325 

© 2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), 

University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons 

Attribution 4.0 International License 

 
Improving Modularity Regularization Techniques for Estimating 

Structures in Complex Network 

 

Samaa F. Ibraheem
1*

 and Basad Al-Sarray
 2 

  
1
Department of Mathematics, College of Applied Science, University of Technology, Baghdad, Iraq. 

2
Department of Computer Science, College of Science, University of Baghdad, Baghdad, Iraq. 

*Corresponding Author.  

 

Received: 17 September 2023           Accepted: 19 February 2024                    Published: 20 July 2025 

doi.org/10.30526/38.3.3728 

 
Abstract 

Complex systems in the real world have networks differ significantly from random graphs 

and have non-trivial structures. In fact, they have a community structure that needs to be 

recognized and recovered. The stochastic block models (SBMs) are popular models for 

community detection in networks, where nodes are divided into groups based on their 

connectivity patterns. Maximum likelihood estimation is a common method for estimating 

the parameters of SBMs. In this paper, a model selection for stochastic block models is 

presented based on the optimization of the log-likelihood function to find the best number of 

communities K detected by the regularized convex modularity maximization method. This 

work deals with many assumptions on K because it is necessary to study the behavior of the 

network and the best optimum K is assumed to select the best partition over the AIC, BIC 

metric. The proposed model selection method is presented in an algorithm that is 

implemented for both real and synthetic networks. This method enables the detection of 

networks with small communities that are more likely to provide a better fit to the observed 

data. 

Keywords: Community detection, likelihood function, model selection, Bayesian 

Information Criterion, Stochastic Block Models. 

 

1. Introduction 

The stochastic block model (SBM) in its standard form is a popular tool for detecting 

communities in networks. It is based on the assumption that the nodes in a network can be 

divided into groups or blocks, so that the probability of an edge between two nodes depends 

only on which block they belong to. This enables the estimation of block membership and the 

detection of communities within a network (1). The SBM has been used to detect 

communities in social networks, biological networks and other types of networks (2) and (3). 

More specifically, it can be used to estimate the number of communities in a network as well 

as their size and connectivity patterns. In addition, it can be used to identify outliers or 

anomalous nodes that do not fit into any of the detected clusters (4).  

https://orcid.org/0000-0002-0836-5173
mailto:samaa.f.ibraheem@uotechnology.edu.iq
https://orcid.org/0000-0003-2419-0710
mailto:basad.a.alsarray@gmail.com
https://orcid.org/0000-0002-0836-5173
mailto:samaa.f.ibraheem@uotechnology.edu.iq
https://orcid.org/0000-0003-2419-0710
mailto:basad.a.alsarray@gmail.com
https://orcid.org/0000-0002-0836-5173
mailto:samaa.f.ibraheem@uotechnology.edu.iq
https://orcid.org/0000-0003-2419-0710
mailto:basad.a.alsarray@gmail.com
https://orcid.org/0000-0002-0836-5173
mailto:samaa.f.ibraheem@uotechnology.edu.iq
https://orcid.org/0000-0003-2419-0710
mailto:basad.a.alsarray@gmail.com
https://orcid.org/0000-0002-0836-5173
mailto:samaa.f.ibraheem@uotechnology.edu.iq
https://orcid.org/0000-0003-2419-0710
mailto:basad.a.alsarray@gmail.com
https://orcid.org/0000-0002-0836-5173
mailto:samaa.f.ibraheem@uotechnology.edu.iq
https://orcid.org/0000-0003-2419-0710
mailto:basad.a.alsarray@gmail.com


IHJPAS. 2025, 38(3) 

326 
 

The SBM model works by randomly assigning nodes to different blocks and then connecting 

nodes within each block with a certain probability, while connecting nodes between blocks 

with a different probability. In this way, networks can be created with different communities 

that are connected to different degrees (5, 6). Model selection is an important task in SBM 

analysis as it helps to select the most accurate model from a set of candidates (7). This 

includes selecting the number of groups or communities as well as the parameters that 

determine the edge probabilities within and between groups. There are several approaches to 

model selection for SBM, including likelihood-based methods (8, 9), information-theoretic 

methods (10), spectral clustering (11), and cross-validation (12). 

Likelihood-based methods are commonly used for model selection in SBM analysis. Here, 

the maximum likelihood estimation (MLE) method is used to estimate the parameters of the 

SBM to find the parameter values that maximize the likelihood function. MLE can also be 

used to compare different SBMs by comparing their likelihood values using different 

penalized likelihood information criteria, such as the Akaike’s Information Criterion (AIC), 

the Consistent AIC, the Bayesian Information Criterion (BIC) and the Adjusted BIC (13). 

Thus, AIC is an estimate of a constant plus the relative distance between the fitted likelihood 

function of the model and the unknown true likelihood function of the data, while BIC is an 

estimate of a function of the posterior probability of a model being true under a given 

Bayesian setup, such that a low AIC and BIC mean that the model is considered closer to the 

truth (14,15). However, optimization of the likelihood function can be practically performed 

by optimizing the profile likelihood criteria over all possible partitions for each K. 

Subsequently, the best model can be selected using AIC or BIC. 

Overall, likelihood-based methods provide a rigorous framework for model selection in SBM 

analysis and help researchers to select the best-fitting model based on statistical criteria. 

Moreover, model selection for SBM can be challenging due to the high-dimensional 

parameter space and non-convex optimization problem (16). Therefore, it is important to use 

multiple approaches and carefully evaluate their results before drawing conclusions about the 

underlying structure of the network.  

The stochastic block models (SBMs) are popular probabilistic models for community 

detection in networks. They provide a powerful and modern way to define and understand 

network structure (17). There are many types of SBMs, including symmetric SBMs and 

overlapping SBMs (18), but the focus is on SBMs in standard form and SBMs with corrected 

degree. To represent the standard SBM, an undirected and unweighted random graph G with 

n nodes is considered, where A = [aij]∈R
n×n

  is the adjacency matrix of G that characterizes 

the relation between each pair of vertices i and j. A is symmetric. To model A by SBM, 

assume that aij s are independent Bernoulli variables, that is 

       (   )     ∈ (   )                                                                                        (1) 

and the vertices in G are partitioned into K groups (communities) such that the vertex i 

belongs to the community         , …, K, according to the map C : {1, …, n} → {1, …, 

K} such that            . Let Z ∈ {1, …, K}
n
 be the vector of labels that members the 

vertices i′s to one of K communities, i.e.,     *          +, where       ( )    

        The membership label Z that defines community assignment is drawn independently 

from a multinomial distribution on K communities with probability 

  (       ) ∑   
 
     , that is    

i i d
multinomial (   )                                            (2) 

In SBM, the probability that vertices i and j are connected (  (       )       ) depends on 

their community membership; therefor,     is identical for any i and j that belong to the same 



IHJPAS. 2025, 38(3) 

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community. By this way, a symmetric K × K, matrix B is formed such that              

where   ∈      , and K ≪ n. This probability matrix is called a stochastic block matrix or 

connectivity matrix, so there are K (K + 1)/2 connectivity parameters to estimate in this 

model (19). In networks with community structures, the edge probabilities within a same 

block exceed the probabilities that vertices of different blocks are connected, i.e., the vertices 

within a same block are more likely to be connected, so in community detection, we seek for 

       , ∀k, q, r=1, 2, · · · , K, with qr. 

The degree-corrected stochastic block model is a probabilistic model for network data that 

extends the traditional stochastic block model (SBM) by incorporating node-specific degree 

information. In the degree corrected SBM, each node is assigned to one of K blocks, and the 

probability of an edge between two nodes depends on their block assignments as well as their 

individual degrees. So the probability matrix B, with size K × K, is formed as      

              where θi>0 is an additional parameter assigned to each node to control its 

expected degree (1). Moreover, the value aij is a Poisson-distributed random variable with 

mean      , i.e., 

             (          )                                                                                                      (3) 

In this paper, a model selection for stochastic block models is performed based on optimizing 

the log likelihood function to find the best number K of communities. The main aim is to 

select the model for communities that are detected using a convex optimization approach 

under degree corrected SBM, which is regularized convex modularity maximization method 

(RCMM) that is suggested in (20), and compare this result with others detected using various 

algorithms, which are convex modularity maximization CMM (4, 21), modularity 

maximization algorithm (22), Danon algorithm (23) and spectral clustering (11). The article 

is organized as follows: Section 2 introduces stochastic block models. Section 3 presents the 

objective, and some details on algorithmic settings are also given. Numerical results are 

presented in Section 4 that show the performance of the model selection algorithm on the 

RCMM method compared with other methods. Finally, concluding remarks are given in 

Section 5. 

 

2.Optimizing Likelihood Function for Model Selection and Estimation 

In community detection, the goal of optimization methods with stochastic block models is 

to estimate the parameters of the model, such as the number of communities and the 

probability of connections between nodes within and between communities, in order to get 

the optimal membership label vector Z. This is done by maximizing a likelihood function that 

measures how well the model fits the observed network data, therefore detecting the network 

community. 

In the stochastic block model SBM, the    ’s are independent Bernoulli variables with 

probability pij, which is the expectation of       That is 

   (   )       which performs a     symmetric matrix (connectivity matrix)      
 

Given the node labels (the vector Z), the likelihood of observing A with probability B under 

SBM is: 

 ( | )  ∏      
        (       )

(     )                                                                            (4) 

whereas the DCSBM, which limits the applicability of the SBM, has an expected value of the 

adjacency matrix element      equal to: 

  (   )                                                                                                                            (5) 



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328 
 

but a Poisson distribution with this mean is used to determine the actual number of edges 

between any two vertices. The likelihood function is defined as 

 ( |     )  ∏
(         )

   

    
      (          )  ∏

(
 

 
  

      )
     

(     ) 
    ( 

 

 
  
      )     (6) 

To find the optimal partition, the likelihood function is maximized over B, giving the profile 

likelihood criteria to be optimized over all possible partitions. Actually, the Bernoulli 

likelihood is replaced by the Poisson likelihood, to simplify derivations. So for these two 

models, a profile likelihood was driven from the log-likelihood function of each of them to 

give the following criteria (1): 

     ∑        
   

    
                                                                                                            (7) 

       ∑        
   

    
                                                                                                         (8) 

where     ∑       *          is the number of edges from community k to 

community l,    .    ∑    
 
    be the node degrees in community k. In fact     gives 

twice the number of edgeswithin community k, and    is the number of nodes of community 

k, also 
   

    
 and     give the maximum likelihood values  ̂   of the SBM and DCSBM model 

parameters respectively. 

 

3.Enhancement of CMM for Community Detection 

There are many methods for detecting communities, distinguishing between traditional 

methods and optimization- and model-based methods. The optimization algorithm tries to 

identify communities within the network (graph) based on the optimization of some quality 

functions. Depending on the representation of the network, the modularity function, denoted 

by Q, can be defined as follows: It is a function that measures the strength of the partition of 

a network into communities (subgraphs) considering the degree distribution, such that 

networks with a high modularity value have dense connections between nodes within a 

cluster but sparse connections between nodes of different clusters. The modularity function is 

given by:  

  
 

  
∑ .    

    

  
/  (     )                                                                                              (9) 

where ,   .=  the adjacency matrix and Kronecker delta ( (  ,  ) is a function that yield 0 or 

1 if vertices i of degree ,  .and j of degree ,  . are in the same community or not). So, the 

possible existing communities are gained by balancing between the actual density of edges 

and the expected density that would appear in the communities if the vertices were connected 

regardless of community structure. Many methods for community detection are used in this 

work for in comparison, these methods are summarized as follows: 

Newman Modularity Maximization: this is an agglomerative hierarchical clustering technique 

in which smaller communities of vertices are gradually merged into larger ones, leading to an 

increase in modularity (22). The algorithm works as follows: 

- Step 1: Assign each of the n vertices to a single cluster. 

- Step 2: Reduce the number of clusters by adding an edge to the set of unconnected 

vertices such that the resulting partition has a modularity Equation (9) greater than the 

value of the previous configuration. 

- Step 3: Choose the optimal fusion by calculating the variation ΔQ 

- Step 4: Repeat steps 2 and 3 until you have added all edges and obtained one cluster. 

Save the modularity score for each partition. 

- Step 5: Select the partition with the highest modularity score. 



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329 
 

3.1. Danon Method 

In this method, (23) proposed to normalize the modularity variation Q resulting from the 

union of two communities by the proportion of edges belonging to one of the two 

communities, since the Newman technique tends to form large communities at the expense of 

small ones. Thus, this approach leads to larger modularity optima, especially when the 

communities have very different sizes. 

3.2.Convex Modularity Maximization 

The CMM algorithm makes use of convex optimization, where it is based on a convex 

programming relaxation of the modularity optimization and works under block models (21). 

Thus, so this algorithm tends to maximize modularity by solving the following optimization 

problem: 

    ⟨        ⟩

                               ( )   
                                                                    (10) 

where X is the partitioning matrix for the network nodes. The CMM approach combines 

convex optimization techniques with modularity maximization and a weighted ℓ1-norm k-

medoids as follows. After obtaining the optimal solution to Equation (10),  ̂, by the 

alternating direction method of multipliers (ADMM), the communities is extracted from 

weighted  ̂ by weighted ℓ1-norm k-medoids, i.e by solving the optimization: 

                 
∑ ∑   ‖ ̂    ‖  ∈       

           ∈     ( ̂)       ∀ ∈ *     +
                                                                              (11) 

where  ̂    ̂,       ( ) and    are the centers, so the clustering is performed on the 

rows of  ̂ instead of  ̂, thus, this procedure is summarized as follows: 

- Step 1: Solve the optimization problem (10) for X using the alternate direction method of 

multipliers (22). 

- Step 2: Find the k-partition of n nodes by the k-medoids approach, weighted by node 

degrees. 

3.3. Regularized Convex Modularity Maximization 

The RCMM algorithm is considered a regularization of the CMM algorithm. Where the 

common neighbor’s similarity measurement is used to weight the ℓ1-norm of k-mediod 

clustering (20) instead by node degrees. Where the weight was given twice, once to the rows 

of the partition matrix X and secondly to the terms of the ℓ1-norm of k-medoides. The 

number     of common neighbors between two vertices i and j is defined as  

    ∑    
 
                                                                                                                      (12) 

So the weighted ℓ1-norm k-medoids clustering will be done for the matrix  ̂    ̂ and the 

partition yields by solving the optimization problem: 

                 
∑ ∑    ‖ ̂    ‖  ∈       

           ∈     ( ̂)  ∈ *     +
                                                                              (13) 

3.4. Spectral Clustering 

In general it is a method uses of information which is taken from the eigenvalues of special 

matrices like Affinity Matrix, Degree Matrix and Laplacian Matrix derived from the data set 

(graph). It takes the matrix of eigenvectors associated with the K largest eigenvalues of one 

of these special matrices and performs one of the distance-based clustering methods on it 

(25). So mathematically, it is summarized in the following steps: 

- Step 1: Take the leading eigenvectors ,           - ∈      of the adjacency A or its 

graph normalized Laplacian L(A) = D
−1/2

AD
−1/2

, where D = diag(di) is the diagonal 



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matrix of degrees and the eigenvectors are associated with the K largest eigenvalues of 

the chosen matrix. 

- Step 2: Normalize each rows of the matrix   ,           - to find    with elements 

   
      (∑    

 
 )   . 

- Step 3: Run any distance clustering method (e.g., K-means) on the rows of the resulted 

vectors    to partition them into desired number of clusters (K), where the output 

        and the node i is assigned to cluster r if the i'’th row of   is assigned to cluster 

  . 

 

4. Model Selection 

Model selection for SBMs involves choosing not only the marginal probabilities but also the 

number of communities K in order to determine the best  fitting model from a set of 

candidate models (26). This is done in two stages: In the first, the parameters for the 

likelihood function are estimated for many chosen K, while in the second, the model that best 

fits the observed data is selected, i.e.,. the model that has the minimum AIC or BIC value 

(27-29). 

In likelihood-based methods, the likelihood of the observed network is maximized among 

different SBM models with different numbers of communities and parameter values. The 

Akaike information criterion (AIC) and Bayesian information criterion (BIC) are commonly 

used to compare models with different numbers of parameters and select an appropriate 

model. Let r be the number of estimated parameters in the model 

               ( )                                                                                                          (14) 

and 

          ( )       ( )                                                                                                   (15) 

The model that has the lowest AIC and BIC values from a group of possible models for the 

data is the one that should be selected. In other words, the AIC or BIC can be used to 

compare different models with different numbers of parameters and select the one that best 

fits the data, where the lower the AIC or BIC value, the better the model fits the data (30). 

Therefore, two stages are needed to estimate the parameters , .and choose the optimum 

model. The first is to choose the best membership label Zbest, which represents the partition of 

the network, using any community detection algorithm for many values K<<n such that, 

                 ( | ). and find the likelihood for every partition using Equation (7) or 

(8). This probability indicates how likely it is that the observed data A matches the model 

specified by the optimized parameters B for the best label Z. In the second phase, the 

corresponding AIC or BIC values are determined, Equations (6) and (7) respectively. As a 

result, the model that best fits the data and has a lower AIC or BIC value is selected. This 

procedure can be summarized using the flowchart in Figure (1) with its Algorithm (1). 

Algorithm (1) can be used with any community detection algorithm of a network with n 

nodes for any chosen number of communities K to identify the model that is more likely to fit 

the observed network or discover K for a model that is more likely to do so. 

 



IHJPAS. 2025, 38(3) 

331 
 

 
Figure1: Flowchart for Model Selection. 

 

Algorithm 1.  Model selection algorithm for SBMs 

Step1: Input: Network adjacency matrix A of n by n elements. 

Step 2: For K=2 to N , where N<<n 

Find the label vector Z by using any community detection algorithm. 

Find its likelihood using Equation (7) or (8).  

Step3: Put the likelihood values in statistical model selection criteria AIC or BIC Equation 

(14) or (15). 

Step4: Pick up the model with smallest AIC or BIC. 

Step5:Output:The selected model with communities K and its associated likelihood for 

given Z. 

 

5.Empirical Example and Results 

Algorithm (1) was implemented on various real and synthetic networks using a dataset from 

the Stanford Network Analysis Platform (SNAP) network analysis library. Table (1) shows 

the total number of nodes and edges in each network analyzed. 

 

Table 1. Datasets for various networks from Stanford large network dataset collection 

No. Network Vertices No. Edges No. 

1 Dolphin 62 159 

2 Karate 34 78 

3 American Football College (AFC) 115 613 

6 Chesapeake Synthetic 39 170 

7 Delaunay Synthetic 1024 3056 

 

As an example, the experiment is performed with a dolphin network (62 nodes and 159 

edges), as shown in Table (1). First, the detection is performed at K=4 with many algorithms 

including RCMM, CMM, Danon method, Newman modularity maximization, and spectral 

clustering with the corresponding likelihood values shown in Figure (2). Then, as mentioned 



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earlier, the best label vector corresponding to the minimum likelihood value is chosen. All 

calculations are performed with a degree-corrected model, as it is more suitable for modeling 

networks. 

 

 
Log-likelihood=-1.6117e+03                                               Log-likelihood=-1.6118e+03 

 
Log-likelihood=-1.6218e+03                                              Log-likelihood=-1.6386e+03 

Figure 2. Dolphin network partition for K=4, using different community detection algorithms under degree 

corrected model with each log-likelihood values. 

 

For model selection, the above experiment is applied for many K and on each network with 

the same community detection algorithm (see Algorithm 1), selecting the model that has the 

lower likelihood criteria for model selection, and here BIC is chosen. The applied community 

detection algorithm is RCMM to test whether it is sufficient to detect the correct model 

(Figure 3). These results are compared with the results of other community detection 

methods, namely: Danon algorithm, CMM algorithm, modularity maximization algorithm 

and spectral clustering for many networks (see Table 2). 

In Table (2), the comparison is done for model selection between the suggested method 

RCMM and the other methods, including: Newman modularity maximization method, Danon 

method, CMM algorithm, and spectral clustering.  

  



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333 
 

  
a b 

  
c d 

Figure 3. Model selection for RCMM community detection method: a) Dolphin network, b) Karate c)AFC 

d) Delaunay Synthetic 

 

Table 2. Model selection for various networks with theirs minimum BIC and associated log-likelihood value 

using various community detection algorithms. 

Network 
RCMM Danon method 

K BIC Log-likelihood K BIC Log-likelihood 

Dolphin 4 3.3555×10
3 

-1.6117×10
3
 4 3.3757×10

3
 -1.6218×10

3
 

Karate 2 1.5177×10
3
 -7.3943×10

2
 4 1.5464×10

3
 -7.1679×10

2
 

American Football College 

(AFC) 
10 1.5450×10

4
 -7.3286e×10

3
 7 1.5662×10

4
 -7.6268×10

3
 

Delaunay Synthetic 19 8.1922×10
4
 -3.8976×10

4
 9 8.6799×10

4
 -4.2925×10

4
 

Chesapeake Synthetic 2 3.9148×10
3
 -1.9373×10

3
 4 4.0696×10

3
 -1.9762×10

3
 

Network 
CMM Newman Modularity Maximization 

K BIC Log-likelihood K BIC Log-likelihood 

Dolphin 4 3.3557×10
3
 -1.6118×10

3
 4 3.4093×10

3
 -1.6386×10

3
 

Karate 2 1.5177×10
3
 -7.3943×10

2
 4 1.5735×10

3
 -7.30324×10

2
 

American Football College 

(AFC) 
10 1.5450×10

4
 -7.3286×10

3
 7 1.5662×10

4
 -7.6268×10

3
 

Delaunay Synthetic 19 8.1885×10
4
 -3.8942×10

4
 7 8.9932×10

4
 -4.4668×10

4
 

Chesapeake Synthetic 3 3.9262×10
3
 -1.9265×10

3
 3 3.9294×10

3
 -1.9281×10

3
 

Network 
Spectral clustering 

K BIC Log-likelihood 

Dolphin 4 3.3940×10
3
 -1.6310 ×10

3
 

Karate 3 1.5119×10
3
 -7.2069×10

2
 

American Football College (AFC) 8 1.5686×10
4 -7.5821×10

3
 

Delaunay Synthetic 19 8.268×10
4 -3.9358×10

4 

Chesapeake Synthetic 4 3.9553×10
3 -1.9190×10

3 



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6. Conclusion 

A model selection for stochastic block models based on the optimization of the 

log_likelihood function is given for the communities detected by the RCMM method under 

degree-corrected SBM to find the best number K of communities that have the minimum 

BIC. The results are compared with the results of other different methods, including CMM 

under degree-corrected SBM, Danon method, Newman modularity maximization, and 

spectral clustering, and applied to both real and synthetic networks. The results show that 

model selection for many networks partitioned with RCMM using degree- corrected SBM is 

identical to many results obtained with other community detection methods, and that in most 

cases it is better because it has a lower BIC value, so it fits the observed data better. 

Moreover, RCMM is able to detect small communities as it analyzes the networks with a 

larger value of K. However, the goal of model selection is to find a model that provides both 

a good fit to the existing observations and reasonable predictions for future observations. 

 

Acknowledgment 

Our researcher extends his Sincere thanks to the editor and members of the preparatory 

committee of the Ibn AL-Haitham Journal of Pure and Applied Sciences. 

 

Conflict of Interest  

There are no conflicts of interest. 

 

Funding  

There is no funding for the article. 

 

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http://arxiv.org/abs/1605.03000
https://doi.org/10.1016/0047-259X(88)90137-6
https://doi.org/10.1109/ICACCS51430.2021.9441868
https://doi.org/10.24996/ijs.2020.61.10.27
https://doi.org/10.1371/journal.pone.0215296
https://doi.org/10.1109/FOCS.2015.47
https://doi.org/10.1063/5.0209446
https://doi.org/10.1214/17-AOS1595
https://doi.org/10.1103/PhysRevE.69.066133
https://doi.org/10.1088/1742-5468/2006/11/P11010
https://doi.org/10.1214/14-AOS1290
https://doi.org/10.1109/TKDE.2021.3104155
http://arxiv.org/abs/2101.03280