blockmodel.py 89.6 KB
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#! /usr/bin/env python
# -*- coding: utf-8 -*-
#
# graph_tool -- a general graph manipulation python module
#
Tiago Peixoto's avatar
Tiago Peixoto committed
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# Copyright (C) 2006-2013 Tiago de Paula Peixoto <tiago@skewed.de>
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#
# 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

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from .. import _degree, _prop, Graph, GraphView, libcore, _get_rng, PropertyMap
from .. stats import label_self_loops
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import random
from numpy import *
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import numpy
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from scipy.optimize import fsolve, fminbound
import scipy.special
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from collections import defaultdict
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import copy
import heapq
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from .. dl_import import dl_import
dl_import("from . import libgraph_tool_community as libcommunity")


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def get_block_graph(g, B, b, vcount, ecount):
    cg, br, vcount, ecount = condensation_graph(g, b,
                                                vweight=vcount,
                                                eweight=ecount,
                                                self_loops=True)[:4]
    cg.vp["count"] = vcount
    cg.ep["count"] = ecount
    cg = Graph(cg, vorder=br)

    cg.add_vertex(B - cg.num_vertices())
    return cg

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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`
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        Graph to be modelled.
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    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.
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    max_BE : ``int`` (optional, default: ``1000``)
        If the number of blocks exceeds this number, a sparse representation of
        the block graph is used, which is slightly less efficient, but uses less
        memory,
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    """

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    def __init__(self, g, eweight=None, vweight=None, b=None,
                 B=None, clabel=None, deg_corr=True, max_BE=1000):
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        self.g = g
        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

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        self.E = int(self.eweight.fa.sum())
        self.N = int(self.vweight.fa.sum())
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        self.deg_corr = deg_corr

        if b is None:
            if B is None:
                B = get_max_B(self.N, self.E, directed=g.is_directed())
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            ba = random.randint(0, B, g.num_vertices())
            ba[:B] = arange(B)        # avoid empty blocks
            random.shuffle(ba)
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            b = g.new_vertex_property("int")
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            b.fa = ba
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            self.b = b
        else:
            if B is None:
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                B = int(b.fa.max()) + 1
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            self.b = b = b.copy(value_type="int32_t")

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        if b.fa.max() >= B:
            raise ValueError("Maximum value of b is larger or equal to B!")

        # Construct block-graph
        self.bg = get_block_graph(g, B, b, vweight, eweight)
        self.bg.set_fast_edge_removal()

        self.mrs = self.bg.ep["count"]
        self.wr = self.bg.vp["count"]
        del self.bg.ep["count"]
        del self.bg.vp["count"]

        self.mrp = self.bg.degree_property_map("out", weight=self.mrs)

        if g.is_directed():
            self.mrm = self.bg.degree_property_map("in", weight=self.mrs)
        else:
            self.mrm = self.mrp
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        self.vertices = libcommunity.get_vector(B)
        self.vertices.a = arange(B)
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        self.B = B
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        self.clabel = clabel
        if self.clabel is None:
            self.clabel = self.g.new_vertex_property("int")

        self.bclabel = self.bg.new_vertex_property("int")
        libcommunity.vector_rmap(self.b.a, self.bclabel.a)
        libcommunity.vector_map(self.bclabel.a, self.clabel.a)

        self.emat = None
        if max_BE is None:
            max_BE = 1000
        self.max_BE = max_BE

        # used by mcmc_sweep()
        self.egroups = None
        self.nsampler = None
        self.sweep_vertices = None

        # used by merge_sweep()
        self.bnsampler = None
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        libcommunity.init_safelog(int(2 * max(self.E, self.N)))
        libcommunity.init_xlogx(int(2 * max(self.E, self.N)))
        libcommunity.init_lgamma(int(3 * max(self.E, self.N)))
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    def __get_emat(self):
        if self.emat is None:
            self.__regen_emat()
        return self.emat
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    def __regen_emat(self):
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        if self.B <= self.max_BE:
            self.emat = libcommunity.create_emat(self.bg._Graph__graph)
        else:
            self.emat = libcommunity.create_ehash(self.bg._Graph__graph)
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    def __build_egroups(self, empty=False):
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        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,
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                                                  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),
                                                  empty)

    def __build_nsampler(self):
        self.nsampler = libcommunity.init_neighbour_sampler(self.g._Graph__graph,
                                                            _prop("e", self.g, self.eweight))
    def __build_bnsampler(self):
        self.bnsampler = libcommunity.init_neighbour_sampler(self.bg._Graph__graph,
                                                             _prop("e", self.bg, self.mrs))

    def __cleanup_bg(self):
        emask = self.bg.new_edge_property("bool")
        emask.a = self.mrs.a[:len(emask.a)] > 0
        self.bg.set_edge_filter(emask)
        self.bg.purge_edges()
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    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():
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            sl = label_self_loops(self.bg, mark_only=True)
            eweight.a[sl.a > 0] /= 2
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        return eweight

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    def entropy(self, complete=False, random=False, dl=False, dense=False,
                multigraph=False):
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        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.
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        dense : ``bool`` (optional, default: ``False``)
            If ``True``, the "dense" variant of the entropy will be computed.
        multigraph : ``bool`` (optional, default: ``False``)
            If ``True``, the multigraph entropy will be used. Only has an effect
            if ``dense == True``.
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        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.

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        If the "dense" entropies are requested, they will be computed as

        .. math::

            \mathcal{S}_t  &= \sum_{r>s} \ln{\textstyle {n_rn_s \choose e_{rs}}} + \sum_r \ln{\textstyle {{n_r\choose 2}\choose e_{rr}/2}}\\
            \mathcal{S}^d_t  &= \sum_{rs} \ln{\textstyle {n_rn_s \choose e_{rs}}},

        for simple graphs, and

        .. math::

            \mathcal{S}_m  &= \sum_{r>s} \ln{\textstyle \left(\!\!{n_rn_s \choose e_{rs}}\!\!\right)} + \sum_r \ln{\textstyle \left(\!\!{\left(\!{n_r\choose 2}\!\right)\choose e_{rr}/2}\!\!\right)}\\
            \mathcal{S}^d_m  &= \sum_{rs} \ln{\textstyle \left(\!\!{n_rn_s \choose e_{rs}}\!\!\right)},

        for multigraphs (i.e. ``multigraph == True``).

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

        """

        E = self.E
        N = self.N

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        if dense:
            if random:
                bg = get_block_graph(self.bg, 1,
                                     self.bg.new_vertex_property("int"),
                                     self.wr, self.mrs)
                S = libcommunity.entropy_dense(bg._Graph__graph,
                                               _prop("e", bg, bg.ep["count"]),
                                               _prop("v", bg, bg.vp["count"]),
                                               multigraph)
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            else:
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                S = libcommunity.entropy_dense(self.bg._Graph__graph,
                                               _prop("e", self.bg, self.mrs),
                                               _prop("v", self.bg, self.wr),
                                               multigraph)
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        else:
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            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)
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            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 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
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        if dl:
            if random:
                S += model_entropy(1, N, E, directed=self.g.is_directed()) * E
            else:
                S += model_entropy(self.B, N, E, directed=self.g.is_directed(), nr=self.wr.a) * E

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

        return S / E

    def remove_vertex(self, v):
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        r"""Remove vertex ``v`` from its current block."""
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        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))
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        self.egroups = None
        self.nb_list = None
        self.nb_count = None
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    def add_vertex(self, v, r):
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        r"""Add vertex ``v`` to block ``r``."""
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        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))
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        self.egroups = None
        self.nb_list = None
        self.nb_count = None
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    def move_vertex(self, v, nr):
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        r"""Move vertex ``v`` to block ``nr``, and return the entropy difference."""
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        dS = libcommunity.move_vertex(self.g._Graph__graph,
                                      self.bg._Graph__graph,
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                                      self.__get_emat(),
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                                      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))
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        self.egroups = None
        self.nb_list = None
        self.nb_count = None
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        return dS / float(self.E)

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    def get_matrix(self, reorder=False, niter=0, ret_order=False):
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        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.
        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.

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

        if reorder:
            if niter == 0:
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                niter = 10
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            states = []

            label = None
            states = [self]
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            Bi = self.B // 2
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            while Bi > 1:
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                state = BlockState(states[-1].bg,
                                   b=states[-1].bg.vertex_index.copy("int"),
                                   B=states[-1].bg.num_vertices(),
                                   clabel=states[-1].bclabel,
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                                   vweight=states[-1].wr,
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                                   eweight=states[-1].mrs,
                                   deg_corr=states[-1].deg_corr,
                                   max_BE=states[-1].max_BE)

                state = greedy_shrink(state, B=Bi, nsweeps=niter,
                                      epsilon=1e-3, c=0,
                                      nmerge_sweeps=niter, sequential=True)
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                for i in range(niter):
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                    mcmc_sweep(state, c=0, beta=float("inf"))
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                states.append(state)

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                Bi //= 2
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                if Bi < self.bclabel.a.max() + 1:
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                    break

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            vorder = list(range(len(states[-1].vertices)))
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            for state in reversed(states[1:]):
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                norder = [[] for i in range(state.B)]
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                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]

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        for r in range(B):
            if r not in rmap:
                rmap[r] = r

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        if ret_order:
            return m, rmap
        else:
            return m


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def model_entropy(B, N, E, directed=False, nr=None):
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    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::

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       \mathcal{L}_t = \ln\Omega_m + \ln\left(\!\!{B \choose N}\!\!\right) + \ln N! - \sum_r \ln n_r!,

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    where the remaining term is the information necessary to describe the
    block partition, where :math:`n_r` is the number of nodes in block :math:`r`.
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    If ``nr`` is ``None``, it is assumed :math:`n_r=N/B`.

    The value returned corresponds to the information per edge, i.e.
    :math:`\mathcal{L}_t/E`.
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    References
    ----------

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    .. [peixoto-parsimonious-2013] Tiago P. Peixoto, "Parsimonious module inference in large networks",
       Phys. Rev. Lett. 110, 148701 (2013), :doi:`10.1103/PhysRevLett.110.148701`, :arxiv:`1212.4794`.
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    .. [peixoto-hierarchical-2013] Tiago P. Peixoto, "Hierarchical block structures and high-resolution
       model selection in large networks ", :arxiv:`1310.4377`.
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    """

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    if directed:
        x = (B * B);
    else:
        x = (B * (B + 1)) / 2;
    L = lbinom(x + E - 1, E) + partition_entropy(B, N, nr)
    return L / E
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def Sdl(B, S, N, E, directed=False):
    return S + model_entropy(B, N, E, directed)

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def lbinom(n, k):
    return scipy.special.gammaln(n + 1) - scipy.special.gammaln(n - k + 1) - scipy.special.gammaln(k + 1)

def partition_entropy(B, N, nr=None):
    if nr is None:
        S = N * log(B) + log1p(-(1 - 1./B) ** N)
    else:
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        S = lbinom(B + N - 1, N) + scipy.special.gammaln(N + 1) - scipy.special.gammaln(nr + 1).sum()
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    return S
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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
    ----------
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    .. [peixoto-parsimonious-2013] Tiago P. Peixoto, "Parsimonious module inference in large networks",
       Phys. Rev. Lett. 110, 148701 (2013), :doi:`10.1103/PhysRevLett.110.148701`, :arxiv:`1212.4794`.
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    """

    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):
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Tiago Peixoto committed
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    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 description length criterion.
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    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)
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    2.414413200430159
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    References
    ----------
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    .. [peixoto-parsimonious-2013] Tiago P. Peixoto, "Parsimonious module inference in large networks",
       Phys. Rev. Lett. 110, 148701 (2013), :doi:`10.1103/PhysRevLett.110.148701`, :arxiv:`1212.4794`.
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    """
    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

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def mcmc_sweep(state, beta=1., random_move=False, c=1., dense=False,
               multigraph=False, sequential=True, vertices=None,
               verbose=False):
    r"""Performs a Markov chain Monte Carlo 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.
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    beta : ``float`` (optional, default: `1.0`)
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        The inverse temperature parameter :math:`\beta`.
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    random_move : ``bool`` (optional, default: ``False``)
        If ``True``, the proposed moves will attempt to place the vertices in
        fully randomly-chosen blocks. If ``False``, the proposed moves will be
        chosen with a probability depending on the membership of the neighbours
        and the currently-inferred block structure.
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    c : ``float`` (optional, default: ``1.0``)
        This parameter specifies how often fully random moves are attempted,
        instead of more likely moves based on the inferred block partition.
        For ``c == 0``, no fully random moves are attempted, and for ``c == inf``
        they are always attempted.
    dense : ``bool`` (optional, default: ``False``)
        If ``True``, the "dense" variant of the entropy will be computed.
    multigraph : ``bool`` (optional, default: ``False``)
        If ``True``, the multigraph entropy will be used. Only has an effect
        if ``dense == True``.
    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.
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    vertices: ``list of ints`` (optional, default: ``None``)
        A list of vertices which will be attempted to be moved. If ``None``, all
        vertices will be attempted.
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    verbose : ``bool`` (optional, default: ``False``)
        If ``True``, verbose information is displayed.

    Returns
    -------

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


    Notes
    -----

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    This algorithm performs a Markov chain Monte Carlo sweep on the network,
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    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
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    the Markov chain by proposing membership moves :math:`r\to s` with
    probability :math:`p(r\to s|t) \propto e_{ts} + c`, where :math:`t` is the
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    block label of a random neighbour of the vertex being moved. See
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    [peixoto-efficient-2013]_ for more details.
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    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`.
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    .. [peixoto-parsimonious-2013] Tiago P. Peixoto, "Parsimonious module inference in large networks",
       Phys. Rev. Lett. 110, 148701 (2013), :doi:`10.1103/PhysRevLett.110.148701`, :arxiv:`1212.4794`.
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    .. [peixoto-efficient-2013] Tiago P. Peixoto, "Efficient Monte Carlo and greedy
       heuristic for the inference of stochastic block models", :arxiv:`1310.4378`.
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    """

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    if state.B == 1:
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        return 0., 0

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    if vertices is not None:
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        vlist = libcommunity.get_vector(len(vertices))
        vlist.a = vertices
        vertices = vlist
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        state.sweep_vertices = vertices
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    if state.sweep_vertices is None:
        vertices = libcommunity.get_vector(state.g.num_vertices())
        vertices.a = state.g.vertex_index.copy("int").fa
        state.sweep_vertices = vertices

    if random_move:
        state._BlockState__build_egroups(empty=True)
    elif state.egroups is None:
        state._BlockState__build_egroups(empty=False)

    if state.nsampler is None:
        state._BlockState__build_nsampler()

    state.bnsampler = None

    try:
        dS, nmoves = libcommunity.move_sweep(state.g._Graph__graph,
                                             state.bg._Graph__graph,
                                             state._BlockState__get_emat(),
                                             state.nsampler,
                                             _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.bg, state.bclabel),
                                             state.sweep_vertices,
                                             state.deg_corr, dense, multigraph,
                                             _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, random_move,
                                             c, verbose, _get_rng())
    finally:
        if random_move:
            state.egroups = None
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    return dS / state.E, nmoves


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def merge_sweep(state, bm, nmerges, nsweeps=10, dense=False, multigraph=False,
                random_moves=False, verbose=False):
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    if state.B == 1:
        return 0., 0
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    if state.bnsampler is None:
        state._BlockState__build_bnsampler()

    state.egroups = None
    state.nsampler = None

    dS, nmoves = libcommunity.merge_sweep(state.bg._Graph__graph,
                                          state._BlockState__get_emat(),
                                          state.bnsampler,
                                          _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.bg, bm),
                                          _prop("v", state.bg, state.bclabel),
                                          state.deg_corr, dense, multigraph,
                                          nsweeps, nmerges, random_moves,
                                          verbose, _get_rng())
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def greedy_shrink(state, B, nsweeps=10, adaptive_sweeps=True, nmerge_sweeps=None,
                  epsilon=0, r=2, greedy=True, anneal=(1, 1), c=1, dense=False,
                  multigraph=False, random_move=False, verbose=False,
                  sequential=True):
    if B > state.B:
        raise ValueError("Cannot shrink to a larger size!")
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    if nmerge_sweeps is None:
        nmerge_sweeps = max((2 * state.g.num_edges()) // state.g.num_vertices(), 1)
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    nmerged = 0
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    state = BlockState(state.g, b=state.b, B=state.B,
                       clabel=state.clabel, vweight=state.vweight,
                       eweight=state.eweight, deg_corr=state.deg_corr,
                       max_BE=state.max_BE)

    cg = state.bg.copy()
    cg_vweight = cg.own_property(state.wr.copy())
    cg_eweight = cg.own_property(state.mrs.copy())
    cg_clabel = cg.own_property(state.bclabel.copy())

    # merge according to indirect neighbourhood
    bm = state.bg.vertex_index.copy("int")
    random = random_move
    while nmerged < state.B - B:
        dS, nmoves = merge_sweep(state, bm, nmerges=state.B - B - nmerged,
                                 nsweeps=nmerge_sweeps, random_moves=random)
        nmerged += nmoves
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        if verbose:
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            print("merging", dS, nmoves, nmerged)
        if nmoves == 0:
            random = True
            if verbose:
                print("can't merge... switching to random")

    # Merged block-level state
    bmap = -ones(len(bm.a), dtype=bm.a.dtype)
    libcommunity.vector_map(bm.a, bmap)

    bm = cg.own_property(bm)
    bg_state = BlockState(cg, b=bm, B=B, clabel=cg_clabel,
                          vweight=cg_vweight, eweight=cg_eweight,
                          deg_corr=state.deg_corr, max_BE=state.max_BE)

    if bg_state.g.num_vertices() != state.g.num_vertices() and nsweeps > 0:
        # Perform block-level moves
        if verbose:
            print("Performing block-level moves...")
        multilevel_minimize(bg_state, B=B, nsweeps=nsweeps,
                            adaptive_sweeps=adaptive_sweeps,
                            epsilon=epsilon, r=r, greedy=greedy,
                            anneal=anneal, c=c, dense=dense,
                            multigraph=multigraph, random_move=random_move,
                            sequential=sequential, verbose=verbose)

    bm = bg_state.b
    libcommunity.vector_map(state.b.a, bm.a)

    state = BlockState(state.g, b=state.b, B=B, clabel=state.clabel,
                       vweight=state.vweight, eweight=state.eweight,
                       deg_corr=state.deg_corr, max_BE=state.max_BE)
    return state

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