Uncertain network reconstruction
--------------------------------
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, or is not
even observed at all. In this situation, we can use statistical
inference to reconstruct the original network. Following
[peixoto-reconstructing-2018]_, if :math:`\boldsymbol{\mathcal{D}}` is
the observed data, the network can be reconstructed according to the
posterior distribution,
.. math::
:label: posterior-reconstruction
P(\boldsymbol A, \boldsymbol b | \boldsymbol{\mathcal{D}}) =
\frac{P(\boldsymbol{\mathcal{D}} | \boldsymbol A)P(\boldsymbol A, \boldsymbol b)}{P(\boldsymbol{\mathcal{D}})}
where the likelihood :math:`P(\boldsymbol{\mathcal{D}}|\boldsymbol A)`
models the measurement process, and for the prior :math:`P(\boldsymbol
A, \boldsymbol b)` we use the SBM as before. This means that when
performing reconstruction, we sample both the community structure
:math:`\boldsymbol b` and the network :math:`\boldsymbol A` itself from
the posterior distribution. From it, we can obtain the marginal probability
of each edge,
.. math::
\pi_{ij} = \sum_{\boldsymbol A, \boldsymbol b}A_{ij}P(\boldsymbol A, \boldsymbol b | \boldsymbol{\mathcal{D}}).
Based on the marginal posterior probabilities, the best estimate for the
whole underlying network :math:`\boldsymbol{\hat{A}}` is given by the
maximum of this distribution,
.. math::
\hat A_{ij} =
\begin{cases}
1 & \text{ if } \pi_{ij} > \frac{1}{2},\\
0 & \text{ if } \pi_{ij} < \frac{1}{2}.\\
\end{cases}
We can also make estimates :math:`\hat y` of arbitrary scalar network
properties :math:`y(\boldsymbol A)` via posterior averages,
.. math::
\begin{align}
\hat y &= \sum_{\boldsymbol A, \boldsymbol b}y(\boldsymbol A)P(\boldsymbol A, \boldsymbol b | \boldsymbol{\mathcal{D}}),\\
\sigma^2_y &= \sum_{\boldsymbol A, \boldsymbol b}(y(\boldsymbol A)-\hat y)^2P(\boldsymbol A, \boldsymbol b | \boldsymbol{\mathcal{D}})
\end{align}
with uncertainty given by :math:`\sigma_y`. This is gives us a complete
probabilistic reconstruction framework that fully reflects both the
information and the uncertainty in the measurement data. Furthermore,
the use of the SBM means that the reconstruction can take advantage of
the *correlations* observed in the data to further inform it, which
generally can lead to substantial improvements
[peixoto-reconstructing-2018]_ [peixoto-network-2019]_.
In graph-tool there is support for reconstruction with the above
framework for three measurement processes: 1. Repeated measurements with
uniform errors (via
:class:`~graph_tool.inference.MeasuredBlockState`), 2. Repeated
measurements with heterogeneous errors (via
:class:`~graph_tool.inference.MixedMeasuredBlockState`),
and 3. Extraneously obtained edge probabilities (via
:class:`~graph_tool.inference.UncertainBlockState`),
which we describe in the following.
In addition, it is also possible to reconstruct networks from observed dynamics
and behavioral observation, as described in :ref:`reconstruction_dynamics`.
.. _measured_networks:
Measured networks
+++++++++++++++++
This model assumes that the node pairs :math:`(i,j)` were measured
:math:`n_{ij}` times, and an edge has been recorded :math:`x_{ij}`
times, where a missing edge occurs with probability :math:`p` and a
spurious edge occurs with probability :math:`q`, uniformly for all node
pairs, yielding a likelihood
.. math::
P(\boldsymbol x | \boldsymbol n, \boldsymbol A, p, q) =
\prod_{i`__, which
specify the amount of prior knowledge we have on the noise
parameters. An important special case, which is the default unless
otherwise specified, is when we are completely agnostic *a priori* about
the noise magnitudes, and all hyperparameters are unity,
.. math::
P(\boldsymbol x | \boldsymbol n, \boldsymbol A) \equiv
P(\boldsymbol x | \boldsymbol n, \boldsymbol A, \alpha=1,\beta=1,\mu=1,\nu=1).
In this situation the priors :math:`P(p|\alpha=1,\beta=1)` and
:math:`P(q|\mu=1,\nu=1)` are uniform distribution in the interval :math:`[0,1]`.
.. note::
Since this approach also makes use of the *correlations* between
edges to inform the reconstruction, as described by the inferred SBM,
this means it can also be used when only single measurements have
been performed, :math:`n_{ij}=1`, and the error magnitudes :math:`p`
and :math:`q` are unknown. Since every arbitrary adjacency matrix can
be cast in this setting, this method can be used to reconstruct
networks for which no error assessments of any kind have been
provided.
Below, we illustrate how the reconstruction can be performed with a
simple example, using
:class:`~graph_tool.inference.MeasuredBlockState`:
.. testsetup:: measured
import os
try:
os.chdir("demos/reconstruction_direct")
except FileNotFoundError:
pass
np.random.seed(43)
gt.seed_rng(42)
.. testcode:: measured
g = gt.collection.data["lesmis"].copy()
# pretend we have measured and observed each edge twice
n = g.new_ep("int", 2) # number of measurements
x = g.new_ep("int", 2) # number of observations
e = g.edge(11, 36)
x[e] = 1 # pretend we have observed edge (11, 36) only once
e = g.add_edge(15, 73)
n[e] = 2 # pretend we have measured non-edge (15, 73) twice,
x[e] = 1 # but observed it as an edge once.
# We inititialize MeasuredBlockState, assuming that each non-edge has
# been measured only once (as opposed to twice for the observed
# edges), as specified by the 'n_default' and 'x_default' parameters.
state = gt.MeasuredBlockState(g, n=n, n_default=1, x=x, x_default=0)
# We will first equilibrate the Markov chain
gt.mcmc_equilibrate(state, wait=100, mcmc_args=dict(niter=10))
# Now we collect the marginals for exactly 100,000 sweeps, at
# intervals of 10 sweeps:
u = None # marginal posterior edge probabilities
bs = [] # partitions
cs = [] # average local clustering coefficient
def collect_marginals(s):
global u, bs, cs
u = s.collect_marginal(u)
bstate = s.get_block_state()
bs.append(bstate.levels[0].b.a.copy())
cs.append(gt.local_clustering(s.get_graph()).fa.mean())
gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10),
callback=collect_marginals)
eprob = u.ep.eprob
print("Posterior probability of edge (11, 36):", eprob[u.edge(11, 36)])
print("Posterior probability of non-edge (15, 73):", eprob[u.edge(15, 73)] if u.edge(15, 73) is not None else 0.)
print("Estimated average local clustering: %g ± %g" % (np.mean(cs), np.std(cs)))
Which yields the following output:
.. testoutput:: measured
Posterior probability of edge (11, 36): 0.849284...
Posterior probability of non-edge (15, 73): 0.061706...
Estimated average local clustering: 0.573465 ± 0.005395...
We have a successful reconstruction, where both ambiguous adjacency
matrix entries are correctly recovered. The value for the average
clustering coefficient is also correctly estimated, and is compatible
with the true value :math:`0.57313675`, within the estimated error.
Below we visualize the maximum marginal posterior estimate of the
reconstructed network:
.. testcode:: measured
# The maximum marginal posterior estimator can be obtained by
# filtering the edges with probability larger than .5
u = gt.GraphView(u, efilt=u.ep.eprob.fa > .5)
# Mark the recovered true edges as red, and the removed spurious edges as green
ecolor = u.new_ep("vector", val=[0, 0, 0, .6])
for e in u.edges():
if g.edge(e.source(), e.target()) is None or (e.source(), e.target()) == (11, 36):
ecolor[e] = [1, 0, 0, .6]
for e in g.edges():
if u.edge(e.source(), e.target()) is None:
ne = u.add_edge(e.source(), e.target())
ecolor[ne] = [0, 1, 0, .6]
# Duplicate the internal block state with the reconstructed network
# u, for visualization purposes.
bstate = state.get_block_state()
bstate = bstate.levels[0].copy(g=u)
# Disambiguate partitions and obtain marginals
pmode = gt.PartitionModeState(bs, converge=True)
pv = pmode.get_marginal(u)
edash = u.new_ep("vector")
edash[u.edge(15, 73)] = [.1, .1, 0]
bstate.draw(pos=u.own_property(g.vp.pos), vertex_shape="pie", vertex_pie_fractions=pv,
edge_color=ecolor, edge_dash_style=edash, edge_gradient=None,
output="lesmis-reconstruction-marginals.svg")
.. figure:: lesmis-reconstruction-marginals.*
:align: center
:width: 450px
Reconstructed network of characters in the novel Les Misérables,
assuming that each edge has been measured and recorded twice, and
each non-edge has been measured only once, with the exception of edge
(11, 36), shown in red, and non-edge (15, 73), shown in green, which
have been measured twice and recorded as an edge once. Despite the
ambiguity, both errors are successfully corrected by the
reconstruction. The pie fractions on the nodes correspond to the
probability of being in group associated with the respective color.
Heterogeneous errors
^^^^^^^^^^^^^^^^^^^^
In a more general scenario the measurement errors can be different for
each node pair, i.e. :math:`p_{ij}` and :math:`q_{ij}` are the missing
and spurious edge probabilities for node pair :math:`(i,j)`. The
measurement likelihood then becomes
.. math::
P(\boldsymbol x | \boldsymbol n, \boldsymbol A, \boldsymbol p, \boldsymbol q) =
\prod_{i`__, like
before. Instead of pre-specifying the hyperparameters, we include them
from the posterior distribution
.. math::
P(\boldsymbol A, \boldsymbol b, \alpha,\beta,\mu,\nu | \boldsymbol x, \boldsymbol n) =
\frac{P(\boldsymbol x | \boldsymbol n, \boldsymbol A, \alpha,\beta,\mu,\nu)P(\boldsymbol A, \boldsymbol b)P(\alpha,\beta,\mu,\nu)}{P(\boldsymbol x| \boldsymbol n)},
where :math:`P(\alpha,\beta,\mu,\nu)\propto 1` is a uniform hyperprior.
Operationally, the inference with this model works similarly to the one
with uniform error rates, as we see with the same example:
.. testcode:: measured
state = gt.MixedMeasuredBlockState(g, n=n, n_default=1, x=x, x_default=0)
# We will first equilibrate the Markov chain
gt.mcmc_equilibrate(state, wait=200, mcmc_args=dict(niter=10))
# Now we collect the marginals for exactly 100,000 sweeps, at
# intervals of 10 sweeps:
u = None # marginal posterior edge probabilities
bs = [] # partitions
cs = [] # average local clustering coefficient
gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10),
callback=collect_marginals)
eprob = u.ep.eprob
print("Posterior probability of edge (11, 36):", eprob[u.edge(11, 36)])
print("Posterior probability of non-edge (15, 73):", eprob[u.edge(15, 73)] if u.edge(15, 73) is not None else 0.)
print("Estimated average local clustering: %g ± %g" % (np.mean(cs), np.std(cs)))
Which yields:
.. testoutput:: measured
Posterior probability of edge (11, 36): 0.701270...
Posterior probability of non-edge (15, 73): 0.052905...
Estimated average local clustering: 0.568815 ± 0.010106...
The results are very similar to the ones obtained with the uniform model
in this case, but can be quite different in situations where a large
number of measurements has been performed (see
[peixoto-reconstructing-2018]_ for details).
Extraneous error estimates
++++++++++++++++++++++++++
In some situations the edge uncertainties are estimated by means other
than repeated measurements, using domain-specific models. Here we
consider the general case where the error estimates are extraneously
provided as independent edge probabilities :math:`\boldsymbol Q`,
.. math::
P_Q(\boldsymbol A | \boldsymbol Q) = \prod_{i .5)
# Mark the recovered true edges as red, and the removed spurious edges as green
ecolor = u.new_ep("vector", val=[0, 0, 0, .6])
edash = u.new_ep("vector")
for e in u.edges():
if g.edge(e.source(), e.target()) is None or (e.source(), e.target()) == (11, 36):
ecolor[e] = [1, 0, 0, .6]
for e in g.edges():
if u.edge(e.source(), e.target()) is None:
ne = u.add_edge(e.source(), e.target())
ecolor[ne] = [0, 1, 0, .6]
if (e.source(), e.target()) == (15, 73):
edash[ne] = [.1, .1, 0]
bstate = state.get_block_state()
bstate = bstate.levels[0].copy(g=u)
# Disambiguate partitions and obtain marginals
pmode = gt.PartitionModeState(bs, converge=True)
pv = pmode.get_marginal(u)
bstate.draw(pos=u.own_property(g.vp.pos), vertex_shape="pie", vertex_pie_fractions=pv,
edge_color=ecolor, edge_dash_style=edash, edge_gradient=None,
output="lesmis-uncertain-reconstruction-marginals.svg")
.. figure:: lesmis-uncertain-reconstruction-marginals.*
:align: center
:width: 450px
Reconstructed network of characters in the novel Les Misérables,
assuming that each edge as a measurement probability of
:math:`.98`. Edge (11, 36), shown in red, and non-edge (15, 73),
shown in green, both have probability :math:`0.5`. Despite the
ambiguity, both errors are successfully corrected by the
reconstruction. The pie fractions on the nodes correspond to the
probability of being in group associated with the respective color.
Latent Poisson multigraphs
++++++++++++++++++++++++++
Even in situations where measurement errors can be neglected, it can
still be useful to assume a given network is the outcome of a "hidden"
multigraph model, i.e. more than one edge between nodes is allowed, but
then its multiedges are "erased" by transforming them into simple
edges. In this way, it is possible to construct generative models that
can better handle situations where the underlying network possesses
heterogeneous density, such as strong community structure and broad
degree distributions [peixoto-latent-2020]_. This can be incorporated
into the scheme of Eq. :eq:`posterior-reconstruction` by considering
the data to be the observed simple graph,
:math:`\boldsymbol{\mathcal{D}} = \boldsymbol G`. We proceed in same
way as in the previous reconstruction scenarios, but using instead
:class:`~graph_tool.inference.LatentMultigraphBlockState`.
For example, in the following we will obtain the community structure and
latent multiedges of a network of political books:
.. testcode:: erased-poisson
g = gt.collection.data["polbooks"]
state = gt.LatentMultigraphBlockState(g)
# We will first equilibrate the Markov chain
gt.mcmc_equilibrate(state, wait=100, mcmc_args=dict(niter=10))
# Now we collect the marginals for exactly 100,000 sweeps, at
# intervals of 10 sweeps:
u = None # marginal posterior multigraph
bs = [] # partitions
def collect_marginals(s):
global bs, u
u = s.collect_marginal_multigraph(u)
bstate = state.get_block_state()
bs.append(bstate.levels[0].b.a.copy())
gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10),
callback=collect_marginals)
# compute average multiplicities
ew = u.new_ep("double")
w = u.ep.w
wcount = u.ep.wcount
for e in u.edges():
ew[e] = (wcount[e].a * w[e].a).sum() / wcount[e].a.sum()
bstate = state.get_block_state()
bstate = bstate.levels[0].copy(g=u)
# Disambiguate partitions and obtain marginals
pmode = gt.PartitionModeState(bs, converge=True)
pv = pmode.get_marginal(u)
bstate.draw(pos=u.own_property(g.vp.pos), vertex_shape="pie", vertex_pie_fractions=pv,
edge_pen_width=gt.prop_to_size(ew, .1, 8, power=1), edge_gradient=None,
output="polbooks-erased-poisson.svg")
.. figure:: polbooks-erased-poisson.*
:align: center
:width: 450px
Reconstructed latent Poisson degree-corrected SBM for a network of
political books, showing the marginal mean edge multiplicities as
line thickness. The pie fractions on the nodes correspond to the
probability of being in group associated with the respective color.
Latent triadic closures
+++++++++++++++++++++++
Another useful reconstruction scenario is when we assume our observed
network is the outcome of a mixture of different edge placement
mechanisms. One example is the combination of triadic closure with
community structure [peixoto-disentangling-2022]_. This approach can be
used to separate the effects of triangle formation from node homophily,
which are typically conflated. We proceed in same way as in the previous
reconstruction scenarios, but using instead
:class:`~graph_tool.inference.LatentClosureBlockState`.
For example, in the following we will obtain the community structure and
latent closure edges of a network of political books:
.. testsetup:: latent-closure
import os
try:
os.chdir("demos/reconstruction_direct")
except FileNotFoundError:
pass
np.random.seed(43)
gt.seed_rng(45)
.. testcode:: latent-closure
g = gt.collection.data["polbooks"]
state = gt.LatentClosureBlockState(g, L=10)
# We will first equilibrate the Markov chain
gt.mcmc_equilibrate(state, wait=100, mcmc_args=dict(niter=10))
# Now we collect the marginals for exactly 100,000 sweeps, at
# intervals of 10 sweeps:
us = None # marginal posterior graphs
bs = [] # partitions
def collect_marginals(s):
global bs, us
us = s.collect_marginal(us)
bstate = state.bstate
bs.append(bstate.levels[0].b.a.copy())
gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10),
callback=collect_marginals)
u = us[0] # marginal seminal edges
# Disambiguate partitions and obtain marginals
pmode = gt.PartitionModeState(bs, converge=True)
pv = pmode.get_marginal(u)
bstate = state.bstate.levels[0].copy(g=u)
# edge width
ew = u.ep.eprob.copy()
ew.a = abs(ew.a - .5)
# get a color map
clrs = [(1, 0, 0, 1.0),
(0, 0, 0, 1.0)]
red_cm = matplotlib.colors.LinearSegmentedColormap.from_list("Set3", clrs)
# draw red edge last
eorder = u.ep.eprob.copy()
eorder.a *= -1
bstate.draw(pos=u.own_property(g.vp.pos), vertex_shape="pie", vertex_pie_fractions=pv,
edge_pen_width=gt.prop_to_size(ew, .1, 4, power=1),
edge_gradient=None, edge_color=u.ep.eprob, ecmap=red_cm,
eorder=eorder, output="polbooks-closure.svg")
.. figure:: polbooks-closure.*
:align: center
:width: 450px
Reconstructed degree-corrected SBM with latent closure edges for a
network of political books, showing the marginal probability of an
edge being due to triadic closure as the color red. The pie fractions
on the nodes correspond to the probability of being in group
associated with the respective color.
Triadic closure can also be used to perform uncertain network
reconstruction, using
:class:`~graph_tool.inference.MeasuredClosureBlockState`,
in a manner analogous to what was done in :ref:`measured_networks`:
.. testsetup:: measured-closure
import os
try:
os.chdir("demos/reconstruction_direct")
except FileNotFoundError:
pass
np.random.seed(43)
gt.seed_rng(45)
.. testcode:: measured-closure
g = gt.collection.data["lesmis"].copy()
# pretend we have measured and observed each edge twice
n = g.new_ep("int", 2) # number of measurements
x = g.new_ep("int", 2) # number of observations
e = g.edge(11, 36)
x[e] = 1 # pretend we have observed edge (11, 36) only once
e = g.add_edge(15, 73)
n[e] = 2 # pretend we have measured non-edge (15, 73) twice,
x[e] = 1 # but observed it as an edge once.
# We inititialize MeasuredBlockState, assuming that each non-edge has
# been measured only once (as opposed to twice for the observed
# edges), as specified by the 'n_default' and 'x_default' parameters.
state = gt.MeasuredClosureBlockState(g, n=n, n_default=1, x=x, x_default=0, L=10)
# We will first equilibrate the Markov chain
gt.mcmc_equilibrate(state, wait=100, mcmc_args=dict(niter=10))
# Now we collect the marginals for exactly 100,000 sweeps, at
# intervals of 10 sweeps:
us = None # marginal posterior edge probabilities
bs = [] # partitions
cs = [] # average local clustering coefficient
def collect_marginals(s):
global us, bs, cs
us = s.collect_marginal(us)
bstate = s.get_block_state()
bs.append(bstate.levels[0].b.a.copy())
cs.append(gt.local_clustering(s.get_graph()).fa.mean())
gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10),
callback=collect_marginals)
u = us[-1]
eprob = u.ep.eprob
print("Posterior probability of edge (11, 36):", eprob[u.edge(11, 36)])
print("Posterior probability of non-edge (15, 73):", eprob[u.edge(15, 73)] if u.edge(15, 73) is not None else 0.)
print("Estimated average local clustering: %g ± %g" % (np.mean(cs), np.std(cs)))
Which yields the following output:
.. testoutput:: measured-closure
Posterior probability of edge (11, 36): 0.992099...
Posterior probability of non-edge (15, 73): 0.013301...
Estimated average local clustering: 0.574592 ± 0.0047688
.. include:: _prediction.rst
References
----------
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networks with unknown and heterogeneous errors", Phys. Rev. X 8
041011 (2018). :doi:`10.1103/PhysRevX.8.041011`, :arxiv:`1806.07956`
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community structure and triadic closure in networks", Phys. Rev. X 12,
011004 (2022), :doi:`10.1103/PhysRevX.12.011004`, :arxiv:`2101.02510`
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:doi:`10.1103/PhysRevE.102.012309`, :arxiv:`2002.07803`
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.. rubric:: Footnotes
.. [#prediction_posterior] Note that the posterior of Eq. :eq:`posterior-missing`
cannot be used to sample the reconstruction :math:`\delta \boldsymbol
G`, as it is not informative of the overall network density
(i.e. absolute number of missing and spurious edges). It can,
however, be used to compare different reconstructions with the same
density.