blockmodel.py 61.9 KB
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#! /usr/bin/env python
# -*- coding: utf-8 -*-
#
# graph_tool -- a general graph manipulation python module
#
# Copyright (C) 2007-2012 Tiago de Paula Peixoto <tiago@skewed.de>
#
# 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
# along with this program.  If not, see <http://www.gnu.org/licenses/>.

from __future__ import division, absolute_import, print_function
import sys
if sys.version_info < (3,):
    range = xrange

from .. import _degree, _prop, Graph, GraphView, libcore, _get_rng
import random
from numpy import *
from scipy.optimize import fsolve, fminbound
import scipy.special

from .. dl_import import dl_import
dl_import("from . import libgraph_tool_community as libcommunity")


class BlockState(object):
    r"""This class encapsulates the block state of a given graph.

    This must be instantiated and used by functions such as :func:`mcmc_sweep`.

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
    eweight : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
        Edge weights (i.e. multiplicity).
    vweight : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
        Vertex weights (i.e. multiplicity).
    b : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
        Initial block labels on the vertices. If not supplied, it will be
        randomly sampled.
    B : ``int`` (optional, default: ``None``)
        Number of blocks. If not supplied it will be either obtained from the
        parameter ``b``, or set to the maximum possible value according to the
        minimum description length.
    clabel : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
        This parameter provides a constraint label, such that vertices with
        different labels will not be allowed to belong to the same block. If not given,
        all labels will be assumed to be the same.
    deg_corr : ``bool`` (optional, default: ``True``)
        If ``True``, the degree-corrected version of the blockmodel ensemble will
        be assumed, otherwise the traditional variant will be used.

    """

    def __init__(self, g, eweight=None, vweight=None, b=None, clabel=None, B=None,
                 deg_corr=True):
        self.g = g
        self.bg = Graph(directed=g.is_directed())
        self.mrs = self.bg.new_edge_property("int")
        self.mrp = self.bg.new_vertex_property("int")
        if g.is_directed():
            self.mrm = self.bg.new_vertex_property("int")
        else:
            self.mrm = self.mrp
        self.wr = self.bg.new_vertex_property("int")

        if eweight is None:
            eweight = g.new_edge_property("int")
            eweight.a = 1
        elif eweight.value_type() != "int32_t":
            eweight = eweight.copy(value_type="int32_t")
        if vweight is None:
            vweight = g.new_vertex_property("int")
            vweight.a = 1
        elif vweight.value_type() != "int32_t":
            vweight = vweight.copy(value_type="int32_t")
        self.eweight = eweight
        self.vweight = vweight

        self.E = self.eweight.a.sum()
        self.N = self.vweight.a.sum()

        self.deg_corr = deg_corr

        if clabel is None:
            self.clabel = g.new_vertex_property("int")
            self.L = 1
        else:
            self.clabel = clabel
            self.L = int(self.clabel.a.max() + 1)

        if b is None:
            if B is None:
                B = get_max_B(self.N, self.E, directed=g.is_directed())
            B = int(ceil(B/float(self.L)) * self.L)
            b = g.new_vertex_property("int")
            b.a = random.randint(0, B / self.L, len(b.a))
            b.a = self.clabel.a + b.a * self.L
            self.b = b
        else:
            if B is None:
                B = int(b.a.max()) + 1
            B = int(ceil(B/float(self.L)) * self.L)
            self.b = b = b.copy(value_type="int32_t")

        cg, br, vcount, ecount = condensation_graph(g, b,
                                                    vweight=vweight,
                                                    eweight=eweight,
                                                    self_loops=True)[:4]
        self.bg.add_vertex(B)
        if b.a.max() >= B:
            raise ValueError("Maximum value of b is largest or equal to B!")

        self.vertices = libcommunity.get_vector(B)
        self.vertices.a = arange(B)

        self.mrp.a = 0
        self.mrm.a = 0
        self.mrs.a = 0
        for e in cg.edges():
            r = self.bg.vertex(br[e.source()])
            s = self.bg.vertex(br[e.target()])
            be = self.bg.add_edge(r, s)
            if self.bg.is_directed():
                self.mrs[be] = ecount[e]
            else:
                self.mrs[be] = ecount[e] if r != s else 2 * ecount[e]
            self.mrp[r] += ecount[e]
            self.mrm[s] += ecount[e]

        self.wr.a = 0
        for v in cg.vertices():
            r = self.bg.vertex(br[v])
            self.wr[r] = vcount[v]

        self.__regen_emat()
        self.__build_egroups()

    def __regen_emat(self):
        self.emat = libcommunity.create_emat(self.bg._Graph__graph, len(self.vertices))

    def __build_egroups(self):
        self.esrcpos = self.g.new_edge_property("vector<int>")
        self.etgtpos = self.g.new_edge_property("vector<int>")
        self.egroups = libcommunity.build_egroups(self.g._Graph__graph,
                                                self.bg._Graph__graph,
                                                _prop("v", self.g, self.b),
                                                _prop("e", self.g, self.eweight),
                                                _prop("e", self.g, self.esrcpos),
                                                _prop("e", self.g, self.etgtpos))

    def get_blocks(self):
        r"""Returns the property map which contains the block labels for each vertex."""
        return self.b

    def get_bg(self):
        r"""Returns the block graph."""
        return self.bg

    def get_ers(self):
        r"""Returns the edge property map of the block graph which contains the :math:`e_{rs}` matrix entries."""
        return self.mrs

    def get_er(self):
        r"""Returns the vertex property map of the block graph which contains the number
        :math:`e_r` of half-edges incident on block :math:`r`. If the graph is
        directed, a pair of property maps is returned, with the number of
        out-edges :math:`e^+_r` and in-edges :math:`e^-_r`, respectively."""
        if self.bg.is_directed():
            return self.mrp. self.mrm
        else:
            return self.mrp

    def get_nr(self):
        r"""Returns the vertex property map of the block graph which contains the block sizes :math:`n_r`."""
        return self.wr

    def get_eweight(self):
        r"""Returns the block edge counts associated with the block matrix
        :math:`e_{rs}`. For directed graphs it is identical to :math:`e_{rs}`,
        but for undirected graphs it is identical except for the diagonal, which
        is :math:`e_{rr}/2`."""
        eweight = self.mrs.copy()
        if not self.g.is_directed():
            for r in self.bg.vertices():
                e = self.bg.edge(r, r)
                if e is not None:
                    eweight[e] /= 2
        return eweight

    def get_clabel(self):
        r"""Obtain the constraint label associated with each block."""
        blabel = self.bg.vertex_index.copy(value_type="int")
        blabel.a = blabel.a % self.L
        return blabel

    def entropy(self, complete=False, random=False, dl=False):
        r"""Calculate the entropy per edge associated with the current block partition.

        Parameters
        ----------
        complete : ``bool`` (optional, default: ``False``)
            If ``True``, the complete entropy will be returned, including constant
            terms not relevant to the block partition.
        random : ``bool`` (optional, default: ``False``)
            If ``True``, the entropy entropy corresponding to an equivalent random
            graph (i.e. no block partition) will be returned.
        dl : ``bool`` (optional, default: ``False``)
            If ``True``, the full description length will be returned.


        Notes
        -----

        For the traditional blockmodel (``deg_corr == False``), the entropy is
        given by

        .. math::

          \mathcal{S}_t &\cong E - \frac{1}{2} \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{n_rn_s}\right), \\
          \mathcal{S}^d_t &\cong E - \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{n_rn_s}\right),

        for undirected and directed graphs, respectively, where :math:`e_{rs}`
        is the number of edges from block :math:`r` to :math:`s` (or the number
        of half-edges for the undirected case when :math:`r=s`), and :math:`n_r`
        is the number of vertices in block :math:`r` .


        For the degree-corrected variant with "hard" degree constraints the
        equivalent expressions are

        .. math::

            \mathcal{S}_c &\cong -E -\sum_kN_k\ln k! - \frac{1}{2} \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{e_re_s}\right), \\
            \mathcal{S}^d_c &\cong -E -\sum_{k^+}N_{k^+}\ln k^+!  -\sum_{k^-}N_{k^-}\ln k^-! - \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{e^+_re^-_s}\right),

        where :math:`e_r = \sum_se_{rs}` is the number of half-edges incident on
        block :math:`r`, and :math:`e^+_r = \sum_se_{rs}` and :math:`e^-_r =
        \sum_se_{sr}` are the number of out- and in-edges adjacent to block
        :math:`r`, respectively.

        If ``complete == False`` only the last term of the equations above will
        be returned. If ``random == True`` it will be assumed that :math:`B=1`
        despite the actual :math:`e_{rs}` matrix.  If ``dl == True``, the
        description length :math:`\mathcal{L}_t` of the model will be returned
        as well, as described in :func:`model_entropy`. Note that for the
        degree-corrected version the description length is

        .. math::

            \mathcal{L}_c = \mathcal{L}_t - N\sum_kp_k\ln p_k,

        where :math:`p_k` is the fraction of nodes with degree :math:`p_k`, and
        we have instead :math:`k \to (k^-, k^+)` for directed graphs.

        Note that the value returned corresponds to the entropy `per edge`,
        i.e. :math:`(\mathcal{S}_{t/c}\; [\,+\, \mathcal{L}_{t/c}])/ E`.

        """

        E = self.E
        N = self.N

        if self.deg_corr:
            if self.g.is_directed():
                S_rand = E * log(E)
            else:
                S_rand = E * log(2 * E)
        else:
            ak = E / float(N) if self.g.is_directed() else  2 * E / float(N)
            S_rand = E * log (N / ak)

        if random:
            S = S_rand
        else:
            S = libcommunity.entropy(self.bg._Graph__graph,
                                   _prop("e", self.bg, self.mrs),
                                   _prop("v", self.bg, self.mrp),
                                   _prop("v", self.bg, self.mrm),
                                   _prop("v", self.bg, self.wr),
                                   self.deg_corr)

        if complete:
            if deg_seq and self.deg_corr:
                S_seq = 0
                hist = defaultdict(int)
                for v in self.g.vertices():
                    hist[(v.in_degree(), v.out_degree())] += 1
                for k, v in hist.iteritems():
                    p = v / float(self.g.num_vertices())
                    S_seq -= p * log(p)
                S_seq *= self.g.num_vertices()
                S += S_seq

            if self.deg_corr:
                S -= E
                for v in self.g.vertices():
                    S -= scipy.special.gammaln(v.out_degree() + 1)
                    if self.g.is_directed():
                        S -= scipy.special.gammaln(v.in_degree() + 1)
            else:
                S += E
        else:
            S -= S_rand


        if dl:
            if random:
                S += model_entropy(1, N, E, directed=self.g.is_directed()) * E
            else:
                S += model_entropy(len(self.vertices), N, E, directed=self.g.is_directed()) * E

        return S / E

    def __min_dl(self):
        return self.entropy(complete=False, dl=True)

    def dist(self, r, s):
        r"""Compute the "distance" between blocks `r` and `s`, i.e. the entropy
        difference after they are merged together."""
        return libcommunity.dist(self.bg._Graph__graph, int(r), int(s),
                                 _prop("e", self.bg, self.mrs),
                                 _prop("v", self.bg, self.mrp),
                                 _prop("v", self.bg, self.mrm),
                                 _prop("v", self.bg, self.wr),
                                 self.deg_corr)


    def join(self, r, s):
        r"""Merge blocks `r` and `s` into a single block."""
        libcommunity.join(self.bg._Graph__graph, int(r), int(s),
                          _prop("e", self.bg, self.mrs),
                          _prop("v", self.bg, self.mrp),
                          _prop("v", self.bg, self.mrm),
                          _prop("v", self.bg, self.wr),
                          self.deg_corr,
                          self.vertices)
        del self.egroups

    def remove_vertex(self, v):
        r"""Remove vertex `v` from its current block."""
        libcommunity.remove_vertex(self.g._Graph__graph,
                                   self.bg._Graph__graph,
                                   int(v),
                                   _prop("e", self.bg, self.mrs),
                                   _prop("v", self.bg, self.mrp),
                                   _prop("v", self.bg, self.mrm),
                                   _prop("v", self.bg, self.wr),
                                   _prop("v", self.g, self.b))
        del self.egroups


    def add_vertex(self, v, r):
        r"""Add vertex `v` to block `r`."""
        libcommunity.add_vertex(v.get_graph()._Graph__graph,
                                self.bg._Graph__graph,
                                int(v), int(r),
                                _prop("e", self.bg, self.mrs),
                                _prop("v", self.bg, self.mrp),
                                _prop("v", self.bg, self.mrm),
                                _prop("v", self.bg, self.wr),
                                _prop("v", self.g, self.b))
        del self.egroups

    def move_vertex(self, v, nr):
        r"""Move vertex `v` to block `r`, and return the entropy difference."""
        dS = libcommunity.move_vertex(self.g._Graph__graph,
                                      self.bg._Graph__graph,
                                      self.emat,
                                      int(v), int(nr),
                                      _prop("e", self.bg, self.mrs),
                                      _prop("v", self.bg, self.mrp),
                                      _prop("v", self.bg, self.mrm),
                                      _prop("v", self.bg, self.wr),
                                      _prop("v", self.g, self.b),
                                      self.deg_corr,
                                      _prop("e", self.bg, self.eweight),
                                      _prop("v", self.bg, self.vweight))
        del self.egroups
        return dS / float(self.E)


    def get_dist_matrix(self):
        r"""Return the distance matrix between all blocks. The distance is
        defined as the entropy difference after two blocks are merged."""
        dist_matrix = zeros((len(self.vertices), len(self.vertices)))
        for ri, r in enumerate(self.vertices):
            for si, s in enumerate(self.vertices):
                if si > ri:
                    dist_matrix[ri, si] = self.dist(r, s)
        for ri, r in enumerate(self.vertices):
            for si, s in enumerate(self.vertices):
                if si < ri:
                    dist_matrix[ri, si] = dist_matrix[si, ri]
        return dist_matrix


    def get_matrix(self, reorder=False, clabel=None, niter=0, ret_order=False):
        r"""Returns the block matrix.

        Parameters
        ----------
        reorder : ``bool`` (optional, default: ``False``)
            If ``True``, the matrix is reordered so that blocks which are
            'similar' are close together.
        clabel : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
            Constraint labels to be imposed during reordering. Only has
            effect if ``reorder == True``.
        niter : ``int`` (optional, default: `0`)
            Number of iterations performed to obtain the best ordering. If
            ``niter == 0`` it will automatically determined. Only has effect
            if ``reorder == True``.
        ret_order : ``bool`` (optional, default: ``False``)
            If ``True``, the vertex ordering is returned. Only has effect if
            ``reorder == True``.

        Examples
        --------

        .. testsetup:: get_matrix

           gt.seed_rng(42)
           np.random.seed(42)
           from pylab import *

        .. doctest:: get_matrix

           >>> g = gt.collection.data["polbooks"]
           >>> state = gt.BlockState(g, B=5, deg_corr=True)
           >>> for i in range(1000):
           ...     ds, nmoves = gt.mcmc_sweep(state)
           >>> m = state.get_matrix(reorder=True)
           >>> figure()
           <...>
           >>> matshow(m)
           <...>
           >>> savefig("bloc_mat.pdf")

        .. testcleanup:: get_matrix

           savefig("bloc_mat.png")

        .. figure:: bloc_mat.*
           :align: center

           A  5x5 block matrix.

       """
        B = len(self.vertices)
        vmap = {}
        for r in range(len(self.vertices)):
            vmap[self.vertices[r]] = r

        if reorder:
            if niter == 0:
                niter = max(10 * len(self.vertices), 100)

            states = []

            label = None
            states = [self]
            Bi = B / 2

            while Bi > 1:
                cblabel = states[-1].get_clabel()
                if clabel is not None and len(states) == 1:
                    clabel.a += (cblabel.a.max() + 1) * clabel.a
                state = BlockState(states[-1].bg, B=Bi,
                                   clabel=clabel,
                                   vweight=states[-1].wr,
                                   eweight=states[-1].get_eweight(),
                                   deg_corr=states[-1].deg_corr)

                for i in range(niter):
                    mcmc_sweep(state, beta=1.)
                for i in range(niter):
                    mcmc_sweep(state, beta=float("inf"))

                states.append(state)

                Bi /= 2

                if Bi < cblabel.a.max() + 1:
                    break


498
            vorder = list(range(len(states[-1].vertices)))
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            for state in reversed(states[1:]):
                norder = [[] for i in range(len(state.vertices))]
                #print(state.L)
                for v in state.g.vertices():
                    pos = vorder.index(state.b[v])
                    norder[pos].append(int(v))
                vorder = [item for sublist in norder for item in sublist]
        else:
            vorder = self.vertices

        order_map = zeros(B, dtype="int")
        for i, v in enumerate(vorder):
            order_map[vmap[v]] = i

        m = zeros((B, B))
        rmap = {}
        for e in self.bg.edges():
            r, s = vmap[int(e.source())], vmap[int(e.target())]
            r = order_map[r]
            s = order_map[s]
            rmap[r] = int(e.source())
            rmap[s] = int(e.target())
            m[r, s] = self.mrs[e]
            if not self.bg.is_directed():
                m[s, r] = m[r, s]

        if ret_order:
            return m, rmap
        else:
            return m

    def modularity(self):
        r"""Computes the modularity of the current block structure."""
        Q = 0
        for vi in self.vertices:
            v = self.bg.vertex(vi)
            err = 0
            for e in v.out_edges():
                if e.target() == v:
                    err = self.mrs[e]
                    break
            er = 0
            if self.bg.is_directed():
                err /= float(self.E)
                er = self.mrp[v] * self.mrm[v] / float(self.E ** 2)
            else:
                err /= float(2 * self.E)
                er = self.mrp[v] ** 2 / float(4 * self.E ** 2)
            Q += err - er
        return Q

def model_entropy(B, N, E, directed):
    r"""Computes the amount of information necessary for the parameters of the traditional blockmodel ensemble, for ``B`` blocks, ``N`` vertices, ``E`` edges, and either a directed or undirected graph.

    A traditional blockmodel is defined as a set of :math:`N` vertices which can
    belong to one of :math:`B` blocks, and the matrix :math:`e_{rs}` describes
    the number of edges from block :math:`r` to :math:`s` (or twice that number
    if :math:`r=s` and the graph is undirected).

    For an undirected graph, the number of distinct :math:`e_{rs}` matrices is given by,

    .. math::

       \Omega_m = \left(\!\!{\left(\!{B \choose 2}\!\right) \choose E}\!\!\right)

    and for a directed graph,

    .. math::
       \Omega_m = \left(\!\!{B^2 \choose E}\!\!\right)


    where :math:`\left(\!{n \choose k}\!\right) = {n+k-1\choose k}` is the
    number of :math:`k` combinations with repetitions from a set of size :math:`n`.

    The total information necessary to describe the model is then,

    .. math::

       \mathcal{L}_t = \ln\Omega_m + N\ln B,


    where :math:`N\ln B` is the information necessary to describe the
    block partition.

    References
    ----------

    .. [peixoto-parsimonious-2012] Tiago P. Peixoto "Parsimonious module inference in large networks",
       (2012), :arxiv:`1212.4794`.

    """
    return libcommunity.SB(float(B), int(N), int(E), directed)


def Sdl(B, S, N, E, directed=False):
    return S + model_entropy(B, N, E, directed)


def get_max_B(N, E, directed=False):
    r"""Return the maximum detectable number of blocks, obtained by minimizing:

    .. math::

        \mathcal{L}_t(B, N, E) - E\ln B

    where :math:`\mathcal{L}_t(B, N, E)` is the information necessary to
    describe a traditional blockmodel with `B` blocks, `N` nodes and `E`
    edges (see :func:`model_entropy`).

    Examples
    --------

    >>> gt.get_max_B(N=1e6, E=5e6)
    1572

    References
    ----------
    .. [peixoto-parsimonious-2012] Tiago P. Peixoto "Parsimonious module inference in large networks",
       (2012), :arxiv:`1212.4794`.


    """

    B = fminbound(lambda B: Sdl(B, -log(B), N, E, directed), 1, E,
                  xtol=1e-6, maxfun=1500, disp=0)
    if isnan(B):
        B = 1
    return max(int(ceil(B)), 2)

def get_akc(B, I, N=float("inf"), directed=False):
    r"""Return the minimum value of the average degree of the network, so that some block structure with :math:`B` blocks can be detected, according to the minimum escription length criterion.

    This is obtained by solving

    .. math::

       \Sigma_b = \mathcal{L}_t(B, N, E) - E\mathcal{I}_{t/c} = 0,

    where :math:`\mathcal{L}_{t}` is the necessary information to describe the
    blockmodel parameters (see :func:`model_entropy`), and
    :math:`\mathcal{I}_{t/c}` characterizes the planted block structure, and is
    given by

    .. math::

        \mathcal{I}_t &= \sum_{rs}m_{rs}\ln\left(\frac{m_{rs}}{w_rw_s}\right),\\
        \mathcal{I}_c &= \sum_{rs}m_{rs}\ln\left(\frac{m_{rs}}{m_rm_s}\right),

    where :math:`m_{rs} = e_{rs}/2E` (or :math:`m_{rs} = e_{rs}/E` for directed
    graphs) and :math:`w_r=n_r/N`. We note that :math:`\mathcal{I}_{t/c}\in[0,
    \ln B]`. In the case where :math:`E \gg B^2`, this simplifies to

    .. math::

       \left<k\right>_c &= \frac{2\ln B}{\mathcal{I}_{t/c}},\\
       \left<k^{-/+}\right>_c &= \frac{\ln B}{\mathcal{I}_{t/c}},

    for undirected and directed graphs, respectively. This limit is assumed if
    ``N == inf``.

    Examples
    --------

    >>> gt.get_akc(10, log(10) / 100, N=100)
    2.4199998721289937

    References
    ----------
    .. [peixoto-parsimonious-2012] Tiago P. Peixoto "Parsimonious module inference in large networks",
       (2012), :arxiv:`1212.4794`.

    """
    if N != float("inf"):
        if directed:
            get_dl = lambda ak: model_entropy(B, N, N * ak, directed) - N * ak * I
        else:
            get_dl = lambda ak: model_entropy(B, N, N * ak / 2., directed) - N * ak * I / 2.
        ak = fsolve(lambda ak: get_dl(ak), 10)
        ak = float(ak)
    else:
        ak = 2 * log(B) / S
        if directed:
            ak /= 2
    return ak

def min_dist(state, n=0):
    r"""Return the minimum distance between all blocks, and the block pair which minimizes it.

    The parameter `state` must be an instance of the
    :class:`~graph_tool.community.BlockState` class, and `n` is the number of
    block pairs to sample. If `n == 0` all block pairs are sampled.


    Examples
    --------

    .. testsetup:: min_dist

       gt.seed_rng(42)
       np.random.seed(42)

    .. doctest:: min_dist

       >>> g = gt.collection.data["polbooks"]
       >>> state = gt.BlockState(g, B=4, deg_corr=True)
       >>> for i in range(1000):
       ...     ds, nmoves = gt.mcmc_sweep(state)
       >>> gt.min_dist(state)
       (795.7694502418635, 2, 3)

    """
    min_d, r, s = libcommunity.min_dist(state.bg._Graph__graph, int(n),
                                        _prop("e", state.bg, state.mrs),
                                        _prop("v", state.bg, state.mrp),
                                        _prop("v", state.bg, state.mrm),
                                        _prop("v", state.bg, state.wr),
                                        state.deg_corr,
                                        state.vertices,
                                        _get_rng())
    return min_d, r, s


def mcmc_sweep(state, beta=1., sequential=True, verbose=False, vertices=None):
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    r"""Performs a Monte Carlo Markov chain sweep on the network, to sample the block partition according to a probability :math:`\propto e^{-\beta \mathcal{S}_{t/c}}`, where :math:`\mathcal{S}_{t/c}` is the blockmodel entropy.
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    Parameters
    ----------
    state : :class:`~graph_tool.community.BlockState`
        The block state.
    beta : `float` (optional, default: `1.0`)
        The inverse temperature parameter :math:`\beta`.
    sequential : ``bool`` (optional, default: ``True``)
        If ``True``, the move attempts on the vertices are done in sequential
        random order. Otherwise a total of `N` moves attempts are made, where
        `N` is the number of vertices, where each vertex can be selected with
        equal probability.
    verbose : ``bool`` (optional, default: ``False``)
        If ``True``, verbose information is displayed.

    Returns
    -------

    dS : `float`
       The entropy difference (per edge) after a full sweep.
    nmoves : ``int``
       The number of accepted block membership moves.


    Notes
    -----

    This algorithm performs a Monte Carlo Markov chain sweep on the network,
    where the block memberships are randomly moved, and either accepted or
    rejected, so that after sufficiently many sweeps the partitions are sampled
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    with probability proportional to :math:`e^{-\beta\mathcal{S}_{t/c}}`, where
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    :math:`\mathcal{S}_{t/c}` is the blockmodel entropy, given by

    .. math::

      \mathcal{S}_t &\cong - \frac{1}{2} \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{n_rn_s}\right), \\
      \mathcal{S}^d_t &\cong - \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{n_rn_s}\right),

    for undirected and directed traditional blockmodels (``deg_corr == False``),
    respectively, where :math:`e_{rs}` is the number of edges from block
    :math:`r` to :math:`s` (or the number of half-edges for the undirected case
    when :math:`r=s`), and :math:`n_r` is the number of vertices in block
    :math:`r`, and constant terms which are independent of the block partition
    were dropped (see :meth:`BlockState.entropy` for the complete entropy). For
    the degree-corrected variant with "hard" degree constraints the equivalent
    expressions are

    .. math::

       \mathcal{S}_c &\cong  - \frac{1}{2} \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{e_re_s}\right), \\
       \mathcal{S}^d_c &\cong - \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{e^+_re^-_s}\right),

    where :math:`e_r = \sum_se_{rs}` is the number of half-edges incident on
    block :math:`r`, and :math:`e^+_r = \sum_se_{rs}` and :math:`e^-_r =
    \sum_se_{sr}` are the number of out- and in-edges adjacent to block
    :math:`r`, respectively.

    The Monte Carlo algorithm employed attempts to improve the mixing time of
    the markov chain by proposing membership moves :math:`r\to s` with
    probability :math:`p(r\to s|t) \propto e_{ts} + 1`, where :math:`t` is the
    block label of a random neighbour of the vertex being moved. See
    [peixoto-parsimonious-2012]_ for more details.

    This algorithm has a complexity of :math:`O(E)`, where :math:`E` is the
    number of edges in the network.

    Examples
    --------
    .. testsetup:: mcmc

       gt.seed_rng(42)
       np.random.seed(42)

    .. doctest:: mcmc

       >>> g = gt.collection.data["polbooks"]
       >>> state = gt.BlockState(g, B=3, deg_corr=True)
       >>> pv = None
       >>> for i in range(1000):        # remove part of the transient
       ...     ds, nmoves = gt.mcmc_sweep(state)
       >>> for i in range(1000):
       ...     ds, nmoves = gt.mcmc_sweep(state)
       ...     pv = gt.collect_vertex_marginals(state, pv)
       >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_shape="pie", vertex_pie_fractions=pv, output="polbooks_blocks_soft.pdf")
       <...>

    .. testcleanup:: mcmc

       gt.graph_draw(g, pos=g.vp["pos"], vertex_shape="pie", vertex_pie_fractions=pv, output="polbooks_blocks_soft.png")

    .. figure:: polbooks_blocks_soft.*
       :align: center

       "Soft" block partition of a political books network with :math:`B=3`.

     References
    ----------

    .. [holland-stochastic-1983] Paul W. Holland, Kathryn Blackmond Laskey,
       Samuel Leinhardt, "Stochastic blockmodels: First steps",
       Carnegie-Mellon University, Pittsburgh, PA 15213, U.S.A., :doi:`10.1016/0378-8733(83)90021-7`
    .. [faust-blockmodels-1992] Katherine Faust, and Stanley
       Wasserman. "Blockmodels: Interpretation and Evaluation." Social Networks
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       14, no. 1-2 (1992): 5-61. :doi:`10.1016/0378-8733(92)90013-W`
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    .. [karrer-stochastic-2011] Brian Karrer, and M. E. J. Newman. "Stochastic
       Blockmodels and Community Structure in Networks." Physical Review E 83,
       no. 1 (2011): 016107. :doi:`10.1103/PhysRevE.83.016107`.
    .. [peixoto-entropy-2012] Tiago P. Peixoto "Entropy of Stochastic Blockmodel
       Ensembles." Physical Review E 85, no. 5 (2012): 056122. :doi:`10.1103/PhysRevE.85.056122`,
       :arxiv:`1112.6028`.
    .. [peixoto-parsimonious-2012] Tiago P. Peixoto "Parsimonious module inference in large networks",
       (2012), :arxiv:`1212.4794`.
    """

    if len(state.vertices) == 1:
        return 0., 0

    if vertices is None:
        u = state.g
        u.stash_filter()
        vertices = libcommunity.get_vector(u.num_vertices())
        u.pop_filter()
        vertices.a = arange(len(vertices.a))

    dS, nmoves = libcommunity.move_sweep(state.g._Graph__graph,
                                         state.bg._Graph__graph,
                                         state.emat,
                                         _prop("e", state.bg, state.mrs),
                                         _prop("v", state.bg, state.mrp),
                                         _prop("v", state.bg, state.mrm),
                                         _prop("v", state.bg, state.wr),
                                         _prop("v", state.g, state.b),
                                         _prop("v", state.g, state.clabel),
                                         state.L,
                                         vertices,
                                         state.deg_corr,
                                         _prop("e", state.g, state.eweight),
                                         _prop("v", state.g, state.vweight),
                                         state.egroups,
                                         _prop("e", state.g, state.esrcpos),
                                         _prop("e", state.g, state.etgtpos),
                                         float(beta), sequential,
                                         verbose, _get_rng())
    return dS / state.E, nmoves




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def mc_get_dl(state, nsweep, greedy, rng, checkpoint=None, anneal=1,
              verbose=False):
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    if len(state.vertices) == 1:
        return state._BlockState__min_dl()

    S = state._BlockState__min_dl()

    if nsweep >= 0:
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        for i in range(nsweep):
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            delta, nmoves = mcmc_sweep(state, beta=1, rng=rng)
            if checkpoint is not None:
                checkpoint(state, S, delta, nmoves)
            S += delta
        if greedy:
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            for i in range(nsweep):
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                delta, nmoves = mcmc_sweep(state, beta=float("inf"))
                if checkpoint is not None:
                    checkpoint(state, S, delta, nmoves)
                    S += delta
    else:
        # adaptive mode
        min_dl = S
        max_dl = S
        count = 0
        time = 0
        bump = False

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        beta = 1.

        last_min = min_dl

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        if verbose:
            print("beta = %g" % beta)
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        while True:
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            delta, nmoves = mcmc_sweep(state, beta=beta)
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            if checkpoint is not None:
                checkpoint(state, S, delta, nmoves)
            S += delta

            if S < min_dl:
                min_dl = S
                count = 0
            elif S > max_dl:
                max_dl = S
                count = 0
            else:
                count += 1

            if count > abs(nsweep):
                if not bump:
                    min_dl = max_dl = S
                    bump = True
                    count = 0
                else:
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                    if anneal <= 1 or min_dl == last_min:
                        break
                    else:
                        beta *= anneal
                        count = 0
                        last_min = min_dl
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                        if verbose:
                            print("beta = %g" % beta)
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        min_dl = S
        count = 0
        while True:
            delta, nmoves = mcmc_sweep(state, beta=float("inf"))
            if checkpoint is not None:
                checkpoint(state, S, delta, nmoves)
            S += delta
            if S < min_dl:
                min_dl = S
                count = 0
            else:
                count += 1
            if count > abs(nsweep):
                break

    return state._BlockState__min_dl()

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def get_b_dl(g, bs, bs_start, B, nsweep, anneal, greedy, clabel, deg_corr, rng,
             checkpoint=None, verbose=False):
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    prev_dl = float("inf")
    if B in bs:
        return bs[B][0]
    elif B in bs_start:
        if verbose:
            print("starting from previous result for B=%d" % B)
        prev_dl, b = bs_start[B]
        state = BlockState(g, b=b.copy(), clabel=clabel, deg_corr=deg_corr)
    else:
        n_iter = 0
        bs_keys = [k for k in bs.keys() if type(k) != str]
        B_sup = max(bs_keys) if len(bs_keys) > 0 else B
        for Bi in bs_keys:
            if Bi > B and Bi < B_sup:
                B_sup = Bi
        if B_sup == B:
            state = BlockState(g, B=B, clabel=clabel, deg_corr=deg_corr)
            b = state.b
        else:
            if verbose:
                print("shrinking from", B_sup, "to", B)
            b = bs[B_sup][1].copy()

            cg, br, vcount, ecount = condensation_graph(g, b, self_loops=True)[:4]

            blabel = cg.new_vertex_property("int")
            if clabel is not None:
                blabel.a = br.a % (clabel.a.max() + 1)

            bg_state = BlockState(cg, B=B, clabel=blabel, vweight=vcount, eweight=ecount,
                                  deg_corr=deg_corr)

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            mc_get_dl(bg_state, nsweep=nsweep, greedy=greedy, rng=rng,
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                      checkpoint=checkpoint, anneal=anneal, verbose=verbose)
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            ### FIXME: the following could be improved by moving it to the C++
            ### side
            bmap = {}
            for v in bg_state.g.vertices():
               bmap[br[v]] = v
            for v in g.vertices():
                b[v] = bg_state.b[bmap[b[v]]]

            state = BlockState(g, b=b, B=B, clabel=clabel, deg_corr=deg_corr)

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    dl = mc_get_dl(state, nsweep=nsweep,  greedy=greedy, rng=rng,
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                   checkpoint=checkpoint, anneal=anneal, verbose=verbose)