# 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 \(\boldsymbol{\mathcal{D}}\) is the observed data, the network can be reconstructed according to the posterior distribution,

where the likelihood \(P(\boldsymbol{\mathcal{D}}|\boldsymbol A)\) models the measurement process, and for the prior \(P(\boldsymbol A, \boldsymbol b)\) we use the SBM as before. This means that when performing reconstruction, we sample both the community structure \(\boldsymbol b\) and the network \(\boldsymbol A\) itself from the posterior distribution. From it, we can obtain the marginal probability of each edge,

Based on the marginal posterior probabilities, the best estimate for the whole underlying network \(\boldsymbol{\hat{A}}\) is given by the maximum of this distribution,

We can also make estimates \(\hat y\) of arbitrary scalar network properties \(y(\boldsymbol A)\) via posterior averages,

with uncertainty given by \(\sigma_y\). 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
`MeasuredBlockState`

), 2. Repeated
measurements with heterogeneous errors (via
`MixedMeasuredBlockState`

),
and 3. Extraneously obtained edge probabilities (via
`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 Network reconstruction from dynamics and behavior.

## Measured networks#

This model assumes that the node pairs \((i,j)\) were measured \(n_{ij}\) times, and an edge has been recorded \(x_{ij}\) times, where a missing edge occurs with probability \(p\) and a spurious edge occurs with probability \(q\), uniformly for all node pairs, yielding a likelihood

In general, \(p\) and \(q\) are not precisely known *a priori*,
so we consider the integrated likelihood

where \(P(p|\alpha,\beta)\) and \(P(q|\mu,\nu)\) are Beta
distributions, 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,

In this situation the priors \(P(p|\alpha=1,\beta=1)\) and \(P(q|\mu=1,\nu=1)\) are uniform distribution in the interval \([0,1]\).

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, \(n_{ij}=1\), and the error magnitudes \(p\)
and \(q\) 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
`MeasuredBlockState`

:

```
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:

```
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 \(0.57313675\), within the estimated error.

Below we visualize the maximum marginal posterior estimate of the reconstructed network:

```
# 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<double>", 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<double>")
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")
```

### Heterogeneous errors#

In a more general scenario the measurement errors can be different for each node pair, i.e. \(p_{ij}\) and \(q_{ij}\) are the missing and spurious edge probabilities for node pair \((i,j)\). The measurement likelihood then becomes

Since the noise magnitudes are *a priori* unknown, we consider the
integrated likelihood

where \(P(p_{ij}|\alpha,\beta)\) and \(P(q_{ij}|\mu,\nu)\) are Beta prior distributions, like before. Instead of pre-specifying the hyperparameters, we include them from the posterior distribution

where \(P(\alpha,\beta,\mu,\nu)\propto 1\) 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:

```
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:

```
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 \(\boldsymbol Q\),

where \(Q_{ij}\) is the estimated probability of edge \((i,j)\). Although in principle we could reconstruct networks directly from the above distribution, we can also incorporate it with SBM inference to take advantage of large-scale structures present in the data. We do so by employing Bayes’ rule to extract the noise model from the provided values [martin-structural-2015] [peixoto-reconstructing-2018],

where \(\bar Q = \sum_{i<j}Q_{ij}/{N\choose 2}\) is the estimated network density, and \(P_Q(\boldsymbol Q)\) is an unknown prior for \(\boldsymbol Q\), which can remain unspecified as it has no effect on the posterior distribution. With the above, we can reconstruct the network based on the posterior distribution,

where \(P(\boldsymbol A, \boldsymbol b)\) is the joint SBM distribution used before. Note that this reconstruction will be different from the one obtained directly from the original estimation, i.e.

This is because the posterior \(P(\boldsymbol A | \boldsymbol Q)\) will take into consideration the correlations found in the data, as captured by the inferred SBM structure, as further evidence for the existence and non-existence of edges. We illustrate this with an example similar to the one considered previously, where two adjacency matrix entries with the same ambiguous edge probability \(Q_{ij}=1/2\) are correctly reconstructed as edge and non-edge, due to the joint SBM inference:

```
g = gt.collection.data["lesmis"].copy()
N = g.num_vertices()
E = g.num_edges()
q = g.new_ep("double", .98) # edge uncertainties
e = g.edge(11, 36)
q[e] = .5 # ambiguous true edge
e = g.add_edge(15, 73)
q[e] = .5 # ambiguous spurious edge
# We inititialize UncertainBlockState, assuming that each non-edge
# has an uncertainty of q_default, chosen to preserve the expected
# density of the original network:
q_default = (E - q.a.sum()) / ((N * (N - 1))/2 - E)
state = gt.UncertainBlockState(g, q=q, q_default=q_default)
# 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 bs, u, 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)))
```

The above yields the output:

```
Posterior probability of edge (11, 36): 0.932693...
Posterior probability of non-edge (15, 73): 0.056705...
Estimated average local clustering: 0.559017 ± 0.018329...
```

The reconstruction is accurate, despite the two ambiguous entries having the same measurement probability. The reconstructed network is visualized below.

```
# 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<double>", val=[0, 0, 0, .6])
edash = u.new_ep("vector<double>")
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")
```

## 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. (1) by considering
the data to be the observed simple graph,
\(\boldsymbol{\mathcal{D}} = \boldsymbol G\). We proceed in same
way as in the previous reconstruction scenarios, but using instead
`LatentMultigraphBlockState`

.

For example, in the following we will obtain the community structure and latent multiedges of a network of political books:

```
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")
```

## 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
`LatentClosureBlockState`

.

For example, in the following we will obtain the community structure and latent closure edges of a network of political books:

```
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")
```

Triadic closure can also be used to perform uncertain network
reconstruction, using
`MeasuredClosureBlockState`

,
in a manner analogous to what was done in Measured networks:

```
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:

```
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
```

## Edge prediction as binary classification#

A more traditional approach to the prediction of missing and spurious
edges formulates it as a supervised binary classification task, where the
edge/non-edge scores are computed by fitting a generative model to the
observed data, and computing their probabilities under that model
[clauset-hierarchical-2008] [guimera-missing-2009]. In this setting,
one typically omits any explicit model of the measurement process (hence
intrinsically assuming it to be uniform), and as a consequence of the
overall setup, only *relative probabilities* between individual missing
and spurious edges can be produced, instead of the full posterior
distribution considered in the last section. Since this limits the
overall network reconstruction, and does not yield confidence
intervals, it is a less powerful approach. Nevertheless, it is a popular
procedure, which can also be performed with graph-tool, as we describe
in the following.

We set up the classification task by dividing the edges/non-edges into two sets \(\boldsymbol A\) and \(\delta \boldsymbol A\), where the former corresponds to the observed network and the latter either to the missing or spurious edges. We may compute the posterior of \(\delta \boldsymbol A\) as [valles-catala-consistencies-2018]

up to a normalization constant ^{1}Note that the posterior of Eq. (2)
cannot be used to sample the reconstruction \(\delta \boldsymbol
G\), 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.. Although the
normalization constant is difficult to obtain in general (since we need
to perform a sum over all possible spurious/missing edges), the
numerator of Eq. (2) 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 \(\{\delta
\boldsymbol A_i\}\) via their likelihood ratios

which do not depend on the normalization constant.

The values \(P(\delta \boldsymbol A | \boldsymbol A, \boldsymbol b)\)
can be computed with
`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 \(\boldsymbol b\) by \(\{\boldsymbol b_l\}\) in the equations above).

```
g = gt.collection.data["football"]
missing_edges = [(101, 102), (17, 56)]
L = 10
state = gt.minimize_nested_blockmodel_dl(g)
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 100,000 sweeps
gt.mcmc_equilibrate(state, force_niter=10000, 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)))
```

```
likelihood-ratio for (101, 102): 0.0201412
likelihood-ratio for (17, 56): 0.979859
```

From which we can conclude that edge \((17, 56)\) is more likely than \((101, 102)\) to be a missing edge.

The prediction using the non-nested model can be performed in an entirely analogous fashion.

# References#

Tiago P. Peixoto, “Reconstructing networks with unknown and heterogeneous errors”, Phys. Rev. X 8 041011 (2018). DOI: 10.1103/PhysRevX.8.041011 [sci-hub, @tor], arXiv: 1806.07956

Tiago P. Peixoto, “Disentangling homophily, community structure and triadic closure in networks”, Phys. Rev. X 12, 011004 (2022), DOI: 10.1103/PhysRevX.12.011004 [sci-hub, @tor], arXiv: 2101.02510

Tiago P. Peixoto, “Latent Poisson models for networks with heterogeneous density”, Phys. Rev. E 102 012309 (2020) DOI: 10.1103/PhysRevE.102.012309 [sci-hub, @tor], arXiv: 2002.07803

Travis Martin, Brian Ball, M. E. J. Newman, “Structural inference for uncertain networks”, Phys. Rev. E 93, 012306 (2016). DOI: 10.1103/PhysRevE.93.012306 [sci-hub, @tor], arXiv: 1506.05490

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 [sci-hub, @tor]

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 [sci-hub, @tor]

Toni Vallès-Català, Tiago P. Peixoto, Roger Guimerà, Marta Sales-Pardo, “Consistencies and inconsistencies between model selection and link prediction in networks”, Phys. Rev. E 97 062316 (2018), DOI: 10.1103/PhysRevE.97.062316 [sci-hub, @tor], arXiv: 1705.07967

Roger Guimerà, Marta Sales-Pardo, and Luís A. Nunes Amaral, “Modularity from fluctuations in random graphs and complex networks”, Phys. Rev. E 70, 025101(R) (2004), DOI: 10.1103/PhysRevE.70.025101 [sci-hub, @tor]

Brian Hayes, “Connecting the dots. can the tools of graph theory and social-network studies unravel the next big plot?”, American Scientist, 94(5):400-404, 2006. http://www.jstor.org/stable/27858828

Robert E. Ulanowicz, and Donald L. DeAngelis. “Network analysis of trophic dynamics in south florida ecosystems.” US Geological Survey Program on the South Florida Ecosystem 114 (2005). https://fl.water.usgs.gov/PDF_files/ofr99_181_gerould.pdf#page=125

Kenneth E. Read, “Cultures of the Central Highlands, New Guinea”, Southwestern J. of Anthropology, 10(1):1-43 (1954). DOI: 10.1086/soutjanth.10.1.3629074 [sci-hub, @tor]

Footnotes