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m_sigma.compute_VN_entropy()
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d = m_sigma.entropy - 0.5 * (li.entropy + lj.entropy)
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d = math.sqrt(d)
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self.JSD[i][j] = d
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self.JSD[j][i] = d
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pass
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def compute_JSD_matrix_approx(self):
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if (self.verb):
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sys.stderr.write("Computing JSD matrix (approx)\n")
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self.JSD_approx = np.zeros((self.M, self.M))
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for i in range(len(self.layers)):
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for j in range(i+1, len(self.layers)):
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li = self.layers[i]
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lj = self.layers[j]
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if not li.entropy_approx:
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li.compute_VN_entropy_approx(self.poly)
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if not lj.entropy_approx:
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lj.compute_VN_entropy_approx(self.poly)
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m_sigma_matr = (li.adj_matr + lj.adj_matr)/2.0
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m_sigma = layer(matrix=m_sigma_matr)
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m_sigma.compute_VN_entropy_approx(self.poly)
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d = m_sigma.entropy_approx - 0.5 * (li.entropy_approx + lj.entropy_approx)
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d = math.sqrt(d)
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self.JSD_approx[i][j] = d
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self.JSD_approx[j][i] = d
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def dump_JSD(self, force_compute=False):
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if self.JSD == None:
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if force_compute:
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self.compute_JSD_matrix()
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else:
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print "Error!!! call to dump_JSD but JSD matrix has not been computed!!!"
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sys.exit(1)
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idx = 0
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for i in range(self.len):
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for j in range(i+1, self.len):
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print i, j, self.JSD[idx]
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idx += 1
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def dump_JSD_approx(self, force_compute=False):
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if self.JSD_approx == None:
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if force_compute:
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self.compute_JSD_matrix_approx()
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else:
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print "Error!!! call to dump_JSD_approx but JSD approximate matrix has not been computed!!!"
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sys.exit(1)
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idx = 0
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for i in range(self.M):
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for j in range(i+1, self.M):
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print i, j, self.JSD_approx[idx]
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idx += 1
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def reduce(self, method="ward"):
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if (self.verb):
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sys.stderr.write("Performing '%s' reduction\n" % method)
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if self.JSD == None:
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self.compute_JSD_matrix()
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self.Z = linkage(self.JSD, method=method)
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return self.Z
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def reduce_approx(self, method="ward"):
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if (self.verb):
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sys.stderr.write("Performing '%s' reduction (approx)\n" % method)
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if self.JSD_approx == None:
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self.compute_JSD_matrix_approx()
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self.Z_approx = linkage(self.JSD_approx, method=method)
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return self.Z_approx
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def get_linkage(self):
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return self.Z
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def get_linkage_approx(self):
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return self.Z_approx
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def __compute_q(self, layers):
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H_avg = 0
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if not self.aggr:
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self.compute_aggregated()
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self.aggr.compute_VN_entropy()
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for l in layers:
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if not l.entropy:
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l.compute_VN_entropy()
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H_avg += l.entropy
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H_avg /= len(layers)
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q = 1.0 - H_avg / self.aggr.entropy
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return q
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def get_q_profile(self):
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mylayers = copy.copy(self.layers)
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rem_layers = copy.copy(self.layers)
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q_vals = []
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if self.Z == None:
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self.reduce()
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q = self.__compute_q(rem_layers)
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q_vals.append(q)
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n = len(self.layers)
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for l1, l2, _d, _x in self.Z:
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l_new = layer(matrix=mylayers[int(l1)].adj_matr)
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l_new.num_layer = n
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n += 1
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l_new.aggregate(mylayers[int(l2)])
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rem_layers.remove(mylayers[int(l1)])
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rem_layers.remove(mylayers[int(l2)])
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rem_layers.append(l_new)
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mylayers.append(l_new)
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q = self.__compute_q(rem_layers)
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q_vals.append(q)
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self.q_vals = q_vals
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return q_vals
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pass
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def __compute_q_approx(self, layers):
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H_avg = 0
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if not self.aggr:
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self.compute_aggregated()
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self.aggr.compute_VN_entropy_approx(self.poly)
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for l in layers:
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if not l.entropy_approx:
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l.compute_VN_entropy_approx(self.poly)
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H_avg += l.entropy_approx
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H_avg /= len(layers)
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q = 1.0 - H_avg / self.aggr.entropy_approx
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return q
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def get_q_profile_approx(self):
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mylayers = copy.copy(self.layers)
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rem_layers = copy.copy(self.layers)
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q_vals = []
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if self.Z_approx == None:
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self.reduce_approx()
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q = self.__compute_q_approx(rem_layers)
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q_vals.append(q)
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n = len(self.layers)
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for l1, l2, _d, _x in self.Z_approx:
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l_new = layer(matrix=mylayers[int(l1)].adj_matr)
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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
|
||||
|
||||
|
||||
|
||||
|
||||
@ -0,0 +1,22 @@
|
||||
import multired as mr
|
||||
import sys
|
||||
|
||||
|
||||
if len(sys.argv) < 2:
|
||||
print "Usage: %s <layer_list>" % sys.argv[0]
|
||||
sys.exit(1)
|
||||
|
||||
print "Loading layers...",
|
||||
m = mr.multiplex_red(sys.argv[1], verbose=True)
|
||||
print "[DONE]"
|
||||
|
||||
print "Getting partitons...",
|
||||
part = m.compute_partitions()
|
||||
print "[DONE]"
|
||||
|
||||
print "Partitions:..."
|
||||
m.dump_partitions()
|
||||
|
||||
m.draw_dendrogram()
|
||||
|
||||
|
||||
@ -0,0 +1,16 @@
|
||||
import multired as mr
|
||||
import sys
|
||||
|
||||
|
||||
if len(sys.argv) < 2:
|
||||
print "Usage: %s <layer_list>" % sys.argv[0]
|
||||
sys.exit(1)
|
||||
|
||||
m = mr.multiplex_red(sys.argv[1], verbose=True, fit_degree=20)
|
||||
part = m.compute_partitions_approx()
|
||||
|
||||
print "Partitions:..."
|
||||
m.dump_partitions_approx()
|
||||
|
||||
m.draw_dendrogram_approx()
|
||||
|
||||
@ -0,0 +1,34 @@
|
||||
import multired as mr
|
||||
import sys
|
||||
|
||||
|
||||
if len(sys.argv) < 2:
|
||||
print "Usage: %s <layer_list>" % sys.argv[0]
|
||||
sys.exit(1)
|
||||
|
||||
print "Loading layers...",
|
||||
m = mr.multiplex_red(sys.argv[1])
|
||||
print "[DONE]"
|
||||
|
||||
print "Computing layer entropies...",
|
||||
m.compute_layer_entropies()
|
||||
print "[DONE]"
|
||||
|
||||
print "Computing JSD matrix...",
|
||||
m.compute_JSD_matrix()
|
||||
print "[DONE]"
|
||||
|
||||
print "Performing reduction...",
|
||||
m.reduce()
|
||||
print "[DONE]"
|
||||
|
||||
print "Getting partitons...",
|
||||
part = m.compute_partitions()
|
||||
print "[DONE]"
|
||||
|
||||
print "Partitions:...",
|
||||
m.dump_partitions()
|
||||
print "[DONE]"
|
||||
|
||||
|
||||
|
||||
@ -0,0 +1,34 @@
|
||||
import multired as mr
|
||||
import sys
|
||||
|
||||
|
||||
if len(sys.argv) < 2:
|
||||
print "Usage: %s <layer_list>" % sys.argv[0]
|
||||
sys.exit(1)
|
||||
|
||||
print "Loading layers...",
|
||||
m = mr.multiplex_red(sys.argv[1])
|
||||
print "[DONE]"
|
||||
|
||||
print "Computing layer entropies (approx)...",
|
||||
m.compute_layer_entropies_approx()
|
||||
print "[DONE]"
|
||||
|
||||
print "Computing JSD matrix (approx)...",
|
||||
m.compute_JSD_matrix_approx()
|
||||
print "[DONE]"
|
||||
|
||||
print "Performing reduction (approx)...",
|
||||
m.reduce_approx()
|
||||
print "[DONE]"
|
||||
|
||||
|
||||
print "Getting partitons...",
|
||||
part = m.compute_partitions_approx()
|
||||
print "[DONE]"
|
||||
|
||||
print "Partitions:..."
|
||||
m.dump_partitions_approx()
|
||||
print "[DONE]"
|
||||
|
||||
|
||||
Loading…
Reference in new issue