#! /usr/bin/env python
# -*- coding: utf-8 -*-
#
# graph_tool -- a general graph manipulation python module
#
# Copyright (C) 2006-2024 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 Lesser 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 Lesser General Public License for more
# details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
from .. import Graph, GraphView, _get_rng, Vector_size_t, PropertyMap, \
group_vector_property
from . blockmodel import DictState, init_q_cache
from . base_states import *
from . util import *
from .. dl_import import dl_import
dl_import("from . import libgraph_tool_inference as libinference")
import numpy as np
import math
[docs]
@entropy_state_signature
class PPBlockState(MCMCState, MultiflipMCMCState, MultilevelMCMCState,
GibbsMCMCState, DrawBlockState):
r"""Obtain the partition of a network according to the Bayesian planted partition
model.
Parameters
----------
g : :class:`~graph_tool.Graph`
Graph to be modelled.
b : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
Initial partition. If not supplied, a partition into a single group will
be used.
entropy_args: ``dict`` (optional, default: ``{}``)
Override default arguments for :meth:`~PPBlockState.entropy()`
method and releated operations.
References
----------
.. [lizhi-statistical-2020] Lizhi Zhang, Tiago P. Peixoto, "Statistical
inference of assortative community structures", Phys. Rev. Research 2
043271 (2020), :doi:`10.1103/PhysRevResearch.2.043271`, :arxiv:`2006.14493`
"""
def __init__(self, g, b=None, entropy_args={}):
EntropyState.__init__(self, entropy_args=entropy_args)
self.g = GraphView(g, directed=False)
if b is None:
self.b = self.g.new_vp("int32_t")
elif isinstance(g, PropertyMap):
self.b = b.copy("int32_t")
else:
self.b = self.g.new_vp("int32_t", vals=b)
self.wr = Vector_size_t()
self.er = Vector_size_t()
self.err = Vector_size_t()
self.eio = Vector_size_t()
init_q_cache(max((2 * max((self.g.num_edges(),
self.g.num_vertices())), 100)))
self.bg = GraphView(self.g)
self.bg.clear_filters()
self._abg = self.bg._get_any()
self._state = libinference.make_pp_state(self)
def __copy__(self):
return self.copy()
def __deepcopy__(self, memo):
g = copy.deepcopy(self.g, memo)
b = copy.deepcopy(self.b, memo)
return self.copy(g=g, b=b)
[docs]
def copy(self, g=None, b=None):
r"""Copies the state. The parameters override the state properties, and
have the same meaning as in the constructor."""
return PPBlockState(g=g if g is not None else self.g,
b=b if b is not None else self.b)
def __getstate__(self):
state = EntropyState.__getstate__(self)
return dict(state, g=self.g, b=self.b)
def __setstate__(self, state):
self.__init__(**state)
def __repr__(self):
return "<PPBlockState object with %d blocks, for graph %s, at 0x%x>" % \
(self.get_B(), str(self.g), id(self))
[docs]
def get_blocks(self):
r"""Returns the property map which contains the block labels for each vertex."""
return self.b
[docs]
def get_state(self):
"""Alias to :meth:`~PPBlockState.get_blocks`."""
return self.get_blocks()
[docs]
def get_B(self):
r"Returns the total number of blocks."
return len(np.unique(self.b.fa))
[docs]
def get_nonempty_B(self):
r"Alias to :meth:`~PPBlockState.get_B`."
return self.get_B()
[docs]
def get_Be(self):
r"""Returns the effective number of blocks, defined as :math:`e^{H}`, with
:math:`H=-\sum_r\frac{n_r}{N}\ln \frac{n_r}{N}`, where :math:`n_r` is
the number of nodes in group r.
"""
w = np.array(np.bincount(self.b.fa), dtype="double")
w = w[w>0]
w /= w.sum()
return np.exp(-(w*np.log(w)).sum())
[docs]
def virtual_vertex_move(self, v, s, **kwargs):
r"""Computes the entropy difference if vertex ``v`` is moved to block ``s``. The
remaining parameters are the same as in
:meth:`~graph_tool.inference.PPBlockState.entropy`."""
return self._state.virtual_move(int(v), self.b[v], s,
self._get_entropy_args(dict(self._entropy_args,
**kwargs)))
[docs]
def move_vertex(self, v, s):
r"""Move vertex ``v`` to block ``s``."""
self._state.move_vertex(int(v), int(s))
@copy_state_wrap
def _entropy(self, uniform=False, degree_dl_kind="distributed"):
r"""Return the model entropy (negative log-likelihood).
Parameters
----------
uniform : ``bool`` (optional, default: ``False``)
If ``True``, the uniform planted partition model is used, otherwise
a non-uniform version is used.
degree_dl_kind : ``str`` (optional, default: ``"distributed"``)
This specifies the prior used for the degree sequence. It must be
one of: ``"uniform"`` or ``"distributed"`` (default).
Notes
-----
The "entropy" of the state is the negative log-likelihood of the
microcanonical SBM, that includes the generated graph
:math:`\boldsymbol{A}` and the model parameters :math:`e_{\text{in}}`,
:math:`e_{\text{out}}`, :math:`\boldsymbol{k}` and
:math:`\boldsymbol{b}`,
.. math::
\Sigma &= - \ln P(\boldsymbol{A},e_{\text{in}},e_{\text{out}},\boldsymbol{k},\boldsymbol{b}) \\
&= - \ln P(\boldsymbol{A}|e_{\text{in}},e_{\text{out}},\boldsymbol{k},\boldsymbol{b}) - \ln P(e_{\text{in}},e_{\text{out}},\boldsymbol{k},\boldsymbol{b}).
This value is also called the `description length
<https://en.wikipedia.org/wiki/Minimum_description_length>`_ of the data,
and it corresponds to the amount of information required to describe it
(in `nats <https://en.wikipedia.org/wiki/Nat_(unit)>`_).
For the uniform version of the model, the likelihood is
.. math::
P(\boldsymbol{A}|\boldsymbol{k},\boldsymbol{b}) = \frac{e_{\text{in}}!e_{\text{out}}!}
{\left(\frac{B}{2}\right)^{e_{\text{in}}}{B\choose 2}^{e_{\text{out}}}(E+1)^{1-\delta_{B,1}}\prod_re_r!}\times
\frac{\prod_ik_i!}{\prod_{i<j}A_{ij}!\prod_i A_{ii}!!}.
where :math:`e_{\text{in}}` and :math:`e_{\text{out}}` are the number of
edges inside and outside communities, respectively, and :math:`e_r` is
the sum of degrees in group :math:`r`.
For the non-uniform model we have instead:
.. math::
P(\boldsymbol{A}|\boldsymbol{k},\boldsymbol{b}) = \frac{e_{\text{out}}!\prod_re_{rr}!!}
{{B\choose 2}^{e_{\text{out}}}(E+1)^{1-\delta_{B,1}}\prod_re_r!}\times{B + e_{\text{in}} - 1 \choose e_{\text{in}}}^{-1}\times
\frac{\prod_ik_i!}{\prod_{i<j}A_{ij}!\prod_i A_{ii}!!}.
Here there are two options for the prior on the degrees:
1. ``degree_dl_kind == "uniform"``
.. math::
P(\boldsymbol{k}|\boldsymbol{e},\boldsymbol{b}) = \prod_r\left(\!\!{n_r\choose e_r}\!\!\right)^{-1}.
This corresponds to a noninformative prior, where the degrees are
sampled from a uniform distribution.
2. ``degree_dl_kind == "distributed"`` (default)
.. math::
P(\boldsymbol{k}|\boldsymbol{e},\boldsymbol{b}) = \prod_r\frac{\prod_k\eta_k^r!}{n_r!} \prod_r q(e_r, n_r)^{-1}
with :math:`\eta_k^r` being the number of nodes with degree
:math:`k` in group :math:`r`, and :math:`q(n,m)` being the number of
`partitions
<https://en.wikipedia.org/wiki/Partition_(number_theory)>`_ of
integer :math:`n` into at most :math:`m` parts.
This corresponds to a prior for the degree sequence conditioned on
the degree frequencies, which are themselves sampled from a uniform
hyperprior. This option should be preferred in most cases.
For the partition prior :math:`P(\boldsymbol{b})` please refer to
:meth:`~graph_tool.inference.BlockState.entropy`.
References
----------
.. [lizhi-statistical-2020] Lizhi Zhang, Tiago P. Peixoto, "Statistical
inference of assortative community structures", Phys. Rev. Research 2
043271 (2020), :doi:`10.1103/PhysRevResearch.2.043271`, :arxiv:`2006.14493`
"""
eargs = self._get_entropy_args(locals())
return self._state.entropy(eargs)
def _gen_eargs(self, args):
return libinference.pp_entropy_args()
def _get_entropy_args(self, kwargs, consume=False):
if not consume:
kwargs = kwargs.copy()
deg_dl_kind = kwargs.get("degree_dl_kind",
self._entropy_args["degree_dl_kind"])
if deg_dl_kind == "entropy":
kind = libinference.deg_dl_kind.ent
elif deg_dl_kind == "uniform":
kind = libinference.deg_dl_kind.uniform
elif deg_dl_kind == "distributed":
kind = libinference.deg_dl_kind.dist
kwargs["degree_dl_kind"] = kind
return super()._get_entropy_args(kwargs, consume)
def _mcmc_sweep_dispatch(self, mcmc_state):
return libinference.pp_mcmc_sweep(mcmc_state, self._state,
_get_rng())
def _multiflip_mcmc_sweep_dispatch(self, mcmc_state):
return libinference.pp_multiflip_mcmc_sweep(mcmc_state, self._state,
_get_rng())
def _multilevel_mcmc_sweep_dispatch(self, mcmc_state):
return libinference.pp_multilevel_mcmc_sweep(mcmc_state, self._state,
_get_rng())
def _gibbs_sweep_dispatch(self, gibbs_state):
return libinference.pp_gibbs_sweep(gibbs_state, self._state,
_get_rng())
