.. _inference-howto:
Inferring network structure
===========================
``graph-tool`` includes algorithms to identify the large-scale structure
of networks in the :mod:`~graph_tool.inference` submodule. Here we
explain the basic functionality with self-contained examples.
Background: Nonparametric statistical inference
-----------------------------------------------
A common task when analyzing networks is to characterize their
structures in simple terms, often by dividing the nodes into modules or
"communities".
A principled approach to perform this task is to formulate `generative
models `_ that include
the idea of "modules" in their descriptions, which then can be detected
by `inferring `_
the model parameters from data. More precisely, given the partition
:math:`\boldsymbol b = \{b_i\}` of the network into :math:`B` groups,
where :math:`b_i\in[0,B-1]` is the group membership of node :math:`i`,
we define a model that generates a network :math:`\boldsymbol G` with a
probability
.. math::
:label: model-likelihood
P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b)
where :math:`\boldsymbol\theta` are additional model parameters. Therefore, if we
observe a network :math:`\boldsymbol G`, the likelihood that it was generated by a
given partition :math:`\boldsymbol b` is obtained via the `Bayesian
`_ posterior
.. math::
:label: model-posterior-sum
P(\boldsymbol b | \boldsymbol G) = \frac{\sum_{\boldsymbol\theta}P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b)P(\boldsymbol\theta, \boldsymbol b)}{P(\boldsymbol G)}
where :math:`P(\boldsymbol\theta, \boldsymbol b)` is the `prior likelihood` of the
model parameters, and
.. math::
:label: model-evidence
P(\boldsymbol G) = \sum_{\boldsymbol\theta,\boldsymbol b}P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b)P(\boldsymbol\theta, \boldsymbol b)
is called the `model evidence`. The particular types of model that will
be considered here have "hard constraints", such that there is only one
choice for the remaining parameters :math:`\boldsymbol\theta` that is compatible
with the generated network, such that Eq. :eq:`model-posterior-sum` simplifies to
.. math::
:label: model-posterior
P(\boldsymbol b | \boldsymbol G) = \frac{P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b)P(\boldsymbol\theta, \boldsymbol b)}{P(\boldsymbol G)}
with :math:`\boldsymbol\theta` above being the only choice compatible with
:math:`\boldsymbol G` and :math:`\boldsymbol b`. The inference procedures considered
here will consist in either finding a network partition that maximizes
Eq. :eq:`model-posterior`, or sampling different partitions according
its posterior probability.
As we will show below, this approach will also enable the comparison of
`different` models according to statistical evidence (a.k.a. `model
selection`).
Minimum description length (MDL)
++++++++++++++++++++++++++++++++
We note that Eq. :eq:`model-posterior` can be written as
.. math::
P(\boldsymbol b | \boldsymbol G) = \frac{\exp(-\Sigma)}{P(\boldsymbol G)}
where
.. math::
:label: model-dl
\Sigma = -\ln P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b) - \ln P(\boldsymbol\theta, \boldsymbol b)
is called the **description length** of the network :math:`\boldsymbol G`. It
measures the amount of `information
`_ required to
describe the data, if we `encode
`_ it using the
particular parametrization of the generative model given by
:math:`\boldsymbol\theta` and :math:`\boldsymbol b`, as well as the parameters
themselves. Therefore, if we choose to maximize the posterior likelihood
of Eq. :eq:`model-posterior` it will be fully equivalent to the
so-called `minimum description length
`_
method. This approach corresponds to an implementation of `Occam's razor
`_, where the `simplest`
model is selected, among all possibilities with the same explanatory
power. The selection is based on the statistical evidence available, and
therefore will not `overfit
`_, i.e. mistake stochastic
fluctuations for actual structure.
The stochastic block model (SBM)
--------------------------------
The `stochastic block model
`_ is arguably
the simplest generative process based on the notion of groups of
nodes [holland-stochastic-1983]_. The `microcanonical
`_ formulation
[peixoto-nonparametric-2016]_ of the basic or "traditional" version takes
as parameters the partition of the nodes into groups
:math:`\boldsymbol b` and a :math:`B\times B` matrix of edge counts
:math:`\boldsymbol e`, where :math:`e_{rs}` is the number of edges
between groups :math:`r` and :math:`s`. Given these constraints, the
edges are then placed randomly. Hence, nodes that belong to the same
group possess the same probability of being connected with other
nodes of the network.
An example of a possible parametrization is given in the following
figure.
.. testcode:: sbm-example
:hide:
import os
try:
os.chdir("demos/inference")
except FileNotFoundError:
pass
g = gt.load_graph("blockmodel-example.gt.gz")
gt.graph_draw(g, pos=g.vp.pos, vertex_size=10, vertex_fill_color=g.vp.bo,
vertex_color="#333333",
edge_gradient=g.new_ep("vector", val=[0]),
output="sbm-example.svg")
ers = g.gp.w
from pylab import *
figure()
matshow(log(ers))
xlabel("Group $r$")
ylabel("Group $s$")
gca().xaxis.set_label_position("top")
savefig("sbm-example-ers.svg")
.. table::
:class: figure
+----------------------------------+------------------------------+
|.. figure:: sbm-example-ers.svg |.. figure:: sbm-example.svg |
| :width: 300px | :width: 300px |
| :align: center | :align: center |
| | |
| Matrix of edge counts | Generated network. |
| :math:`\boldsymbol e` between | |
| groups. | |
+----------------------------------+------------------------------+
.. note::
We emphasize that no constraints are imposed on what `kind` of
modular structure is allowed. Hence, we can detect the putatively
typical pattern of `"community structure"
`_, i.e. when
nodes are connected mostly to other nodes of the same group, if it
happens to be the most likely network description, but we can also
detect a large multiplicity of other patterns, such as `bipartiteness
`_, core-periphery,
and many others, all under the same inference framework.
Although quite general, the traditional model assumes that the edges are
placed randomly inside each group, and as such the nodes that belong to
the same group have very similar degrees. As it turns out, this is often
a poor model for many networks, which possess highly heterogeneous
degree distributions. A better model for such networks is called the
`degree-corrected` stochastic block model [karrer-stochastic-2011]_, and
it is defined just like the traditional model, with the addition of the
degree sequence :math:`\boldsymbol k = \{k_i\}` of the graph as an
additional set of parameters (assuming again a microcanonical
formulation [peixoto-nonparametric-2016]_).
The nested stochastic block model
+++++++++++++++++++++++++++++++++
The regular SBM has a drawback when applied to very large
networks. Namely, it cannot be used to find relatively small groups in
very large networks: The maximum number of groups that can be found
scales as :math:`B_{\text{max}}\sim\sqrt{N}`, where :math:`N` is the
number of nodes in the network, if Bayesian inference is performed
[peixoto-parsimonious-2013]_. In order to circumvent this, we need to
replace the noninformative priors used by a hierarchy of priors and
hyperpriors, which amounts to a `nested SBM`, where the groups
themselves are clustered into groups, and the matrix :math:`e` of edge
counts are generated from another SBM, and so on recursively
[peixoto-hierarchical-2014]_.
.. figure:: nested-diagram.*
:width: 400px
:align: center
Example of a nested SBM with three levels.
In addition to being able to find small groups in large networks, this
model also provides a multilevel hierarchical description of the
network, that describes its structure at multiple scales.
Inferring the best partition
----------------------------
The simplest and most efficient approach is to find the best
partition of the network by maximizing Eq. :eq:`model-posterior`
according to some version of the model. This is obtained via the
functions :func:`~graph_tool.inference.minimize_blockmodel_dl` or
:func:`~graph_tool.inference.minimize_nested_blockmodel_dl`, which
employs an agglomerative multilevel `Markov chain Monte Carlo (MCMC)
`_ algorithm
[peixoto-efficient-2014]_.
We focus first on the non-nested model, and we illustrate its use with a
network of American football teams, which we load from the
:mod:`~graph_tool.collection` module:
.. testsetup:: football
import os
try:
os.chdir("demos/inference")
except FileNotFoundError:
pass
.. testcode:: football
g = gt.collection.data["football"]
print(g)
which yields
.. testoutput:: football
we then fit the `traditional` model by calling
.. testcode:: football
state = gt.minimize_blockmodel_dl(g, deg_corr=False)
This returns a :class:`~graph_tool.inference.BlockState` object that
includes the inference results.
.. note::
The inference algorithm used is stochastic by nature, and may return
a slightly different answer each time it is run. This may be due to
the fact that there are alternative partitions with similar
likelihoods, or that the optimum is difficult to find. Note that the
inference problem here is, in general, `NP-Hard
`_, hence there is no
efficient algorithm that is guaranteed to always find the best
answer.
Because of this, typically one would call the algorithm many times,
and select the partition with the largest posterior likelihood of
Eq. :eq:`model-posterior`, or equivalently, the minimum description
length of Eq. :eq:`model-dl`. The description length of a fit can be
obtained with the :meth:`~graph_tool.inference.BlockState.entropy`
method. See also :ref:`sec_model_selection` below.
We may perform a drawing of the partition obtained via the
:mod:`~graph_tool.inference.BlockState.draw` method, that functions as a
convenience wrapper to the :func:`~graph_tool.draw.graph_draw` function
.. testcode:: football
state.draw(pos=g.vp.pos, output="football-sbm-fit.svg")
which yields the following image.
.. figure:: football-sbm-fit.*
:align: center
:width: 400px
Stochastic block model inference of a network of American college
football teams. The colors correspond to inferred group membership of
the nodes.
We can obtain the group memberships as a
:class:`~graph_tool.PropertyMap` on the vertices via the
:mod:`~graph_tool.inference.BlockState.get_blocks` method:
.. testcode:: football
b = state.get_blocks()
r = b[10] # group membership of vertex 10
print(r)
which yields:
.. testoutput:: football
10
We may also access the matrix of edge counts between groups via
:mod:`~graph_tool.inference.BlockState.get_matrix`
.. testcode:: football
e = state.get_matrix()
matshow(e.todense())
savefig("football-edge-counts.svg")
.. figure:: football-edge-counts.*
:align: center
Matrix of edge counts between groups.
We may obtain the same matrix of edge counts as a graph, which has
internal edge and vertex property maps with the edge and vertex counts,
respectively:
.. testcode:: football
bg = state.get_bg()
ers = bg.ep.count # edge counts
nr = bg.vp.count # node counts
.. _sec_model_selection:
Hierarchical partitions
+++++++++++++++++++++++
The inference of the nested family of SBMs is done in a similar manner,
but we must use instead the
:func:`~graph_tool.inference.minimize_nested_blockmodel_dl` function. We
illustrate its use with the neural network of the `C. elegans
`_ worm:
.. testcode:: celegans
g = gt.collection.data["celegansneural"]
print(g)
which has 297 vertices and 2359 edges.
.. testoutput:: celegans
A hierarchical fit of the degree-corrected model is performed as follows.
.. testcode:: celegans
state = gt.minimize_nested_blockmodel_dl(g, deg_corr=True)
The object returned is an instance of a
:class:`~graph_tool.inference.NestedBlockState` class, which
encapsulates the results. We can again draw the resulting hierarchical
clustering using the
:meth:`~graph_tool.inference.NestedBlockState.draw` method:
.. testcode:: celegans
state.draw(output="celegans-hsbm-fit.svg")
.. figure:: celegans-hsbm-fit.*
:align: center
Most likely hierarchical partition of the neural network of
the C. elegans worm according to the nested degree-corrected SBM.
.. note::
If the ``output`` parameter to
:meth:`~graph_tool.inference.NestedBlockState.draw` is omitted, an
interactive visualization is performed, where the user can re-order
the hierarchy nodes using the mouse and pressing the ``r`` key.
A summary of the inferred hierarchy can be obtained with the
:meth:`~graph_tool.inference.NestedBlockState.print_summary` method,
which shows the number of nodes and groups in all levels:
.. testcode:: celegans
state.print_summary()
.. testoutput:: celegans
l: 0, N: 297, B: 13
l: 1, N: 13, B: 5
l: 2, N: 5, B: 2
l: 3, N: 2, B: 1
The hierarchical levels themselves are represented by individual
:meth:`~graph_tool.inference.BlockState` instances obtained via the
:meth:`~graph_tool.inference.NestedBlockState.get_levels()` method:
.. testcode:: celegans
levels = state.get_levels()
for s in levels:
print(s)
.. testoutput:: celegans
, at 0x...>
, at 0x...>
, at 0x...>
, at 0x...>
This means that we can inspect the hierarchical partition just as before:
.. testcode:: celegans
r = levels[0].get_blocks()[46] # group membership of node 46 in level 0
print(r)
r = levels[0].get_blocks()[r] # group membership of node 46 in level 1
print(r)
r = levels[0].get_blocks()[r] # group membership of node 46 in level 2
print(r)
.. testoutput:: celegans
2
1
0
Model selection
+++++++++++++++
As mentioned above, one can select the best model according to the
choice that yields the smallest description length. For instance, in
case of the `C. elegans` network we have
.. testcode:: model-selection
g = gt.collection.data["celegansneural"]
state_ndc = gt.minimize_nested_blockmodel_dl(g, deg_corr=False)
state_dc = gt.minimize_nested_blockmodel_dl(g, deg_corr=True)
print("Non-degree-corrected DL:\t", state_ndc.entropy())
print("Degree-corrected DL:\t", state_dc.entropy())
.. testoutput:: model-selection
:options: +NORMALIZE_WHITESPACE
Non-degree-corrected DL: 8568.61212614
Degree-corrected DL: 8246.48662192
Since it yields the smallest description length, the degree-corrected
fit should be preferred. The statistical significance of the choice can
be accessed by inspecting the posterior odds ratio
[peixoto-nonparametric-2016]_
.. math::
\Lambda &= \frac{P(\boldsymbol b, \mathcal{H}_\text{NDC} | \boldsymbol G)}{P(\boldsymbol b, \mathcal{H}_\text{DC} | \boldsymbol G)} \\
&= \frac{P(\boldsymbol G, \boldsymbol b | \mathcal{H}_\text{NDC})}{P(\boldsymbol G, \boldsymbol b | \mathcal{H}_\text{DC})}\times\frac{P(\mathcal{H}_\text{NDC})}{P(\mathcal{H}_\text{DC})} \\
&= \exp(-\Delta\Sigma)
where :math:`\mathcal{H}_\text{NDC}` and :math:`\mathcal{H}_\text{DC}`
correspond to the non-degree-corrected and degree-corrected model
hypotheses (assumed to be equally likely `a priori`), respectively, and
:math:`\Delta\Sigma` is the difference of the description length of both
fits. In our particular case, we have
.. testcode:: model-selection
print("ln Λ: ", state_dc.entropy() - state_ndc.entropy())
.. testoutput:: model-selection
:options: +NORMALIZE_WHITESPACE
ln Λ: -322.125504215
The precise threshold that should be used to decide when to `reject a
hypothesis `_ is
subjective and context-dependent, but the value above implies that the
particular degree-corrected fit is around :math:`e^{322} \sim 10^{140}`
times more likely than the non-degree corrected one, and hence it can be
safely concluded that it provides a substantially better fit.
Although it is often true that the degree-corrected model provides a
better fit for many empirical networks, there are also exceptions. For
example, for the American football network above, we have:
.. testcode:: model-selection
g = gt.collection.data["football"]
state_ndc = gt.minimize_nested_blockmodel_dl(g, deg_corr=False)
state_dc = gt.minimize_nested_blockmodel_dl(g, deg_corr=True)
print("Non-degree-corrected DL:\t", state_ndc.entropy())
print("Degree-corrected DL:\t", state_dc.entropy())
print("ln Λ:\t\t\t", state_ndc.entropy() - state_dc.entropy())
.. testoutput:: model-selection
:options: +NORMALIZE_WHITESPACE
Non-degree-corrected DL: 1757.84382615
Degree-corrected DL: 1787.60777164
ln Λ: -29.7639454931
Hence, with a posterior odds ratio of :math:`\Lambda \sim e^{-29} \sim
10^{-13}` in favor of the non-degree-corrected model, it seems like the
degree-corrected variant is an unnecessarily complex description for
this network.
Averaging over models
---------------------
When analyzing empirical networks, one should be open to the possibility
that there will be more than one fit of the SBM with similar posterior
likelihoods. In such situations, one should instead `sample` partitions
from the posterior likelihood, instead of simply finding its
maximum. One can then compute quantities that are averaged over the
different model fits, weighted according to their posterior likelihoods.
Full support for model averaging is implemented in ``graph-tool`` via an
efficient `Markov chain Monte Carlo (MCMC)
`_ algorithm
[peixoto-efficient-2014]_. It works by attempting to move nodes into
different groups with specific probabilities, and `accepting or
rejecting
`_
such moves such that, after a sufficiently long time, the partitions
will be observed with the desired posterior probability. The algorithm
is so designed, that its run-time is independent on the number of groups
being used in the model, and hence is suitable for use on very large
networks.
In order to perform such moves, one needs again to operate with
:class:`~graph_tool.inference.BlockState` or
:class:`~graph_tool.inference.NestedBlockState` instances, and calling
their :meth:`~graph_tool.inference.BlockState.mcmc_sweep` methods. For
example, the following will perform 1000 sweeps of the algorithm with
the network of characters in the novel Les Misérables, starting from a
random partition into 20 groups
.. testcode:: model-averaging
g = gt.collection.data["lesmis"]
state = gt.BlockState(g, B=20) # This automatically initializes the state
# with a random partition into B=20
# nonempty groups; The user could
# also pass an arbitrary initial
# partition using the 'b' parameter.
# If we work with the above state object, we will be restricted to
# partitions into at most B=20 groups. But since we want to consider
# an arbitrary number of groups in the range [1, N], we transform it
# into a state with B=N groups (where N-20 will be empty).
state = state.copy(B=g.num_vertices())
# Now we run 1,000 sweeps of the MCMC
dS, nmoves = state.mcmc_sweep(niter=1000)
print("Change in description length:", dS)
print("Number of accepted vertex moves:", nmoves)
.. testoutput:: model-averaging
Change in description length: -360.18357903823386
Number of accepted vertex moves: 4743
.. note::
Starting from a random partition is rarely the best option, since it
may take a long time for it to equilibrate; It was done above simply
as an illustration on how to initialize
:class:`~graph_tool.inference.BlockState` by hand. Instead, a much
better option in practice is to start from the "ground state"
obtained with :func:`~graph_tool.inference.minimize_blockmodel_dl`,
e.g.
.. testcode:: model-averaging
state = gt.minimize_blockmodel_dl(g)
state = state.copy(B=g.num_vertices())
dS, nmoves = state.mcmc_sweep(niter=1000)
print("Change in description length:", dS)
print("Number of accepted vertex moves:", nmoves)
.. testoutput:: model-averaging
Change in description length: 0.23920882820149814
Number of accepted vertex moves: 4016
Although the above is sufficient to implement model averaging, there is a
convenience function called
:func:`~graph_tool.inference.mcmc_equilibrate` that is intend to
simplify the detection of equilibration, by keeping track of the maximum
and minimum values of description length encountered and how many sweeps
have been made without a "record breaking" event. For example,
.. testcode:: model-averaging
# We will accept equilibration if 10 sweeps are completed without a
# record breaking event, 2 consecutive times.
gt.mcmc_equilibrate(state, wait=10, nbreaks=2, mcmc_args=dict(niter=10), verbose=True)
will output:
.. testoutput:: model-averaging
:options: +NORMALIZE_WHITESPACE
niter: 1 count: 0 breaks: 0 min_S: 699.58882 max_S: 713.12054 S: 713.12054 ΔS: 13.5317 moves: 31
niter: 2 count: 1 breaks: 0 min_S: 699.58882 max_S: 713.12054 S: 711.03345 ΔS: -2.08709 moves: 25
niter: 3 count: 0 breaks: 0 min_S: 699.58882 max_S: 715.72860 S: 715.72860 ΔS: 4.69514 moves: 37
niter: 4 count: 1 breaks: 0 min_S: 699.58882 max_S: 715.72860 S: 704.76394 ΔS: -10.9647 moves: 36
niter: 5 count: 2 breaks: 0 min_S: 699.58882 max_S: 715.72860 S: 706.55192 ΔS: 1.78798 moves: 27
niter: 6 count: 3 breaks: 0 min_S: 699.58882 max_S: 715.72860 S: 706.97865 ΔS: 0.426724 moves: 30
niter: 7 count: 4 breaks: 0 min_S: 699.58882 max_S: 715.72860 S: 706.41383 ΔS: -0.564821 moves: 33
niter: 8 count: 0 breaks: 0 min_S: 699.58882 max_S: 718.47291 S: 718.47291 ΔS: 12.0591 moves: 43
niter: 9 count: 1 breaks: 0 min_S: 699.58882 max_S: 718.47291 S: 709.59053 ΔS: -8.88238 moves: 29
niter: 10 count: 2 breaks: 0 min_S: 699.58882 max_S: 718.47291 S: 700.39505 ΔS: -9.19548 moves: 27
niter: 11 count: 3 breaks: 0 min_S: 699.58882 max_S: 718.47291 S: 710.45317 ΔS: 10.0581 moves: 44
niter: 12 count: 0 breaks: 0 min_S: 699.58882 max_S: 719.51868 S: 719.51868 ΔS: 9.06551 moves: 38
niter: 13 count: 1 breaks: 0 min_S: 699.58882 max_S: 719.51868 S: 711.66691 ΔS: -7.85177 moves: 41
niter: 14 count: 2 breaks: 0 min_S: 699.58882 max_S: 719.51868 S: 701.51609 ΔS: -10.1508 moves: 34
niter: 15 count: 3 breaks: 0 min_S: 699.58882 max_S: 719.51868 S: 705.78796 ΔS: 4.27188 moves: 41
niter: 16 count: 4 breaks: 0 min_S: 699.58882 max_S: 719.51868 S: 715.67960 ΔS: 9.89164 moves: 33
niter: 17 count: 5 breaks: 0 min_S: 699.58882 max_S: 719.51868 S: 703.65838 ΔS: -12.0212 moves: 33
niter: 18 count: 6 breaks: 0 min_S: 699.58882 max_S: 719.51868 S: 709.55912 ΔS: 5.90074 moves: 28
niter: 19 count: 7 breaks: 0 min_S: 699.58882 max_S: 719.51868 S: 718.42521 ΔS: 8.86609 moves: 28
niter: 20 count: 8 breaks: 0 min_S: 699.58882 max_S: 719.51868 S: 711.13784 ΔS: -7.28737 moves: 48
niter: 21 count: 9 breaks: 0 min_S: 699.58882 max_S: 719.51868 S: 706.63047 ΔS: -4.50737 moves: 28
niter: 22 count: 0 breaks: 1 min_S: 707.03211 max_S: 707.03211 S: 707.03211 ΔS: 0.401639 moves: 57
niter: 23 count: 0 breaks: 1 min_S: 707.03211 max_S: 717.50359 S: 717.50359 ΔS: 10.4715 moves: 31
niter: 24 count: 0 breaks: 1 min_S: 707.03211 max_S: 726.72811 S: 726.72811 ΔS: 9.22451 moves: 63
niter: 25 count: 0 breaks: 1 min_S: 701.53898 max_S: 726.72811 S: 701.53898 ΔS: -25.1891 moves: 26
niter: 26 count: 1 breaks: 1 min_S: 701.53898 max_S: 726.72811 S: 710.04615 ΔS: 8.50718 moves: 25
niter: 27 count: 2 breaks: 1 min_S: 701.53898 max_S: 726.72811 S: 710.59565 ΔS: 0.549493 moves: 31
niter: 28 count: 3 breaks: 1 min_S: 701.53898 max_S: 726.72811 S: 713.53473 ΔS: 2.93908 moves: 19
niter: 29 count: 4 breaks: 1 min_S: 701.53898 max_S: 726.72811 S: 707.57709 ΔS: -5.95763 moves: 34
niter: 30 count: 5 breaks: 1 min_S: 701.53898 max_S: 726.72811 S: 712.49104 ΔS: 4.91395 moves: 24
niter: 31 count: 6 breaks: 1 min_S: 701.53898 max_S: 726.72811 S: 716.40137 ΔS: 3.91032 moves: 33
niter: 32 count: 7 breaks: 1 min_S: 701.53898 max_S: 726.72811 S: 717.30606 ΔS: 0.904690 moves: 54
niter: 33 count: 8 breaks: 1 min_S: 701.53898 max_S: 726.72811 S: 712.74418 ΔS: -4.56187 moves: 43
niter: 34 count: 9 breaks: 1 min_S: 701.53898 max_S: 726.72811 S: 711.01120 ΔS: -1.73298 moves: 57
niter: 35 count: 10 breaks: 2 min_S: 701.53898 max_S: 726.72811 S: 717.58446 ΔS: 6.57326 moves: 47
Note that the value of `wait` above was made purposefully low so that
the output would not be overly long. The most appropriate value requires
experimentation, but a typically good value is `wait=1000`.
The function :func:`~graph_tool.inference.mcmc_equilibrate` accepts a
``callback`` argument that takes an optional function to be invoked
after each call to
:meth:`~graph_tool.inference.BlockState.mcmc_sweep`. This function
should accept a single parameter which will contain the actual
:class:`~graph_tool.inference.BlockState` instance. We will use this in
the example below to collect the posterior vertex marginals, i.e. the
posterior probability that a node belongs to a given group:
.. testcode:: model-averaging
# We will first equilibrate the Markov chain
gt.mcmc_equilibrate(state, wait=1000, mcmc_args=dict(niter=10))
pv = None
def collect_marginals(s):
global pv
pv = s.collect_vertex_marginals(pv)
# Now we collect the marginals for exactly 100,000 sweeps
gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10),
callback=collect_marginals)
# Now the node marginals are stored in property map pv. We can
# visualize them as pie charts on the nodes:
state.draw(pos=g.vp.pos, vertex_shape="pie", vertex_pie_fractions=pv,
edge_gradient=None, output="lesmis-sbm-marginals.svg")
.. figure:: lesmis-sbm-marginals.*
:align: center
:width: 450px
Marginal probabilities of group memberships of the network of
characters in the novel Les Misérables, according to the
degree-corrected SBM. The `pie fractions
`_ on the nodes correspond
to the probability of being in group associated with the respective
color.
We can also obtain a marginal probability on the number of groups
itself, as follows.
.. testcode:: model-averaging
h = np.zeros(g.num_vertices() + 1)
def collect_num_groups(s):
B = s.get_nonempty_B()
h[B] += 1
# Now we collect the marginal distribution for exactly 100,000 sweeps
gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10),
callback=collect_num_groups)
.. testcode:: model-averaging
:hide:
figure()
Bs = np.arange(len(h))
idx = h > 0
bar(Bs[idx] - .5, h[idx] / h.sum(), width=1, color="#ccb974")
gca().set_xticks([6,7,8,9])
xlabel("$B$")
ylabel(r"$P(B|\boldsymbol G)$")
savefig("lesmis-B-posterior.svg")
.. figure:: lesmis-B-posterior.*
:align: center
Marginal posterior likelihood of the number of nonempty groups for the
network of characters in the novel Les Misérables, according to the
degree-corrected SBM.
Hierarchical partitions
+++++++++++++++++++++++
We can also perform model averaging using the nested SBM, which will
give us a distribution over hierarchies. The whole procedure is fairly
analogous, but now we make use of
:class:`~graph_tool.inference.NestedBlockState` instances.
.. note::
When using :class:`~graph_tool.inference.NestedBlockState` instances
to perform model averaging, they need to be constructed with the
option `sampling=True`.
Here we perform the sampling of hierarchical partitions using the same
network as above.
.. testcode:: nested-model-averaging
g = gt.collection.data["lesmis"]
state = gt.minimize_nested_blockmodel_dl(g) # Initialize he Markov
# chain from the "ground
# state"
# Before doing model averaging, the need to create a NestedBlockState
# by passing sampling = True.
# We also want to increase the maximum hierarchy depth to L = 10
# We can do both of the above by copying.
bs = state.get_bs() # Get hierarchical partition.
bs += [np.zeros(1)] * (10 - len(bs)) # Augment it to L = 10 with
# single-group levels.
state = state.copy(bs=bs, sampling=True)
# Now we run 1000 sweeps of the MCMC
dS, nmoves = state.mcmc_sweep(niter=1000)
print("Change in description length:", dS)
print("Number of accepted vertex moves:", nmoves)
.. testoutput:: nested-model-averaging
Change in description length: 6.1062707...
Number of accepted vertex moves: 8488
Similarly to the the non-nested case, we can use
:func:`~graph_tool.inference.mcmc_equilibrate` to do most of the boring
work, and we can now obtain vertex marginals on all hierarchical levels:
.. testcode:: nested-model-averaging
# We will first equilibrate the Markov chain
gt.mcmc_equilibrate(state, wait=1000, mcmc_args=dict(niter=10))
pv = [None] * len(state.get_levels())
def collect_marginals(s):
global pv
pv = [sl.collect_vertex_marginals(pv[l]) for l, sl in enumerate(s.get_levels())]
# Now we collect the marginals for exactly 100,000 sweeps
gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10),
callback=collect_marginals)
# Now the node marginals for all levels are stored in property map
# list pv. We can visualize the first level as pie charts on the nodes:
state_0 = state.get_levels()[0]
state_0.draw(pos=g.vp.pos, vertex_shape="pie", vertex_pie_fractions=pv[0],
edge_gradient=None, output="lesmis-nested-sbm-marginals.svg")
.. figure:: lesmis-nested-sbm-marginals.*
:align: center
:width: 450px
Marginal probabilities of group memberships of the network of
characters in the novel Les Misérables, according to the nested
degree-corrected SBM. The pie fractions on the nodes correspond to
the probability of being in group associated with the respective
color.
We can also obtain a marginal probability of the number of groups
itself, as follows.
.. testcode:: nested-model-averaging
h = [np.zeros(g.num_vertices() + 1) for s in state.get_levels()]
def collect_num_groups(s):
for l, sl in enumerate(s.get_levels()):
B = sl.get_nonempty_B()
h[l][B] += 1
# Now we collect the marginal distribution for exactly 100,000 sweeps
gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10),
callback=collect_num_groups)
.. testcode:: nested-model-averaging
:hide:
figure()
f, ax = plt.subplots(1, 5, figsize=(10, 3))
for i, h_ in enumerate(h[:5]):
Bs = np.arange(len(h_))
idx = h_ > 0
ax[i].bar(Bs[idx] - .5, h_[idx] / h_.sum(), width=1, color="#ccb974")
ax[i].set_xticks(Bs[idx])
ax[i].set_xlabel("$B_{%d}$" % i)
ax[i].set_ylabel(r"$P(B_{%d}|\boldsymbol G)$" % i)
locator = MaxNLocator(prune='both', nbins=5)
ax[i].yaxis.set_major_locator(locator)
tight_layout()
savefig("lesmis-nested-B-posterior.svg")
.. figure:: lesmis-nested-B-posterior.*
:align: center
Marginal posterior likelihood of the number of nonempty groups
:math:`B_l` at each hierarchy level :math:`l` for the network of
characters in the novel Les Misérables, according to the nested
degree-corrected SBM.
Below we obtain some hierarchical partitions sampled from the posterior
distribution.
.. testcode:: nested-model-averaging
for i in range(10):
state.mcmc_sweep(niter=1000)
state.draw(output="lesmis-partition-sample-%i.svg" % i, empty_branches=False)
.. image:: lesmis-partition-sample-0.svg
:width: 200px
.. image:: lesmis-partition-sample-1.svg
:width: 200px
.. image:: lesmis-partition-sample-2.svg
:width: 200px
.. image:: lesmis-partition-sample-3.svg
:width: 200px
.. image:: lesmis-partition-sample-4.svg
:width: 200px
.. image:: lesmis-partition-sample-5.svg
:width: 200px
.. image:: lesmis-partition-sample-6.svg
:width: 200px
.. image:: lesmis-partition-sample-7.svg
:width: 200px
.. image:: lesmis-partition-sample-8.svg
:width: 200px
.. image:: lesmis-partition-sample-9.svg
:width: 200px
Model class selection
+++++++++++++++++++++
When averaging over partitions, we may be interested in evaluating which
**model class** provides a better fit of the data, considering all
possible parameter choices. This is done by evaluating the model
evidence [peixoto-nonparametric-2016]_
.. math::
P(\boldsymbol G) = \sum_{\boldsymbol\theta,\boldsymbol b}P(\boldsymbol G,\boldsymbol\theta, \boldsymbol b) = \sum_{\boldsymbol b}P(\boldsymbol G,\boldsymbol b).
This quantity is analogous to a `partition function
`_
in statistical physics, which we can write more conveniently as a
negative `free energy
`_ by taking
its logarithm
.. math::
:label: free-energy
\ln P(\boldsymbol G) = \underbrace{\sum_{\boldsymbol b}q(\boldsymbol b)\ln P(\boldsymbol G,\boldsymbol b)}_{-\left<\Sigma\right>}\;
\underbrace{- \sum_{\boldsymbol b}q(\boldsymbol b)\ln q(\boldsymbol b)}_{\mathcal{S}}
where
.. math::
q(\boldsymbol b) = \frac{P(\boldsymbol G,\boldsymbol b)}{\sum_{\boldsymbol b'}P(\boldsymbol G,\boldsymbol b')}
is the posterior likelihood of partition :math:`\boldsymbol b`. The
first term of Eq. :eq:`free-energy` (the "negative energy") is minus the
average of description length :math:`\left<\Sigma\right>`, weighted
according to the posterior distribution. The second term
:math:`\mathcal{S}` is the `entropy
`_ of the
posterior distribution, and measures, in a sense, the "quality of fit"
of the model: If the posterior is very "peaked", i.e. dominated by a
single partition with a very large likelihood, the entropy will tend to
zero. However, if there are many partitions with similar likelihoods ---
meaning that there is no single partition that describes the
network uniquely well --- it will take a large value instead.
Since the MCMC algorithm samples partitions from the distribution
:math:`q(\boldsymbol b)`, it can be used to compute
:math:`\left<\Sigma\right>` easily, simply by averaging the description
length values encountered by sampling from the posterior distribution
many times.
The computation of the posterior entropy :math:`\mathcal{S}`, however,
is significantly more difficult, since it involves measuring the precise
value of :math:`q(\boldsymbol b)`. A direct "brute force" computation of
:math:`\mathcal{S}` is implemented via
:meth:`~graph_tool.inference.BlockState.collect_partition_histogram` and
:func:`~graph_tool.inference.microstate_entropy`, however this is only
feasible for very small networks. For larger networks, we are forced to
perform approximations. The simplest is a "mean field" one, where we
assume the posterior factorizes as
.. math::
q(\boldsymbol b) \approx \prod_i{q_i(b_i)}
where
.. math::
q_i(r) = P(b_i = r | \boldsymbol G)
is the marginal group membership distribution of node :math:`i`. This
yields an entropy value given by
.. math::
S \approx -\sum_i\sum_rq_i(r)\ln q_i(r).
This approximation should be seen as an upper bound, since any existing
correlation between the nodes (which are ignored here) will yield
smaller entropy values.
A more accurate assumption is called the `Bethe approximation`
[mezard-information-2009]_, and takes into account the correlation
between adjacent nodes in the network,
.. math::
q(\boldsymbol b) \approx \prod_{i`_, :math:`k_i` is the
degree of node :math:`i`, and
.. math::
q_{ij}(r, s) = P(b_i = r, b_j = s|\boldsymbol G)
is the joint group membership distribution of nodes :math:`i` and
:math:`j` (a.k.a. the `edge marginals`). This yields an entropy value
given by
.. math::
S \approx -\sum_{i0` only the mean-field approximation is applicable, since the
adjacency matrix of the higher layers is not constant. We show below the
approach for the same network, using the nested model.
.. testcode:: model-evidence
g = gt.collection.data["lesmis"]
L = 10
for deg_corr in [True, False]:
state = gt.minimize_nested_blockmodel_dl(g, deg_corr=deg_corr) # Initialize the Markov
# chain from the "ground
# state"
bs = state.get_bs() # Get hierarchical partition.
bs += [np.zeros(1)] * (L - len(bs)) # Augment it to L = 10 with
# single-group levels.
state = state.copy(bs=bs, sampling=True)
dls = [] # description length history
vm = [None] * len(state.get_levels()) # vertex marginals
em = None # edge marginals
def collect_marginals(s):
global vm, em
levels = s.get_levels()
vm = [sl.collect_vertex_marginals(vm[l]) for l, sl in enumerate(levels)]
em = levels[0].collect_edge_marginals(em)
dls.append(s.entropy())
# Now we collect the marginal distributions for exactly 200,000 sweeps
gt.mcmc_equilibrate(state, force_niter=20000, mcmc_args=dict(niter=10),
callback=collect_marginals)
S_mf = [gt.mf_entropy(sl.g, vm[l]) for l, sl in enumerate(state.get_levels())]
S_bethe = gt.bethe_entropy(g, em)[0]
L = -mean(dls)
print("Model evidence for deg_corr = %s:" % deg_corr,
L + sum(S_mf), "(mean field),", L + S_bethe + sum(S_mf[1:]), "(Bethe)")
.. testoutput:: model-evidence
Model evidence for deg_corr = True: -339.011500645 (mean field), -665.93103635 (Bethe)
Model evidence for deg_corr = False: -395.273662985 (mean field), -562.873911796 (Bethe)
The results are similar: If we consider the most accurate approximation,
the non-degree-corrected model possesses the largest evidence. Note also
that we observe a better evidence for the nested models themselves, when
comparing to the evidences for the non-nested model --- which is not
quite surprising, since the non-nested model is a special case of the
nested one.
Edge layers and covariates
--------------------------
In many situations, the edges of the network may posses discrete
covariates on them, or they may be distributed in discrete
"layers". Extensions to the SBM may be defined for such data, and they
can be inferred using the exact same interface shown above, except one
should use the :class:`~graph_tool.inference.LayeredBlockState` class,
instead of :class:`~graph_tool.inference.BlockState`. This class takes
two additional parameters: the ``ec`` parameter, that must correspond to
an edge :class:`~graph_tool.PropertyMap` with the layer/covariate
values on the edges, and the Boolean ``layers`` parameter, which if
``True`` specifies a layered model, otherwise one with edge covariates.
If we use :func:`~graph_tool.inference.minimize_blockmodel_dl`, this can
be achieved simply by passing the option ``layers=True`` as well as the
appropriate value of ``state_args``, which will be propagated to
:class:`~graph_tool.inference.LayeredBlockState`'s constructor.
For example, consider again the Les Misérables network, where we
consider the number of co-appearances between characters as edge
covariates.
.. testsetup:: layered-model
import os
try:
os.chdir("demos/inference")
except FileNotFoundError:
pass
.. testcode:: layered-model
g = gt.collection.data["lesmis"]
# Note the different meaning of the two 'layers' parameters below: The
# first enables the use of LayeredBlockState, and the second selects
# the 'edge covariates' version.
state = gt.minimize_blockmodel_dl(g, deg_corr=False, layers=True,
state_args=dict(ec=g.ep.value, layers=False))
state.draw(pos=g.vp.pos, edge_color=g.ep.value, edge_gradient=None,
output="lesmis-sbm-edge-cov.svg")
.. figure:: lesmis-sbm-edge-cov.*
:align: center
:width: 350px
Best fit of the non-degree-corrected SBM with edge covariates for the
network of characters in the novel Les Misérables, using the number
of co-appearances as edge covariates. The edge colors correspond to
the edge covariates.
In the case of the nested model, we still should use the
:class:`~graph_tool.inference.NestedBlockState` class, but it must be
initialized with the parameter ``base_type = LayeredBlockState``. But if
we use :func:`~graph_tool.inference.minimize_nested_blockmodel_dl`, it
works identically to the above:
.. testcode:: layered-model
state = gt.minimize_nested_blockmodel_dl(g, deg_corr=False, layers=True,
state_args=dict(ec=g.ep.value, layers=False))
state.draw(eprops=dict(color=g.ep.value, gradient=None),
output="lesmis-nested-sbm-edge-cov.svg")
.. figure:: lesmis-nested-sbm-edge-cov.*
:align: center
:width: 350px
Best fit of the nested non-degree-corrected SBM with edge covariates
for the network of characters in the novel Les Misérables, using the
number of co-appearances as edge covariates. The edge colors
correspond to the edge covariates.
It is possible to perform model averaging of all layered variants
exactly like for the regular SBMs as was shown above.
Predicting spurious and missing edges
-------------------------------------
An important application of generative models is to be able to
generalize from observations and make predictions that go beyond what
is seen in the data. This is particularly useful when the network we
observe is incomplete, or contains errors, i.e. some of the edges are
either missing or are outcomes of mistakes in measurement. In this
situation, the fit we make of the observed network can help us
predict missing or spurious edges in the network
[clauset-hierarchical-2008]_ [guimera-missing-2009]_.
We do so by dividing the edges into two sets :math:`\boldsymbol G` and :math:`\delta
\boldsymbol G`, where the former corresponds to the observed network and the latter
either to the missing or spurious edges. In the case of missing edges,
we may compute the posterior of :math:`\delta \boldsymbol G` as
.. math::
:label: posterior-missing
P(\delta \boldsymbol G | \boldsymbol G) = \frac{\sum_{\boldsymbol b}P(\boldsymbol G+\delta \boldsymbol G | \boldsymbol b)P(\boldsymbol b | \boldsymbol G)}{P_{\delta}(\boldsymbol G)}
where
.. math::
P_{\delta}(\boldsymbol G) = \sum_{\delta \boldsymbol G}\sum_{\boldsymbol b}P(\boldsymbol G+\delta \boldsymbol G | \boldsymbol b)P(\boldsymbol b | \boldsymbol G)
is a normalization constant. Although the value of :math:`P_{\delta}(\boldsymbol G)` is
difficult to obtain in general (since we need to perform a sum over all
possible spurious/missing edges), the numerator of
Eq. :eq:`posterior-missing` can be computed by sampling partitions from
the posterior, and then inserting or deleting edges from the graph and
computing the new likelihood. This means that we can easily compare
alternative predictive hypotheses :math:`\{\delta \boldsymbol G_i\}` via their
likelihood ratios
.. math::
\lambda_i = \frac{P(\delta \boldsymbol G_i | \boldsymbol G)}{\sum_j P(\delta \boldsymbol G_j | \boldsymbol G)}
= \frac{\sum_{\boldsymbol b}P(\boldsymbol G+\delta \boldsymbol G_i | \boldsymbol b)P(\boldsymbol b | \boldsymbol G)}
{\sum_j \sum_{\boldsymbol b}P(\boldsymbol G+\delta \boldsymbol G_j | \boldsymbol b)P(\boldsymbol b | \boldsymbol G)}
which do not depend on the value of :math:`P_{\delta}(\boldsymbol G)`.
The values :math:`P(\boldsymbol G+\delta \boldsymbol G | \boldsymbol b)`
can be computed with
:meth:`~graph_tool.inference.BlockState.get_edges_prob`. Hence, we can
compute spurious/missing edge probabilities just as if we were
collecting marginal distributions when doing model averaging.
Below is an example for predicting the two following edges in the
football network, using the nested model (for which we need to replace
:math:`\boldsymbol b` by :math:`\{\boldsymbol b_l\}` in the equations
above).
.. testcode:: missing-edges
:hide:
g = gt.collection.data["football"].copy()
color = g.new_vp("string", val="#cccccc")
ecolor = g.new_ep("string", val="#cccccc")
e = g.add_edge(101, 102)
ecolor[e] = "#a40000"
e = g.add_edge(17, 56)
ecolor[e] = "#a40000"
eorder = g.edge_index.copy("int")
gt.graph_draw(g, pos=g.vp.pos, vertex_color=color,
vertex_fill_color=color, edge_color=ecolor,
eorder=eorder,
output="football_missing.svg")
.. figure:: football_missing.*
:align: center
:width: 350px
Two non-existing edges in the football network (in red):
:math:`(101,102)` in the middle, and :math:`(17,56)` in the upper
right region of the figure.
.. testcode:: missing-edges
g = gt.collection.data["football"]
missing_edges = [(101, 102), (17, 56)]
L = 10
state = gt.minimize_nested_blockmodel_dl(g, deg_corr=True)
bs = state.get_bs() # Get hierarchical partition.
bs += [np.zeros(1)] * (L - len(bs)) # Augment it to L = 10 with
# single-group levels.
state = state.copy(bs=bs, sampling=True)
probs = ([], [])
def collect_edge_probs(s):
p1 = s.get_edges_prob([missing_edges[0]], entropy_args=dict(partition_dl=False))
p2 = s.get_edges_prob([missing_edges[1]], entropy_args=dict(partition_dl=False))
probs[0].append(p1)
probs[1].append(p2)
# Now we collect the probabilities for exactly 10,000 sweeps
gt.mcmc_equilibrate(state, force_niter=1000, mcmc_args=dict(niter=10),
callback=collect_edge_probs)
def get_avg(p):
p = np.array(p)
pmax = p.max()
p -= pmax
return pmax + log(exp(p).mean())
p1 = get_avg(probs[0])
p2 = get_avg(probs[1])
p_sum = get_avg([p1, p2]) + log(2)
l1 = p1 - p_sum
l2 = p2 - p_sum
print("likelihood-ratio for %s: %g" % (missing_edges[0], exp(l1)))
print("likelihood-ratio for %s: %g" % (missing_edges[1], exp(l2)))
.. testoutput:: missing-edges
likelihood-ratio for (101, 102): 0.365264
likelihood-ratio for (17, 56): 0.634736
From which we can conclude that edge :math:`(17, 56)` is around twice as
likely as :math:`(101, 102)` to be a missing edge.
The prediction using the non-nested model can be performed in an
entirely analogous fashion.
References
----------
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Samuel Leinhardt, "Stochastic blockmodels: First steps", Social Networks
Volume 5, Issue 2, Pages 109-137 (1983), :doi:`10.1016/0378-8733(83)90021-7`
.. [karrer-stochastic-2011] Brian Karrer, M. E. J. Newman "Stochastic
blockmodels and community structure in networks", Phys. Rev. E 83,
016107 (2011), :doi:`10.1103/PhysRevE.83.016107`, :arxiv:`1008.3926`
.. [peixoto-nonparametric-2016] Tiago P. Peixoto, "Nonparametric
Bayesian inference of the microcanonical stochastic block model"
:arxiv:`1610.02703`
.. [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`.
.. [peixoto-hierarchical-2014] Tiago P. Peixoto, "Hierarchical block
structures and high-resolution model selection in large networks",
Phys. Rev. X 4, 011047 (2014), :doi:`10.1103/PhysRevX.4.011047`,
:arxiv:`1310.4377`.
.. [peixoto-model-2016] Tiago P. Peixoto, "Model selection and hypothesis
testing for large-scale network models with overlapping groups",
Phys. Rev. X 5, 011033 (2016), :doi:`10.1103/PhysRevX.5.011033`,
:arxiv:`1409.3059`.
.. [peixoto-efficient-2014] Tiago P. Peixoto, "Efficient Monte Carlo and
greedy heuristic for the inference of stochastic block models", Phys.
Rev. E 89, 012804 (2014), :doi:`10.1103/PhysRevE.89.012804`,
:arxiv:`1310.4378`
.. [clauset-hierarchical-2008] Aaron Clauset, Cristopher
Moore, M. E. J. Newman, "Hierarchical structure and the prediction of
missing links in networks", Nature 453, 98-101 (2008),
:doi:`10.1038/nature06830`
.. [guimera-missing-2009] Roger Guimerà, Marta Sales-Pardo, "Missing and
spurious interactions and the reconstruction of complex networks", PNAS
vol. 106 no. 52 (2009), :doi:`10.1073/pnas.0908366106`
.. [mezard-information-2009] Marc Mézard, Andrea Montanari, "Information,
Physics, and Computation", Oxford Univ Press, 2009.
:DOI:`10.1093/acprof:oso/9780198570837.001.0001`