# -*- coding: utf-8 -*-
"""
Created on Tue Apr 05 10:00:21 2016
Module:
bipcm - Bipartite Partial Configuration Model
Author:
Mika Straka
Description:
Implementation of the Bipartite Partial Configuration Model with one-sided
constraint (BiPCM) for binary undirected bipartite networks [Saracco2016]_.
Given the biadjacency matrix of a bipartite graph in the form of a binary
array as input, the module allows the user to calculate the biadjacency
matrix of the ensemble average graph :math:`<G>^*` of the BiPCM null model,
where the matrix entries correspond to the link probabilities
:math:`<G>^*_{rc} = p_{rc}` between nodes of the two distinct bipartite
node sets. Subsequently, one can calculate the p-values of the node
similarities for nodes in the same bipartite layer [Saracco2016]_.
The BiPCM derives from the Bipartite Configuration Model as presented in
[Saracco2015]_, in which the degrees of both bipartite layers are fixed
on average. For the BiPCM, the constraints are relaxed and only the degrees
of one bipartite layer are imposed [Saracco2016]_.
Usage:
Be ``mat`` a two-dimensional binary NumPy array. The nodes of the two
bipartite layers are ordered along the columns and rows, respectively. In
the algorithm, the two layers are identified by the boolean values ``True``
for the **row-nodes** and ``False`` for the **column-nodes**.
Import the module and initialize the Bipartite Partial Configuration Model::
>>> from src.bipcm import BiPCM
>>> pcm = BiPCM(bin_mat=mat, constraint=<bool>)
The parameter ``constraint`` specifies whether the degrees of the row-nodes
(``constraint = True``) or the degrees of the column-nodes
(``constraint = False``) should be constrained.
In order to analyze the similarity of the row-layer nodes and to save the
p-values of the corresponding :math:`\\Lambda`-motifs, use::
>>> pcm.lambda_motifs_main(bip_set=True, filename=<filename>)
For the column-layer nodes, use::
>>> pcm.lambda_motifs_main(bip_set=False, filename=<filename>)
``bip_set`` selects the bipartite node set for which the p-values should be
calculated and saved. The filename *<filename>* should contain a relative
path declaration. The default name of the output file is
*pval_constr_<constraint>_proj_<bip_set>.csv*, where *<constraint>* and
*<bip_set>* are either *rows* or *columns* depending on the degree
constraint and the parameter choice in ``lambda_motifs_main``. By default,
the values in the file are separated by tabs, which can be changed using the
``delim`` keyword.
Reference:
[Saracco2015] F. Saracco, R. Di Clemente, A. Gabrielli, T. Squartini,
Randomizing bipartite networks: the case of the World Trade Web, Scientific
Reports 5, 10595 (2015)
[Saracco2016] F. Saracco, M. J. Straka, R. Di Clemente, A. Gabrielli, G.
Caldarelli, T. Squartini, Inferring monopartite projections of bipartite
networks: an entropy-based approach, arXiv preprint arXiv:1607.02481
"""
import numpy as np
from scipy.stats import binom
from poibin.poibin import PoiBin
[docs]class BiPCM:
"""Bipartite Partial Configuration Model for binary bipartite networks.
This class implements the Bipartite Partial Configuration Model (BiPCM),
which can be used as a null model for the analysis of undirected and binary
bipartite networks. The class provides methods to calculate the biadjacency
matrix of the null model and to quantify node similarities in terms of
p-values.
"""
def __init__(self, bin_mat, constraint):
"""Initialize the parameters of the BiPCM.
:param bin_mat: binary input matrix describing the biadjacency matrix
of a bipartite graph with the nodes of one layer along the rows
and the nodes of the other layer along the columns.
:type bin_mat: numpy.array
:param constraint: constrains either the degrees of the row-nodes
(``True``), or the degrees of the column-nodes columns (``False``)
:type constraint: bool
"""
self.bin_mat = np.array(bin_mat)
self.check_constraint(constraint)
self.const_set = constraint
self.check_input_matrix_is_binary()
[self.num_rows, self.num_columns] = self.bin_mat.shape
self.eprob_seq = self.get_edge_prob_seq()
# ------------------------------------------------------------------------------
# Initialization
# ------------------------------------------------------------------------------
@staticmethod
[docs] def check_constraint(constraint):
"""Check that the constraint parameter is either ``True`` of ``False``.
:param constraint: constrains the degrees of either the row-nodes
(``True``) or column-nodes (``False``)
:type constraint: bool
:raise AssertionError: raise an error if the constraint is neither
``True`` nor ``False``
"""
assert constraint in [False, True], \
"Constraint has to be True or False."
[docs] def set_degree_seq(self):
"""Return the node degree sequence of the input matrix.
:returns: node degree sequence [degrees row-nodes, degrees column-nodes]
:rtype: numpy.array
:raise AssertionError: raise an error if the length of the returned
degree sequence does not correspond to the total number of nodes
"""
dseq = np.empty(self.num_rows + self.num_columns)
dseq[self.num_rows:] = np.squeeze(np.sum(self.bin_mat, axis=0))
dseq[:self.num_rows] = np.squeeze(np.sum(self.bin_mat, axis=1))
assert dseq.size == (self.num_rows + self.num_columns)
return dseq
[docs] def get_edge_prob_seq(self):
"""Return an array with the link probabilities of the BiPCM null model.
In the first part of the array, the row degrees are fixed. In the
second part, the column degrees are fixed.
:returns: array of link probabilities
:rtype: numpy.array
"""
dseq = self.set_degree_seq()
eprob_seq = np.zeros(dseq.size)
eprob_seq[:self.num_rows] = dseq[:self.num_rows] / \
self.num_columns
eprob_seq[self.num_rows:] = dseq[self.num_rows:] / \
self.num_rows
return eprob_seq
# ------------------------------------------------------------------------------
# Total log-likelihood of the observed Lambda motifs in the input matrix
# ------------------------------------------------------------------------------
[docs] def lambda_loglike(self, bip_set=False):
"""Return the log-likelihood of the number of :math:`\\Lambda`-motifs.
The total log-likelihood of the number of observed
:math:`\\Lambda`-motifs in the input matrix is calculated according to
the BiPCM null model.
:param bip_set: analyze :math:`\\Lambda`-motifs of row-nodes (``True``)
or column-nodes (``False``)
:type bip_set: bool
"""
plam_mat = self.get_plambda_matrix()
nlam_mat = self.get_lambda_motif_matrix(self.bin_mat, bip_set)
p_mat = self.get_proj_pmat(plam_mat, nlam_mat, bip_set)
logp = np.log(p_mat[np.triu_indices_from(p_mat, k=1)])
loglike = logp.sum()
return loglike
[docs] def get_proj_pmat(self, plam_mat, nlam_mat, bip_set=False):
"""Return the probabilities of the observed :math:`\\Lambda`-motifs.
The probabilities of the :math:`\\Lambda`-motifs between the nodes
specified by ``bip_set`` in the input matrix are calculated and
returned.
If the node set ``bip_set`` is the same as the constrained one, the
:math:`\\Lambda`-motifs follow a Binomial probability distribution.
Otherwise, all the node pairs follow the same Poisson Binomial
distribution.
The probability mass function is given by
.. math::
pmf(k) = Pr(X = k)
.. note::
The lower triangular part including the diagonal is set to 0 since
the matrix is symmetric.
:param plam_mat: matrix of Lambda motif probabilities
:type plam_mat: numpy.array
:param nlam_mat: matrix of observed number of Lambda motifs
:type nlam_mat: numpy.array
:param bip_set: select row-nodes (``True``) or column-nodes (``False``)
:type bip_set: bool
:returns: matrix containing the probabilities of the
:math:`\\Lambda`-motifs
:rtype: numpy.array
:raise NameError: raise an error if the parameter ``bip_set`` is
neither ``True`` nor ``False``
:raise AssertionError: raise an error if shapes of the probability
matrix and the matrix with the number of :math:`\\Lambda`-motifs
are not equal
"""
if bip_set:
m = self.num_columns
elif not bip_set:
m = self.num_rows
else:
errmsg = "'" + str(bip_set) + "' " + 'not supported.'
raise NameError(errmsg)
n = nlam_mat.shape[0]
pmat = np.zeros(nlam_mat.shape)
if bip_set != self.const_set:
pb = PoiBin(plam_mat[np.diag_indices_from(plam_mat)])
for i in xrange(n):
pmat[i, i + 1:] = pb.pmf(nlam_mat[i, i + 1:])
elif bip_set == self.const_set:
# if the sets correspond, the matrix dimensions should be the same
assert plam_mat.shape[0] == nlam_mat.shape[0]
for i in xrange(n):
for j in xrange(i + 1, n):
bn = binom(m, plam_mat[i, j])
pmat[i, j] = bn.pmf(nlam_mat[i, j])
return pmat
# ------------------------------------------------------------------------------
# Lambda motifs
# ------------------------------------------------------------------------------
[docs] def lambda_motifs_main(self, bip_set=False, write=True, filename=None,
delim='\t'):
"""Calculate and save the p-values of the :math:`\\Lambda`-motifs.
For each node couple in the bipartite layer specified by ``bip_set``,
:math:`\\Lambda`-motifs and calculate the corresponding p-value.
:param bip_set: select row-nodes (``True``) or column-nodes (``False``)
:type bip_set: bool
:param write: if ``True``, the p-values are saved in the specified file
:type write: bool
:param filename: name of the file which will contain the p-values,
default is *pval_constr_<constraint>_proj_<rows OR columns>.csv*
:type filename: str
:param delim: delimiter between entries in file, default is tab
:type delim: str
:returns: matrix of p-values if ``write==True``
:rtype: numpy.array
:raise NameError: raise an error if the parameter ``bip_set`` is
neither ``True`` nor ``False``
"""
plam_mat = self.get_plambda_matrix()
nlam_mat = self.get_lambda_motif_matrix(self.bin_mat, bip_set)
pval_mat = self.get_lambda_pvalues(plam_mat, nlam_mat, bip_set)
if write:
if filename is None:
if bip_set:
b = 'rows'
elif not bip_set:
b = 'columns'
else:
errmsg = "'" + str(bip_set) + "' " + 'not supported.'
raise NameError(errmsg)
if self.const_set:
constr = "rows"
else:
constr = "columns"
fname = 'pval_constr_' + constr + '_proj_' + b + '.csv'
else:
fname = filename
self.save_matrix(pval_mat, filename=fname, delim=delim)
else:
return pval_mat
[docs] def get_plambda_matrix(self):
"""Return the :math:`\\Lambda`-motif probability matrix.
Return a square matrix :math:`M` of Lambda probabilities for the nodes
given the degree constraints on the node set self.const_set.
.. note::
If :math:`N_i` are the nodes with constrained degrees,
:math:`M_{ij} = p(\\Lambda_{ij})` is the probability of nodes
:math:`i, j \in N_i` sharing one common neighbor, whereas
:math:`M_{ii}` is the probability that two nodes of the opposite
layer have node :math:`i \in N_i` as a common neighbor.
The lower triangular part of :math:`M` excluding the diagonal is
set to 0 since the matrix is symmetric.
:returns: :math:`\\Lambda`-motif probability matrix
:rtype: numpy.array
"""
if self.const_set:
ps = self.eprob_seq[:self.num_rows]
elif not self.const_set:
ps = self.eprob_seq[self.num_rows:]
mat = np.zeros((ps.size, ps.size))
for i in xrange(ps.size):
mat[i, i:] = ps[i] * ps[i:]
return mat
[docs] def get_lambda_motif_matrix(self, mm, bip_set=False):
"""Return the number of :math:`\\Lambda`-motifs as found in ``mm``.
Given the binary input matrix ``mm``, count the number of
:math:`\\Lambda`-motifs between node couples of the bipartite layer
specified by ``bip_set``.
:param mm: binary matrix
:type mm: numpy.array
:param bip_set: selects row-nodes (``True``) or column-nodes (``False``)
:type bip_set: bool
:returns: square matrix of observed :math:`\\Lambda`-motifs
:rtype: numpy.array
:raise NameError: raise an error if the parameter ``bip_set`` is
neither ``True`` nor ``False``
:raise AssertionError: raise an error if shape of the probability
matrix is not correct
"""
if bip_set:
l2_mat = np.dot(mm, np.transpose(mm))
assert l2_mat.shape == (self.num_rows, self.num_rows)
elif not bip_set:
l2_mat = np.dot(np.transpose(mm), mm)
assert l2_mat.shape == (self.num_columns, self.num_columns)
else:
errmsg = str(bip_set) + 'not supported.'
raise NameError(errmsg)
# set diagonal to zero
di = np.diag_indices_from(l2_mat)
l2_mat[di] = 0
return l2_mat.astype(int)
[docs] def get_lambda_pvalues(self, plam_mat, nlam_mat, bip_set=False):
"""Return the p-values for the :math:`\\Lambda`-motifs in ``nlam_mat``.
Calculate the p-values for the numbers of observed
:math:`\\Lambda`-motifs as given in the parameter ``nlam_mat`` for the
bipartite node layer ``bip_set``. The probabilities for the single
:math:`\\Lambda`-motifs are given in ``plam_mat``.
If ``bip_set`` corresponds to the constrained bipartite node set, the
:math:`\\Lambda`-motifs follow a Binomial probability distribution.
Otherwise, all the node pairs follow the same Poisson Binomial
probability distribution. The p-values are calculated as
.. math::
p_{val}(k) = Pr(X >= k) = 1 - Pr(X < k) = 1 - cdf(k) + pmf(k)
.. note::
The lower triangular part (including the diagonal) of the returned
matrix is set to zero.
:param plam_mat: matrix of :math:`\\Lambda`-motif probabilities
:type plam_mat: numpy.array
:param nlam_mat: matrix of observed number of Lambda motifs
:type nlam_mat: numpy.array
:param bip_set: selects row-nodes (``True``) or column-nodes (``False``)
:type bip_set: bool
:returns: matrix of the p-values for the :math:`\\Lambda`-motifs
:rtype: numpy.array
:raise NameError: raise an error if the parameter ``bip_set`` is
neither ``True`` nor ``False``
:raise AssertionError: raise an error if shapes of the probability
matrix and the matrix with the number of :math:`\\Lambda`-motifs
are not equal
"""
if bip_set:
m = self.num_columns
elif not bip_set:
m = self.num_rows
else:
errmsg = "'" + str(bip_set) + "' " + 'not supported.'
raise NameError(errmsg)
n = nlam_mat.shape[0]
pval_mat = np.zeros(nlam_mat.shape)
if bip_set != self.const_set:
pb = PoiBin(plam_mat[np.diag_indices_from(plam_mat)])
for i in xrange(n):
pval_mat[i, i + 1:] = pb.pval(nlam_mat[i, i + 1:])
elif bip_set == self.const_set:
# if the sets correspond, the matrix dimensions should be the same
assert plam_mat.shape[0] == nlam_mat.shape[0]
for i in xrange(n):
for j in xrange(i + 1, n):
bn = binom(m, plam_mat[i, j])
pval_mat[i, j] = 1. - bn.cdf(nlam_mat[i, j]) \
+ bn.pmf(nlam_mat[i, j])
return pval_mat
# ------------------------------------------------------------------------------
# Probability distributions for Lambda values
# ------------------------------------------------------------------------------
# def save_lambda_probdist(self, bip_set=False, write=True, filename=None,
# delim='\t', binary=True):
# """Obtain and save the probabilities of all possible values of the
# Lambda motifs for the node set defined by bip_set. The matrix can
# either be saved as human-readable ASCII or as a binary Numpy file.
#
# :param bip_set: analyze row-nodes (True) or column-nodes (False)
# :type bip_set: bool
# :param write: if True, the pvalues are saved in an external file
# :type write: bool
# :param filename: name of the output file, default name is
# bipcm_pval_constr_<constraint>_proj_<rows OR columns>.csv
# :param delim: delimiter to use if file is saved as .csv
# :param binary: if true, save as binary .npy file. Otherwise as .csv
# file
# """
# plam_mat = self.get_plambda_matrix()
# lambdaprobs_mat = self.get_lambda_probdist(plam_mat, bip_set)
# if write:
# if filename is None:
# if bip_set:
# b = 'rows'
# elif not bip_set:
# b = 'columns'
# else:
# errmsg = "'" + str(bip_set) + "' " + 'not supported.'
# raise NameError(errmsg)
# if self.const_set:
# constr = "rows"
# else:
# constr = "columns"
# fname = 'bipcm_lambdaprob_constr_' + constr + '_layer_' + b
# else:
# fname = filename
# if binary:
# fname += '.npy'
# else:
# fname += '.csv'
# self.save_matrix(lambdaprobs_mat, filename=fname, delim=delim,
# binary=binary)
# else:
# return lambdaprobs_mat
#
# def get_lambda_probdist(self, plam_mat, bip_set=False):
# """Return a matrix which contains the probabilities for each node pair
# (i, j) in the bipartite node set bip_set of observing every possible
# number of common neighbors in the opposite bipartite layer.
#
# :param plam_mat: matrix of Lambda motif probabilities
# :type plam_mat: numpy.array
# :param bip_set: selects row-nodes (True) or column-nodes (False)
# :type bip_set: bool
# """
# if bip_set:
# n = self.num_rows
# m = self.num_columns
# elif not bip_set:
# n = self.num_columns
# m = self.num_rows
# else:
# errmsg = "'" + str(bip_set) + "' " + 'not supported.'
# raise NameError(errmsg)
#
# lambda_values = np.arange(m + 1)
#
# if bip_set != self.const_set:
# lambdaprobs_mat = np.zeros([1, m + 1])
# pb = PoiBin(plam_mat[np.diag_indices_from(plam_mat)])
# lambdaprobs_mat[0, :] = pb.pmf(lambda_values)
# elif bip_set == self.const_set:
# lambdaprobs_mat = np.zeros([n * (n - 1) / 2, m + 1])
# # if the sets correspond, the matrix dimensions should be the same
# for i in xrange(n):
# for j in xrange(i + 1, n):
# bn = binom(m, plam_mat[i, j])
# k = self.triumat2flat_idx(i, j, n)
# lambdaprobs_mat[k, :] = bn.pmf(lambda_values)
# else:
# errmsg = "'" + str(bip_set) + "' " + 'not supported.'
# raise NameError(errmsg)
# return lambdaprobs_mat
# ------------------------------------------------------------------------------
# Auxiliary methods
# ------------------------------------------------------------------------------
@staticmethod
[docs] def triumat2flat_idx(i, j, n):
"""Convert an matrix index couple to a flattened array index.
Given a square matrix of dimension :math:`n` and an index couple
:math:`(i, j)` *of the upper triangular part* of the matrix, the
function returns the index which the matrix element would have in a
flattened array.
.. note::
* :math:`i \\in [0, ..., n - 1]`
* :math:`j \\in [i + 1, ..., n - 1]`
* returned index :math:`\\in [0,\\, n (n - 1) / 2 - 1]`
:param i: row index
:type i: int
:param j: column index
:type j: int
:param n: dimension of the square matrix
:type n: int
:returns: flattened array index
:rtype: int
"""
return int((i + 1) * n - (i + 2) * (i + 1) / 2. - (n - (j + 1)) - 1)
# @staticmethod
# def triumat2flat_idx(idx_i, idx_j, n):
# """Convert index couple (i, j) into index in one-dimensional array index
# for the upper triangular part of a square matrix with dimension n. I.e.
# idx_i runs from 0, ..., n and idx_j runs from idx_i + 1, ..., n
#
# NB: the returned indices start from 0.
# """
# return int((idx_i + 1) * n - (idx_i + 2) * (idx_i + 1) / 2.
# - (n - (idx_j + 1)) - 1)
# @staticmethod
# def get_main_dir(main_dir_name='bipcm'):
# """Return the absolute path to the main directory which contains the
# folders "src" and "output".
# Note that the default directory name is "bipcm".
#
# :param main_dir_name: name of the main directory of the program.
# :type main_dir_name: string
# """
# s = os.getcwd()
# dirpath = s[:s.index(main_dir_name) + len(main_dir_name) + 1]
# return dirpath
# def save_matrix(self, mat, filename, delim='\t', binary=False):
# """Save the input matrix in a csv-file.
#
# :param mat: two-dimensional matrix
# :param filename: name of the output file
# :param delim: delimiter between values in file.
# :param binary: if true, save as binary .npy file. Otherwise as .csv
# file
# """
# fname = ''.join([self.main_dir, '/output/', filename])
# if binary:
# np.save(fname, mat)
# else:
# np.savetxt(fname, mat, delimiter=delim)
@staticmethod
[docs] def save_matrix(mat, filename, delim='\t', binary=False):
"""Save the matrix ``mat`` in the file ``filename``.
The matrix can either be saved as a binary NumPy ``.npy`` file or as a
human-readable CSV file.
.. note:: The relative path has to be provided in the filename, e.g.
*../data/pvalue_matrix.csv*
:param mat: two-dimensional matrix
:type mat: numpy.array
:param filename: name of the output file
:type filename: str
:param delim: delimiter between values in file
:type delim: str
:param binary: if ``True``, save as binary ``.npy``, otherwise as a
CSV file
:type binary: bool
"""
if binary:
np.save(filename, mat)
else:
np.savetxt(filename, mat, delimiter=delim)
################################################################################
# Main
################################################################################
if __name__ == "__main__":
pass