Statistical clustering of temporal networks through a dynamic stochastic block model

TL;DR: This work explores statistical properties and frequentist inference in a model that combines a stochastic block model for its static part with independent Markov chains for the evolution of the nodes groups through time and proposes an inference procedure based on a variational expectation–maximization algorithm.

Abstract: Statistical node clustering in discrete time dynamic networks is an emerging field that raises many challenges. Here, we explore statistical properties and frequentist inference in a model that combines a stochastic block model (SBM) for its static part with independent Markov chains for the evolution of the nodes groups through time. We model binary data as well as weighted dynamic random graphs (with discrete or continuous edges values). Our approach, motivated by the importance of controlling for label switching issues across the different time steps, focuses on detecting groups characterized by a stable within group connectivity behavior. We study identifiability of the model parameters , propose an inference procedure based on a variational expectation maximization algorithm as well as a model selection criterion to select for the number of groups. We carefully discuss our initialization strategy which plays an important role in the method and compare our procedure with existing ones on synthetic datasets. We also illustrate our approach on dynamic contact networks, one of encounters among high school students and two others on animal interactions. An implementation of the method is available as a R package called dynsbm.

Summary

1. Introduction

• Statistical network analysis has become a major field of research, with applications as diverse as sociology, ecology, biology, internet, etc.
• While static approaches have been developed as early as in the 60’s (mostly in the field of sociology), the literature concerning dynamic models is much more recent.
• An important part of the literature on static network analysis is dedicated to clustering methods, with both aims of taking into account the intrinsic heterogeneity of the data and summarizing this data through node classification.
• This kind of assumption has never been discussed in the literature.
• The authors are interested in statistical models for discrete time dynamic random graphs, with the aim of providing a node classification varying with time, while controlling for label switching issues across the different time steps.

2.2. Varying connectivity parameters vs varying group membership

• The authors give some intuition on why it is not possible to let both connectivity parameters and group membership vary through time without entering into label switching issues between time steps.
• Thus it is not always possible to label the groups so that between two successive time steps, estimation of the transition parameters would be constrained to have large diagonal elements.
• In the above toy example, this corresponds to the first interpretation rather than the second.
• Other choices could be made and the authors believe that this one is particularly suited to model social networks or contact data where the groups are defined as structures exhibiting a stable within group connectivity behavior and individuals may change groups through time (see Section 5 for applications on real datasets).

2.3. Parameters identifiability

• This property is not satisfactory since clustering in models that only satisfy a local identifiability of SBM part of the parameter prevents from obtaining a picture of the evolution of the groups across time.
• The authors stress that this example implies that dynamic affiliation SBM (or planted partition model) does not have identifiable parameters and groups may not be recovered consistently across time.
• This is an important point as previous authors have tried to recover groups from this type of synthetic datasets and evaluated their estimated classification in a non natural way.
• The authors prove below that these constraints, combined with the same conditions used for identifiability in the static case, are sufficient to ensure identifiability of the parametrization in their dynamic setup.
• In particular for the binary case, assuming that the matrix of Bernoulli parameters β has distinct rows is a generic constraint (meaning that it removes a subset of zero Lebesgue measure of the parameter set).

3.1. General description

• As usual with latent variables, the log-likelihood logPθ(Y) contains a sum over all possible latent configurations Z and thus may not be computed except for small values of N and T .
• A classical solution is to rely on expectation-maximisation (EM) algorithm (Dempster et al., 1977), an iterative procedure that finds local maxima of the log-likelihood.
• This distribution has not a factored form and thus may not be computed efficiently.
• The authors refer to the review by Matias and Robin (2014) for more details about VEM algorithm (in particular a presentation of EM viewed as a special instance of VEM) and its comparison to other estimation procedures in SBM.

3.2. Algorithm initialization

• All EM based procedures look for local maxima of their objective function and careful initialization is a key in their success.
• For static SBM, VEM procedures often rely on a k-means algorithm on the adjacency matrix to obtain an initial clustering of the individuals.
• In their context, the dynamic aspect of the data needs to be properly handled.
• As a result, their initial clustering of the individuals is constant across time (namely Zti does not depend on t).
• The authors initialization is performant in these cases.

4. Synthetic experiments

• The methods presented in this manuscript are implemented into a R package and available at http: //lbbe.univ-lyon1.fr/dynsbm.
• While the complexity of the estimation algorithm is O(TQ2N2),.

4.1. Clustering performances

• The authors explore the performances of their method for clustering the nodes across the different time steps.
• As for the Bernoulli parameters β, the authors explore 4 different cases (see Table 4.1) representing different difficulty levels, plus a specific example of affiliation for which they recall that parameters are not identifiable in the dynamic setting.
• For each combination of (π,β), the authors generate 100 datasets, estimate their parameters, cluster their nodes and report in Figure 3 boxplots of a global and of an averaged ARI value.
• The authors believe that this is due to the initialization of their procedure: with T = 10 time points, it is more likely that the groups membership differ from their initial value.
• The authors note that the authors do not discuss initialization and simply propose to start with a random partition of the nodes, which proves to be a bad strategy.

4.2. Model selection

• The authors generate 100 datasets under this model and estimate the number of groups relying on ICL criterion.
• The authors observe that the correct number of groups is recovered in 88% of the cases (left panel).
• Moreover, the right panel shows that when ICL selects only 3 groups, ARI of the classification with 4 groups is rather low (less than 80%).
• This shows that in those cases, classification with 4 groups is not the correct one, so that VEM algorithm seems responsible for bad results (optimum has not been reached) more than the penalization term.

5. Revealing social structure in dynamic contact networks

• Dynamic network analysis has recently emerged as an efficient method for revealing social structure and organization in humans and animals.
• Indeed, many studies are now beyond the analysis of static networks and take advantage of longitudinal data on the long term, for instance during days or years of observations, that allow for constructing dynamic social networks.
• In particular, contact networks built from field observations of association between animals or from sensors-based measurements, are now currently available in Ecology or Sociology.
• The authors show that their statistical approach is a suitable tool to analyze dynamic contact networks from the literature.

5.1. Encounters among high school students

• Describing face-to-face contacts in a population (in their case, a classroom) can play an important role in 1/ understanding if there is a peculiar non-random mixing of individuals that would be a sign for a social organization and 2/ predicting how infectious diseases can spread, by studying the crosslink between the contacts dynamics and the disease dynamics.
• Interaction times were aggregated by days to form Clustering dynamic random graphs via SBM 15 a sequence of 4 different networks.
• The method selects Q = 4 groups and the authors now present the results obtained with their model fitted with Q = 4 groups.
• The authors observe that groups 2 and 3 are composed by students that are likely to interact together .
• Group 4 displays a similar pattern of community structure, with much less interaction (intermediate value of β̂44) but also a significant level of interaction with group 2 .

5.2. Social interactions between animals

• Examination of the estimated parameters β̂ and γ̂ reveals that groups 2, 3 and 4 are clear communities (with different intra-group behaviors) that eventually correspond to those revealed by Shizuka et al.
• Clearly, the authors observe some stability across years with individuals staying in communities 2, 3 and 4 over time and that are joined by incoming birds .
• Again, the structure of the onagers social network remains persistent over time (see similar conclusions in Rubenstein et al., 2015) and their model is therefore particularly adapted and efficient in this case.
• The authors would like to thank Tianbao Yang for making his code available from his web page.

A. Counter example of identifiability when groups memberships and connectivity parameters vary freely

• Here, the authors exhibit an example where the parameters are non identifiable when both groups memberships and connectivity parameters may vary across time without any constraint.
• Id the size-two identity matrix and (βt, γt) = (β, γ) are chosen constant with t.
• Finally, the across group parameter is not modified through time and the authors set φ̃t12 = φ12.
• Thus the two parameters θ, θ̃ are not equal up to label switching while they produce the same distribution on the observations.

B. Non identifiability in affiliation case (planted partition)

• Identifying the whole parameters from a binary affiliation SBM is a difficult task, as may be seen for instance by the many different but always partial results obtained by Allman et al. (2011).
• In their Corollary 7, the authors establish that when group proportions are known, the parameters βin(:= βqq for all q) and βout(:= βql for all q 6= l) of a binary affiliation static SBM are identifiable.
• This may be seen for instance from the example constructed in Section A that remains valid in the affiliation case.
• While static affiliation often relies on an assumption of equal group proportions, there is no simple parallel situation for the transition matrix π in the dynamic case (the trivial assumption π = Empirical evidence for label switching between time steps in the affiliation setup is given in Section 4 from the Main Manuscript.

D. Estimation of γ and model selection: specific examples

• The M-step equations concerning γ differ depending on the specific choice of the parametric family {f(·, γ), γ ∈ Γ}. Remember that the resulting conditional distribution on the observations is a mixture between an element from this family and the Dirac mass at zero.
• The authors also provide expressions for ICL criterion in these different setups.
• The parameter θ reduces to (π,β) for which updating expressions at the M-step have already been given (see Proposition 2).
• These equations remain valid when considering a set of disjoint bins {Im}m instead of pointwise values {am}m.

E. Extension to varying number of nodes

• However in real data applications it may happen that some actors enter or leave the study during the analysis.
• Note that the whole chain Zi is not stationary anymore.
• As such, a node that would not be present at each time point contributes to the likelihood only through the part of the trajectory where it is present.
• Generalization of their VEM algorithm easily follows.

Figures

Figure 1. Dependency structures of the model. Top: general view corresponding to hidden Markov model (HMM) structure; Middle: details on latent structure organization corresponding to N different iid Markov chains Zi = (Zti )1≤t≤T across individuals; Bottom: details for fixed time point t corresponding to SBM structure.

Figure 4. Estimation of the number of groups via ICL criterion. Left panel shows the frequency of the selected number of groups. Right panel shows ARI of the classification obtained with 4 groups depending on the selected number of groups.

Figure 2. Connectivity parameters or group membership variation: a toy example.

Figure 3. Boxplots of global ARI (white, left) and averaged ARI (grey, right) in different setups. From left to right: the three panels correspond to π = πlow (panels A,D), πmedium (panels B,E) and πhigh (panels C,F), respectively. In each panel, from left to right: results corresponding to β = low−, low+,medium−,medium+ and affiliation case, respectively. First row: T = 5 time points, second row: T = 10.

Figure 7. Summary of the interaction parameters β̂ and γ̂ estimated by our model with Q = 4 groups on the dataset of sparrows (left panel, Shizuka et al., 2014) and Q = 3 groups on dataset of onagers (right panel, Rubenstein et al., 2015), respectively. Same principle as in Figure 5.

Table 1. Bernoulli parameter values in 4 different cases, plus an affiliation example.

Figure 8. Alluvial plot showing the dynamics of the group membership estimated by our model on the datasets of interactions between 69 sparrows (left, Shizuka et al. (2014)) and 23 onagers (right, Rubenstein et al. (2015)) respectively. Same principle as in Figure 6 (with Si− k,Mi− k indicating group k at Season or Month i). A fake group (group 0) gathers absent animals at a specific time step and fuzzy fluxes represent arrival/departure to/from a group from/to group 0.

Figure 6. Alluvial plot showing the dynamics of the group membership estimated by our model on the dataset of interactions in the ’PC’ class (Fournet and Barrat, 2014). Each line is a flux that represents the move of one or more students from a group to another group (Di − k indicates group k for day i). The thickness of the lines is proportional to the number of students and the total height represents the 27 students.

Figure 5. Summary of the interaction parameters β̂ and γ̂ estimated by our model with Q = 4 grous on the dataset of interactions in the ’PC’ class (Fournet and Barrat, 2014). In each cell (q, l) with 1 ≤ q ≤ l ≤ 4, there are T = 4 barplots corresponding to the 4 measurements (Tuesday to Friday). Each barplot represents the distribution of the parameter γtql for the three categories of interaction frequency (low, medium and high). The width of each barplot is proportional to the sparsity parameter βtql. We recall that when considering the diagonal cells (q, q), parameters do not depend on t anymore.

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Statistical clustering of temporal networks through a
dynamic stochastic block model
Catherine Matias, Vincent Miele
To cite this version:
Catherine Matias, Vincent Miele. Statistical clustering of temporal networks through a dynamic
stochastic block model. Journal of the Royal Statistical Society: Series B, Royal Statistical Society,
2017, 79 (4), pp.1119-1141. �10.1111/rssb.12200�. �hal-01167837v2�

Statistical clustering of temporal networks through a dy-
namic stochastic block model
Catherine Matias†
Sorbonne Universit
´
es, UPMC Univ Paris 06, Univ Paris Diderot, Sorbonne Paris Cit
´
e, CNRS, Labora-
toire de Probabilit
´
es et Mod
`
eles Al
´
eatoires (LPMA), 75005 Paris, France.
Vincent Miele
Universit
´
e de Lyon, F-69000 Lyon; Universit
´
e Lyon 1; CNRS, UMR5558,
Laboratoire de Biom
´
etrie et Biologie
´
Evolutive, F-69622 Villeurbanne, France.
Abstract. Statistical node clustering in discrete time dynamic networks is an emerging field that
raises many challenges. Here, we explore statistical properties and frequentist inference in a model
that combines a stochastic block model (SBM) for its static part with independent Markov chains for the
evolution of the nodes groups through time. We model binary data as well as weighted dynamic ran-
dom graphs (with discrete or continuous edges values). Our approach, motivated by the importance
of controlling for label switching issues across the different time steps, focuses on detecting groups
characterized by a stable within group connectivity behavior. We study identifiability of the model pa-
rameters, propose an inference procedure based on a variational expectation maximization algorithm
as well as a model selection criterion to select for the number of groups. We carefully discuss our
initialization strategy which plays an important role in the method and compare our procedure with
existing ones on synthetic datasets. We also illustrate our approach on dynamic contact networks, one
of encounters among high school students and two others on animal interactions. An implementation
of the method is available as a R package called dynsbm.
Keywords: contact network, dynamic random graph, graph clustering, stochastic block model, varia-
tional expectation maximization
1. Introduction
Statistical network analysis has become a major field of research, with applications as diverse as
sociology, ecology, biology, internet, etc. General references on statistical modeling of random
graphs include the recent book by Kolaczyk (2009) and the two reviews by Goldenberg et al.
(2010) and Snijders (2011). While static approaches have been developed as early as in the 60’s
(mostly in the field of sociology), the literature concerning dynamic models is much more recent.
Modeling discrete time dynamic networks is an emerging field that raises many challenges and we
refer to Holme (2015) for a most recent review.
An important part of the literature on static network analysis is dedicated to clustering meth-
ods, with both aims of taking into account the intrinsic heterogeneity of the data and summarizing
this data through node classification. Among clustering approaches, community detection meth-
ods form a smaller class of methods that aim at finding groups of highly connected nodes. Our
focus here is not only on community detection but more generally on node classification based on
connectivity behaviors, with a particular interest on model-based approaches (see e.g. Matias and
Robin, 2014). When considering a sequence of snapshots of a network at different time steps, these
static clustering approaches will give rise to classifications that are difficult to compare through
time and thus difficult to interpret. An important thing to note is that label switching between
two successive time steps may not be solved without an extra assumption e.g. that most of the
nodes do not change group across two different time steps. However to our knowledge, this kind of
assumption has never been discussed in the literature. In this work, we are interested in statistical
models for discrete time dynamic random graphs, with the aim of providing a node classification
varying with time, while controlling for label switching issues across the different time steps. Our
answer to this challenge will be to focus on the detection of groups characterized by a stable within
†Corresponding author.

2 C. Matias and V. Miele
group connectivity behavior. We believe that this is particularly suited for dynamic contact net-
works.
Stochastic block models (SBM) form a widely used class of statistical (and static) random
graphs models that provide a clustering of the nodes. SBM introduces latent (i.e. unobserved)
random variables on the nodes of the graph, taking values in a finite set. These latent variables
represent the nodes groups and interaction between two nodes is governed by these corresponding
groups. The model includes (but is not restricted to) the specific case of community detection,
where within groups connections have higher probability than across groups ones. Combining SBM
with a Markov structure on the latent part of the process (the nodes classification) is a natural way
of ensuring a smooth evolution of the groups across time and has already been considered in the
literature. In Yang et al. (2011), the authors consider undirected, either binary or finitely valued,
discrete time dynamic random graphs. The static aspect of the data is handled through SBM,
while its dynamic aspect is as follows. For each node, its group membership forms a Markov chain,
independent of the values of the other nodes memberships. There, only the group membership is
allowed to vary across time while connectivity parameters among groups stay constant through
time. The authors propose a method to infer these parameters (either online or offline), based
on a combination of Gibbs sampling and simulated annealing. For binary random graphs, Xu
and Hero (2014) propose to introduce a state-space model through time on (the logit transform
of) the probability of connection between groups. Contrarily to the previous work, both group
membership and connectivity parameters across groups may vary through time. As such, we will
see below that this model has a strong identifiability problem. Their (online) iterative estimation
procedure is based on alternating two steps: a label-switching method to explore the space of
node groups configuration, and the (extended) Kalman filter that optimizes the likelihood when
the groups memberships are known. Note that neither Yang et al. (2011) nor Xu and Hero (2014)
propose to infer the number of clusters. Bayesian variants of these dynamic SB models may be
found for instance in Ishiguro et al. (2010); Herlau et al. (2013).
Surprisingly, we noticed that the above mentioned methods were evaluated on synthetic datasets
in terms of averaged value over the time steps of a clustering quality index computed at fixed time
step. Naturally, those indexes do not penalize for label switching and two classifications that are
identical up to a permutation have the highest quality index value. Computing an index for each
time step, the label switching issue between different time steps disappears and the classification
task becomes easier. Indeed, such criteria do not control for a smoothed recovery of groups along
different time points. It should also be noted that the synthetic experiments from these works were
performed under the dynamic version of the binary affiliation SBM, which has non identifiable
parameters. The affiliation SBM, also known as planted partition model, corresponds to the case
where the connectivity parameter matrix has only two different values: a diagonal one that drives
within groups connections and an off-diagonal one for across groups connections. In particular, the
label switching issue between different time steps may not be easily removed in this particular case.
Other approaches for model-based clustering of dynamic random graphs do not rely directly on
SBM but rather on variants of SBM. We mention the random subgraph model (RSM) that combines
SBM with the a priori knowledge of a nodes partition (inducing subgraphs), by authorizing the
groups proportions to differ in the different subgraphs. A dynamic version of RSM that builds
upon the approach of Xu and Hero (2014) appears in Zreik et al. (2015). Detection of persistent
communities has been proposed in Liu et al. (2014) for directed and dynamic graphs of call counts
between individuals. Here the static underlying model is a time and degree-corrected SBM with
Poisson distribution on the call counts. Groups memberships are independent through time instead
of Markov, but smoothness in the classification is obtained by imposing that within groups expected
call volumes are constant through time. Inference is performed through a heuristic greedy search
in the space of groups memberships. Note that only real datasets and no synthetic experiments
have been explored in this latter work.
Another very popular statistical method for analyzing static networks is based on latent space
models. Each node is associated to a point in a latent space and probability of connection is higher
for nodes whose latent points are closer (Hoff et al., 2002). In Sarkar and Moore (2005), a dynamic
version of the latent space model is proposed, where the latent points follow a (continuous state

Clustering dynamic random graphs via SBM 3
space) Markov chain, with transition kernel given by a Gaussian perturbation on current position
with zero mean and small variance. Latent position inference is performed in two steps: a first
initial guess is obtained through multi dimensional scaling. Then, nonlinear optimization is used
to maximize the model likelihood. The work by Xu and Zheng (2009) is very similar, adding a
clustering step on the nodes. Finally, Heaukulani and Ghahramani (2013) rely on Monte Carlo
Markov Chain methods to perform a Bayesian inference in a more complicate setup where the
latent positions of the nodes are not independent.
Mixed membership models (Airoldi et al., 2008) are also explored in a dynamic context. The
work by Xing et al. (2010) relies on a state space model for the evolution of the parameters of the
priors of both the mixed membership vector of a node and the connectivity behavior. Inference is
carried out through a variational Bayes expectation maximisation (VBEM) algorithm (e.g. Jordan
et al., 1999).
This non exhaustive bibliography on model-based clustering methods for dynamic random
graphs shows both the importance and the huge interest in the topic.
In the present work, we explore statistical properties and frequentist inference in a model
that combines SBM for its static part with independent Markov chains for the evolution of the
nodes groups through time. Our approach aims at achieving both interpretability and statistical
accuracy. Our setup is very close to the ones of Yang et al. (2011); Xu and Hero (2014), the first
and main difference being that we allow for both groups memberships and connectivity parameters
to vary through time. By focusing on groups characterized by a stable within group connectivity
behavior, we are able to ensure parameter identifiability and thus valid statistical inference. Indeed,
while Yang et al. (2011) use the strong constraint of fixed connectivity parameters through time, Xu
and Hero (2014) entirely relax this assumption at the (not acknowledged) cost of a label switching
issue between time steps. Second, we model binary data as well as weighted random graphs, should
they be dense or sparse, with discrete or continuous edges. Third, we propose a model selection
criterion to choose the number of clusters. To simplify notation, we restrict our model to undirected
random graphs with no self-loops but easy generalizations would handle directed datasets and/or
including self-loops.
The manuscript is organized as follows. Section 2.1 describes the model and sets notation. Sec-
tion 2.2 gives intuition on the identifiability issues raised by authorizing both groups memberships
and connectivity parameters to freely vary with time. This was not pointed out by Xu and Hero
(2014) despite they worked in this context. The section motivates our focus on groups character-
ized by a stable within group connectivity behavior. Section 2.3 then establishes our identifiability
results. To our knowledge, it is the first dynamic random graph model where parameters identifia-
bility (up to label switching) is discussed and established. Then, Section 3 describes a variational
expectation maximization (VEM) procedure for inferring the model parameters and clustering the
nodes. The VEM procedure works with a fixed number of groups and an Integrated Classifica-
tion Likelihood (ICL, Biernacki et al., 2000) criterion is proposed for estimating the number of
groups. We also discuss initialization of the algorithm - an important but rarely discussed step,
in Section 3.2. Synthetic experiments are presented in Section 4. There, we discuss classification
performances without neglecting the label switching issue that may occur between time steps. In
Section 5, we illustrate our approach with the analysis of real-life contact networks: a dataset of
encounters among high school students and two other datasets of animal interactions. We believe
that our model is particularly suited to handle this type of data. We mention that the methods are
implemented into a R package available at http://lbbe.univ-lyon1.fr/dynsbm and will be soon
available on the CRAN. Supplementary Materials (available at the end of this article) complete
the main manuscript.
2. Setup and notation
2.1. Model description
We consider weighted interactions between N individuals recorded through time in a set of data
matrices Y = (Y
t
)
1≤t≤T
. Here T is the number of time points and for each value t ∈ {1, . . . , T },
the adjacency matrix Y
t
= (Y
t
ij
)
1≤i6=j≤N
contains real values measuring interactions between

4 C. Matias and V. Miele
individuals i, j ∈ {1, . . . , N }
2
. Without loss of generality, we consider undirected random graphs
without self-loops, so that Y
t
is a symmetric matrix with no diagonal elements.
We assume that the N individuals are split into Q latent (unobserved) groups that may vary
through time, as encoded by the random variables Z = (Z
t
i
)
1≤t≤T,1≤i≤N
with values in Q
NT
:=
{1, . . . , Q}
NT
. This process is modeled as follows. Across individuals, random variables (Z
i
)
1≤i≤N
are independent and identically distributed (iid). Now, for each individual i ∈ {1, . . . , N }, the
process Z
i
= (Z
t
i
)
1≤t≤T
is an irreducible, aperiodic stationary Markov chain with transition matrix
π = (π
qq
0
)
1≤q,q
0
≤Q
and initial stationary distribution α = (α
1
, . . . , α
Q
). When no confusion occurs,
we may alternatively consider Z
t
i
as a value in Q or as a random vector Z
t
i
= (Z
t
i1
, . . . , Z
t
iQ
) ∈
{0, 1}
Q
constrained to
P
q
Z
t
iq
= 1.
Given latent groups Z, the time varying random graphs Y = (Y
t
)
1≤t≤T
are independent, the
conditional distribution of each Y
t
depending only on Z
t
. Then, for fixed 1 ≤ t ≤ T , random graph
Y
t
follows a stochastic block model. In other words, for each time t, conditional on Z
t
, random
variables (Y
t
ij
)
1≤i<j≤N
are independent and the distribution of each Y
t
ij
only depends on Z
t
i
, Z
t
j
.
For now, we assume a very general parametric form for this distribution on R. Following Ambroise
and Matias (2012), in order to take into account possible sparse weighted graphs, we explicitly
introduce a Dirac mass at 0, denoted by δ
0
, as a component of this distribution. More precisely,
we assume
Y
t
ij
|{Z
t
iq
Z
t
jl
= 1} ∼ (1 − β
t
ql
)δ
0
(·) + β
t
ql
F (·, γ
t
ql
), (1)
where {F (·, γ), γ ∈ Γ} is a parametric family of distributions with no point mass at 0 and densities
(with respect to Lebesgue or counting measure) denoted by f(·, γ). This could be the Gaussian
family with unknown mean and variance, the truncated Poisson family on N \ {0} (leading to a
0-inflated or 0-deflated distribution on the edges of the graph), a finite space distribution on M
values (a case which comprises nonparametric approximations of continuous distributions through
discretization into a finite number of M bins), etc. Note that the binary case is encompassed
in this setup with F (·, γ) = δ
1
(·), namely the parametric family of laws is reduced to a single
point, the Dirac mass at 1 and conditional distribution of Y
t
ij
is simply a Bernoulli B(β
t
ql
). In
the following and by opposition to the ’binary case’, we will call ’weighted case’ any setup where
the set of distributions F is parametrized and not reduced to a single point. Here, the sparsity
parameters β
t
= (β
t
ql
)
1≤q,l≤Q
satisfy β
t
ql
∈ [0, 1], with β
t
≡ 1 corresponding to the particular case
of a complete weighted graph. As a result of considering undirected graphs, the parameters β
t
ql
, γ
t
ql
moreover satisfy β
t
ql
= β
t
lq
and γ
t
ql
= γ
t
lq
for all 1 ≤ q, l ≤ Q. Note that for the moment, SBM
parameters may be different across time points. We will go back to this point in the next sections.
The model is thus parameterized by
θ = (π, β, γ) = (π, {β
t
, γ
t
}
1≤t≤T
) = ({π
qq
0
}
1≤q,q
0
≤Q
, {β
t
ql
, γ
t
ql
}
1≤t≤T,1≤q≤l≤Q
) ∈ Θ,
and we let P
θ
denote the probability distribution on the whole space Q
N
×R
N
. We also let φ(·; β, γ)
denote the density of the distribution given by (1), namely
∀y ∈ R, φ(y; β, γ) = (1 − β)1{y = 0} + βf(y, γ)1{y 6= 0},
where 1{A} is the indicator function of set A. With some abuse of notation and when no confusion
occurs, we shorten φ(·; β
t
ql
, γ
t
ql
) to φ
t
ql
(·) or φ
t
ql
(·; θ). Directed acyclic graphs (DAGs) describing
the dependency structure of the variables in the model with different levels of detail are given in
Figure 1. Note that the model assumes that the individuals are present at any time in the dataset.
An extension that covers for the case where some nodes are not present at every time point is given
in Section E from the Supplementary Materials and used in analyzing the animal datasets from
Section 5.2.
2.2. Varying connectivity parameters vs varying group membership
In this section, we give some intuition on why it is not possible to let both connectivity parameters
and group membership vary through time without entering into label switching issues between
time steps. To this aim, let us consider the toy example from Figure 2.
This figure shows a graph between N = 12 nodes at two different time points t
1
, t
2
. Node 1 is
a hub (namely a highly connected node), nodes 2 to 6 form a community at time t
1
(they tend to

Frequently asked questions

Q1. What are the contributions in "Statistical clustering of temporal networks through a dynamic stochastic block model" ?

Their approach, motivated by the importance of controlling for label switching issues across the different time steps, focuses on detecting groups characterized by a stable within group connectivity behavior. The authors study identifiability of the model parameters, propose an inference procedure based on a variational expectation maximization algorithm as well as a model selection criterion to select for the number of groups. The authors carefully discuss their initialization strategy which plays an important role in the method and compare their procedure with existing ones on synthetic datasets. The authors also illustrate their approach on dynamic contact networks, one of encounters among high school students and two others on animal interactions.