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TABLE 1: Summary of SBM-based community detection, where ”AD k” describes whether the approach can automatically
determine the number of communities, i.e., Yes or No.
Categories Approaches Sketches Overlapping AD k
Basic SBM (1983) [28] Propose a stochastic model for social network. No No
MMSB
MMSB (2008) [29]
Extend blockmodels for relational data to ones that capture
mixed membership latent relational structure.
Yes No
cMMSB (2016) [47]
Combine an individual Copula function with MMSB with
improved in capturing group interactions.
Yes No
MMDCB (2019) [49]
Propose a mixed membership degree-correct SBM and develop
an inference method of the posterior distribution with Markov
chain Monte Carlo (MCMC).
Yes No
DCSBM
DCSBM (2012) [50]
Introduce expected values to basic SBM to adapt multi-edges
and self-edges contained in social networks.
Yes No
sparseDCSBM (2017) [51]
Propose a spectral clustering algorithm with normalized adja-
cency matrix based on DCSBM.
No Yes
CMM (2018) [52]
Establish a convexified modularity maximization approach for
estimating the hidden communities based on DCSBM.
No No
NCV (2018) [53]
Provide a network cross-validation approach to determine the
hidden communities based on DCSBM.
No No
DynSBM
dMMSB (2009) [54]
Propose a state space MMSB which can track across time
dynamic evolution.
Yes No
DynamicSBM (2010) [55]
Propose a novel Bayesian approach for network tomographic
inference buliding on MMSB and apply in dynamic network.
No No
DSBM (2011) [56]
Capture the evolution of c ommunity by modeling the transi-
tion of community memberships for individual nodes.
No No
DBTDP (2014) [57]
Propose a dynamic stochastic block model with te mporal
Dirichlet process for hidden community.
Yes No
SBTM (2015) [58]
Provide a local search algorithm for the inference procedure of
time evolution.
No No
dDCSBM (2016) [59]
Propose a dynamic DCSBM to model and monitor dynamic
networks that undergo a significant structural change.
No Yes
DPSBM (2019) [60]
Establish a fully Bayesian generation model with the hetero-
geneity of node degree.
No Yes
SNR-DSBM/ER (2020) [61]
Focus on estimating the location of a single change point in
a dynamic stochastic block model and take a least squares
criterion function for evaluating each point in time.
No Yes
OSBM
OSBM (2011) [62]
Provide a global and local variational technique for discover-
ing community.
Yes No
K-LAFTER (2018) [63]
Present a small-variance asymptotics based SBM for overlap-
ping c ommunity det e ction.
Yes Yes
MNPAOCD (2020) [64]
Optimize th e inference process and expect parameters in pro-
ceeding.
Yes Yes
LSBM LMBP (2015) [18]
Combine heterogeneous distribution with SBM to link com-
munity detection.
Yes Yes
GNNSBM DGLRFM (2019) [65]
Design a GNN-based overlapping SBM framework and can be
adapted readily for othe r types of SBMs.
Yes Yes
ference. Chen et al. [52] present a convexified modularity
maximization approach for estimating hidden communities
under DCSBM. Chen et al. [53] provide a network cross-
validation (NCV) approach based on a block-wise node-pair
splitting to determine the hidden communities for DCSBM.
Dynamic SBMs. Different from the above methods, an-
alyzing dynamic networks based on SBM is also a relatively
active field. Yang et al. [56] suggest a dynamic stochastic
block model named DSBM, which progressively updates
the probabilistic model to find community in large dynamic
sparse network. Specifically, DSBM u ses the distribution of
model parameter instead of the most likely values for the
model paramet e rs in prediction, and provides an offline
inference and an online inference to estimate the param-
eters. DS BM assumes that nodes in a dynamic network
remain unchanged. Letting C
T
= {C
(1)
, C
(2)
, ..., C
(T )
} be
the collection of community assignments of all nodes over T
discrete time steps, and the likelihood function of the model
is as follows:
P (W
(T )
, C
(T )
|ω, π, A) =
T
Y
t=1
P (W
(t)
|C
(t)
, π)
T
Y
t=2
P (C
(t)
|C
(t−1)
, A)P (C
(1)
|ω),
(6)
where W
(t)
and C
(t)
denote the snapshot of a network and
the community assignments of nodes at a given time step t.
The time complexity of DSBM is O(nT + mT + k
2
T ), and
its generation process is illust rated in Appendix B [56],
Following DSBM, Tang et al. [57] introduce the Dirichlet
process to SBM, which can find the optimal number of
communities of evolution, and in turn, alleviate the problem
of fixed community number of dynamic social networks. Xu
et al. [66] propose a new approach named stochastic block
transition model (SBTM) that includes two hidden Markov
assumptions for dynamic network. Wu et al. [67] propose
a full Bayesian generation model, which incorporates the