440 © 2024 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 Enhanced Multi-Objective Evolutionary Algorithm for Community Detection Using a Community Strength-Based Mutation Strategy Asal Jameel Khudhair 1* , Amenah Dahim Abbood 2 1,2 Department of Computer Science, College of Science, University of Baghdad, Baghdad, Iraq‎. *Corresponding Author. Received: 8 April 2025 Accepted: 17 June 2025 Published: 20 October 2025 doi.org/10.30526/38.4.4143 Abstract Community structures are fundamental in understanding the structure and functionality of complex networks. Different optimization algorithms, including both single-objective and multi- objective approaches, have been employed to address the challenge of community detection. Recently, multi-objective evolutionary algorithms (MOEAs) have attracted many researchers to identify communities in static networks. Many algorithms have been proposed to find a solution that achieves a trade-off between exploring new areas of the solution space and improving the quality of existing solutions. In this trade-off is crucial; whereas exploitation improves existing solutions, it may fail to find better solutions from insufficiently explored regions of the solution space. Therefore, mutation in evolutionary algorithms greatly impacts community detection within social networks. Conventional mutation methods usually tend to apply too much randomness, which results in convergence being less precise about finding a suitable optimum solution. This paper introduces a new mutation called community strength enhancement (CSE) to enhance the search efficiency of the Multi-Objective Evolutionary Algorithm with Decomposition (MOEA/D) and speed up the convergence of the suggested algorithm. Moreover, the proposed algorithm overcomes the limitations of traditional MOEA/D by accurately and effectively identifying communities across a wide range of social networks. The enhanced algorithm was evaluated on two groups of datasets (twenty synthetic and four real-world) using normalized mutual information (NMI) and modularity (Q) across five baseline models. Integrating the CSE mutation strategy led to significant improvements in performance, particularly under high mixing parameters and in large-scale networks, as evidenced by increased NMI and modularity scores. Keywords: Multi-objective optimization, Evolutionary algorithms, Community detection, Metaheuristic, Hybrid, Social network. 1.Introduction Networks represent a robust framework for modeling and analyzing various real-world systems, such as social network platforms, biological systems, transportation infrastructure, and https://orcid.org/0009-0006-3032-1376 mailto:asal.jameel2201m@sc.uobaghdad.edu.iq https://orcid.org/0000-0001-7873-3712 mailto:amenah.abbood@sc.uobaghdad.edu.iq https://orcid.org/0009-0006-3032-1376 mailto:asal.jameel2201m@sc.uobaghdad.edu.iq https://orcid.org/0000-0001-7873-3712 mailto:amenah.abbood@sc.uobaghdad.edu.iq https://orcid.org/0009-0006-3032-1376 mailto:asal.jameel2201m@sc.uobaghdad.edu.iq https://orcid.org/0000-0001-7873-3712 mailto:amenah.abbood@sc.uobaghdad.edu.iq https://orcid.org/0009-0006-3032-1376 mailto:asal.jameel2201m@sc.uobaghdad.edu.iq https://orcid.org/0000-0001-7873-3712 mailto:amenah.abbood@sc.uobaghdad.edu.iq https://orcid.org/0009-0006-3032-1376 mailto:asal.jameel2201m@sc.uobaghdad.edu.iq https://orcid.org/0000-0001-7873-3712 mailto:amenah.abbood@sc.uobaghdad.edu.iq https://orcid.org/0009-0006-3032-1376 mailto:asal.jameel2201m@sc.uobaghdad.edu.iq https://orcid.org/0000-0001-7873-3712 mailto:amenah.abbood@sc.uobaghdad.edu.iq IHJPAS. 2025,38(4) 441 information systems (1). These frameworks ‎effectively capture both the components of these systems and their interactions (2). When identifying communities, the number of connections within a community is denser than those outside the communities (3-5). Therefore, this study employs two objective functions in the MOEA/D algorithm: the first objective function aims to maximize the intra connections within a community and the second objective function aims to minimize the inter connections connecting a node to other communities. To build interconnected communities, increasing internal connections and reducing external connections is necessary (6, 7). Evolutionary optimization algorithms inspired by the principles of natural selection, due to the iterative and adaptive exploration of large solution spaces, can carry out this task according to (8). Among the core components of evolutionary algorithms, mutation operators play a crucial role in introducing diversity into the population and preventing premature convergence (9). Many social network analysis studies have aimed to capture the evolution of the community structure (11). For example, (11) unsigned and unweighted networks, (12) with the signed network, and (13) with the positive‎and negative weight networks. Despite the success of current evolutionary-based identification algorithms, they still exhibit a speed problem and face the challenge of assigning the weak node to the true community. Several researchers have attempted to improve the performance of MOEA/D algorithms by developing mutation strategies. A need for techniques that depend more on the structural properties of the network remains. The study in this context introduces the new mutation strategy Community Strength Enhancement (CSE), which focuses on intra-neighbor characteristics of nodes, thus helping the effectiveness of community detection. The contributions in this paper are listed below.  We state the problem as a multi-objective community detection algorithm based on the MOEA/D framework. Two conflicting objective functions are employed to solve the problem of single-objective functions, which most existing state-of-the-art methods adopt.  We introduce a new mutation operator based on the initialization process of neighbor nodes and their relationships to assign nodes to more suitable communities. The latest mutation strategy can improve the efficiency of the mutation process and make the algorithm converge faster.  We conducted a comparative evaluation by applying the traditional and the proposed algorithms across five models. The results revealed that integrating the proposed strategy consistently enhanced the performance of all models, achieving superior results on real-world networks compared to existing state-of-the-art methods. We organize the paper as follows. Section 2 introduces the problem of community detection; Section 4 presents the original framework of the MOEA/D algorithm and the proposed mutation operator to identify communities. Section 5 shows the results of the experiments on the proposed method and its comparison with other methods; Section 6 covers the paper's conclusion. 1.1. Preliminaries This section presents the basic concepts of community detection in complex networks. In complex networks, communities present a vital substructure, depicting a set of nodes with significantly greater internal connections than external connections with the rest of the network. The network structure represents the system as a graph (G). For a static, unweighted, and undirected network, , where is the set of nodes, is the set of edges connecting these nodes, is the number of nodes in the network | |, is a number of subgroups and IHJPAS. 2025,38(4) 442 is the number of all edges | |. An adjacency matrix ( ) can also represent the graph; let where is given by: { t r s n tw n no s n ot rw s You can use the adjacency matrix to calculate a community or a node's internal and external edges. The primary objective of community detection is to divide the network into subgroups so that each community demonstrates significantly stronger internal connections than external ones. In other words, nodes that belong to the same community are very densely connected, whereas the links between different communities are comparatively less dense. To determine whether a subset represents a valid community (Measuring Community Strength), we compute the number of internal edges and the number of external edges for this subset: ∑ , ∑ . For node represents the number of internal edges of a node within the same community and represents the number of edges connecting the node to nodes outside the community. If , the subset can be considered a well-defined community. 1.2. The Objective Function The purpose of partitioning a network into groups of vertices is to ensure that the edge density within each group exceeds the edge density between groups. For this purpose, we require an objective function that increases the number of connections within each cluster and another objective function that reduces the number of connections between the other clusters. Evaluate how well the generated communities capture the original structure and properties of the data. Additionally, any problem with a single objective function has a unique solution. At the same time, the Pareto front is the result of multiobjective optimization efforts to find exact solutions under many conflicting objectives simultaneously. This study presents the formulation for identifying communities as a task involving two optimization objectives. It was formulates the community detection problem as a maximization problem with two objectives (14). The first objective is the community score (CS), and the second is the community fitness (CF). ∑ | | ∑ ( | | ) (1) The parameter controls the size of the communities to increase the weight of the degree of the internal node within a community and community size| |. Thus, the CS calculated by the summation of a local score for each community Ck. ∑ ( ) (2) Where is a positive number that determines the size of communities. If is significant, the network will be split into small communities; otherwise, large communities will be the most prevalent. Where α is a positive number that determines the size of communities. If α is significant, the network will be split into small communities; otherwise, large communities will be the most prevalent. For example, in 2020, (15) and (16) incorporated these two objectives into their research as a multiobjective optimization model. It was formulated the community detection problem as a multi-objective minimization problem by (17). They considered modularity (Q) as two conflicting objectives that measure the degree of internal connectivity and interconnectivity. IHJPAS. 2025,38(4) 443 Therefore, they define two objectives that should be minimized as follows. The first measures the intra-connection ∑ (3) While the second measures the degree of inter-connections: ∑ * + (4) They have shown that optimizing these two objectives simultaneously can produce a wide range of possible community structures, with more or less emphasis on connections within and between communities. Since modularity is a weighted sum of two objectives, it is clear that the partition that maximizes modularity must be a member of the Pareto set. The two objectives are also adopted by other researchers. For example, in 2015, (18) and (19) minimize the kernel k-means (KKM) and the ratio cut (RC). ∑ | | (5) ∑ | | (6) Which is used additionally in recent literary research (19-22). It was proposed a local information-based multiobjective evolutionary algorithm (LMOEA) by (23). A similar framework as MOEA/D is adopted to simultaneously optimize the two conflicting objectives of the negative ratio association (NRA) and ratio cut (RC). In addition, (24, 25) used the same objectives. ∑ | | (7) It was proposed a new model that is based on the Hidden Markov Model (HMM- MODCD) consisting from fintra and finter with the Multi-Objective Evolutionary Algorithm and it proved effective in enhancing the performance of dynamic community detection by (7). ntr ∑ | | ∑ ( n n out ) (8) nt r ∑ | | ∑ out( ) n (9) 1.3 Evolutionary Algorithm with Decomposition for Multiple Objectives (MOEA/D) The MOEA/D is used for solving multi-objective optimization by maximizing a limited set of scalar optimization sub-problems to get uniformly spread Pareto solutions. MOEA / D algorithm achieves this objective by distributing resources using pre-specified weight vectors (w). The decomposition and selection of the weight vector will collaborate to promote diversity in the population (P), thus improving the quality of the solution. MOEA/D also studies the effect of constraints on mating coordination on the relationship between exploration and exploitation. MOEA/D ensures that solutions are exchanged synchronously through local interactions and safeguards the optimization process against being stopped prematurely. In this, the MOEA/D procedure presents a convenient optimization process where innovation flourishes in complexity (C) and methodological composition delivers solutions (26). Many studies have shown that MOEA / D-based algorithms show a very high level of success in solving MOP (27, 28). Because of this, researchers have applied it to the problem of detecting a community (CD), in which at least two target functions (f1, f2, . . . , fn) either need to be maximized or minimized. The algorithm below briefly describes the principal MOEA/D phases. IHJPAS. 2025,38(4) 444 Algorithm 1: General framework to MOEA/D Algorithm for Community Detection Input: • Number of objectives (Objective Dimension) • Population size (Pop Size) • Number of generations (Generations) • Mutation Probability (pm) • Crossover Probability (px) • Initial Ideal Point based on Fitness Model (minimization or maximization) Output: • Near Pareto Optimal Sets (Non-dominated solutions) • All solutions detected MOEA/D Loop For Generation Counter = 1 to Generations do Step 1: Update Reference Points (Ideal Point, IndivPoint); Step 2: Initialize Results and Near Pareto Optimal Set; Iteration Loop (ProblemCounter = 1 to PopSize) Step 3: Selection Operator – Choose two parent solutions (Parent1, Parent2); Step 4: Crossover Operator – Generate child solution from parents; Step 5: Mutation Operator – Apply mutation to the child solution; End loop Step 6: Decode Child Population – Convert child solutions to meaningful representations (Individual2ClusterDecoding); Step 7: Evaluate Fitness – Calculate fitness values for the child population (ComputeFitnessCollectionParallel); Step 8: Update Reference Points – Update IdealPoint and IndivPoint based on child population; Step 9: Update Population – Combine parent and child populations based on fitness (UpdateProblem); Step 10: Update Near Pareto Optimal Set – Add non-dominated solutions from child population (UpdateNearParetoOptimalSet); Step 11: Store Results – Save Near Pareto Optimal Set for current generation; End for End loop 1.4. Related work Various evolutionary algorithms and optimization frameworks have been used to address the problem of community detection in complex networks. Current mutation strategies may fail to adequately explore solution spaces or preserve meaningful community structures, which can negatively impact the quality and efficiency of the detection process. Researchers address this issue by integrating a local search strategy with the evolutionary algorithm, which can make evolutionary algorithms much better at finding communities. Given the importance of mutation, several studies have proposed improvements to mutation operators designed explicitly for community detection. It was presented a new formulation for the intra-neighbor community detection score and the intra-neighbor score, and a migration operator method with MOEA/D (29). It was suggested a new multi-‎objective generational genetic algorithm (MOGGA+) integrated with three types of mutation operators (30). In each execution, choose a random mutation from one of these. This algorithm optimized two objectives, the modularity and the ‎conductance measure. RAMESH and SRIVATSUN proposed a framework that ‎integrates the MOEA/D algorithm with a new mutation operator that uses community labels of cliques as IHJPAS. 2025,38(4) 445 genes rather than cliques (31). This algorithm optimizes Kernal K-mean (KKM) and Ratio Cut (RC) objectives. It was. suggested a multi-objective community identification method based on a pigeon-inspired optimization algorithm, MOPIO-Net, and this algorithm minimizes two objectives, RC and Negative Ratio Association (NRA). The algorithm was combined with the suggested boundary node variation strategy to enhance the detection effectiveness (25). Despite many approaches that have led to meaningful progress, there are still challenges, particularly in designing mutation strategies that can be deeply aligned with the structural characteristics of the networks. Most existing methods involve general or random mutation mechanisms, which are not conducive to effectively preserving meaningful community structures. Thus, more targeted strategies that consider the local topology of nodes are called for. Table 9 summarizes the difference between the state-of-the-art and our proposed CSE mutation strategies. 2. Materials and Methods In this section, we present the framework of our algorithm and introduce a proposed mutation that would improve its efficiency and effectiveness. This mutation encourages the diversity of efficient solutions that contribute positively to the community detection task across different kinds of networks. 2.1. Framework structure Figure 1. The framework of the proposed Algorithm In Figure 1, the graph is first transformed into an adjacency matrix to represent the relationships between nodes. The next step is the genetic representation. The initialization step in which an IHJPAS. 2025,38(4) 446 initial set of potential solutions is generated to ensure diversity of solutions and a broader exploration of the solution space. It will be attained and evaluated against the chosen objective function, which helps to point out the solution capable of achieving the most initial balance with objects. In the next stage, solutions are selected based on neighborhood relation- ships, and then genetic operations such as cross-over and mutation are performed to introduce new solutions and increase diversity within the population. The population is updated based on fitness evaluations performed over multiple generations, culminating in the final network partition, which can be visualized as a Pareto front of optimal solutions. 2.1.1. Chromosome Representation (chromosome encoding) The representation of the chromosome is crucial to the effectiveness of evolutionary algorithms (EA). The genotype encoding used here follows a locus-based adjacency representation (14). It is the most commonly used approach and has been proven to be effective in previous research. 2.1.2. Population Initialization (Decoding) In the first phase of population decoding, the genetic representation of each individual is converted into a community structure. The individual chromosomes represent the relationships between the nodes and should be used to assign the nodes to different clusters. 2.1.3. Uniform Crossover It generates offspring by exchanging genes between two parents. In this method, each gene in an offspring chromosome is determined with some given prespecified probability of recombination. This operation is repeated for all genes on the chromosome, where, for each position, a random number between 0 and 1 is generated. If it is less than or equal to Pc, the gene in the position copies its allele from Parent1. Otherwise, it copies its allele from Parent2. The genes thus selected are then joined in order of selection itself to form the offspring chromosome. 2.1.4. Enhanced Mutation Operator (Community Strength Enhancement (CSE)) The mutation operator in the traditional version of MOEA/D is employed to enhance the current optimal solution in a random manner. In this case, the exchange of node location between communities occurs randomly, meaning each node of the current solution is replaced by a random selected node from the neighboring nodes. The exchange occurs when the randomly generated value less than the mutation probability (pm). Randomly detecting nodes reduces the effectiveness of the search. To improve the efficiency of community evaluation solutions in networks, we propose to implement a Community Strength Enhancement (CSE) strategy that provides greater precision in assigning nodes to their actual communities. This strategy illustrates the interconnection of nodes, both within and among communities. The CSE strategy is calculated as follows: n our ∑ n our (10) n our , defined by: n our ∑ | | (11) refers to the computation of the reassignment rate of the current vulnerable node in the neighboring community , where n our represent the neighbor nodes of node in the community and represents the internal connections of each neighbor nodes of the vulnerable node with each community. The CSE reassignment strategy is partitioned into three separate scenarios, see Figure 2.  In the first scenario, if the community to which the vulnerable node is assigned has a lower IHJPAS. 2025,38(4) 447 reassignment rate than other communities in its vicinity, the node will be reassigned to the community with the highest reassignment rate.  In contrast, the second scenario occurs when multiple communities have a shared maximum occupancy rate across all surrounding communities of the vulnerable community. In this circumstance, we assign the node to one of these communities at random based on the probability of mutation (Pm).  In the third scenario, when a node has equal internal and external connections, it will be reassigned to the community with the highest reassignment rate. This is because the strategy looks at the neighborhood strength and the internal connections of neighborhood, providing a better understanding of the structural context. Thus, even if the internal and external degrees look seemingly equal, this additional information layer enables more accurate and context- aware community assignments. This study examines the distinction between canonical mutation operators and enhanced mutation operators. The key difference lies in the enhanced mutation strategy ability to potentially migrate a specific node to another suitable cluster according to the above three scenarios. In contrast, the canonical mutation selects and relocates a random node towards a randomly selected cluster. The procedure of the suggested mutation strategy is provided in l or t m 2. In or r to n nc t commun ty’s loc l s rc c p l ty n ncr s t qu l ty of the network community division, therefore, the (MOEA/D) can be applied to address the challenge of community detection as well as other optimization problems. In this work, we used the canonical and enhanced mutation strategy independently. Figure 2. Explain application of Community Strength Enhancement Mutation. IHJPAS. 2025,38(4) 448 Algorithm 2. The Proposed Community Strength Enhancement Mutation Input: Number of nodes in the network, degree of each node, mutation probability, individual (partition) Output: Child (partition) For Node i= 1 to N do For Node j = 1 to NumConnectedNodes (Node i) do Calculate intra-connection for nodes; End for End for For Node i = 1 to N do If NumConnectedNodes (Node i) > 0 and rand ≤ Pm then If k Node i in ≤ k Node i out then For ConnectedNodeCounter = 1 to NumConnectedNodes (Node i) do Calculate Vneighbor from Equation (10) for each community; Calculate the intra-connection for Vneighbor (11) for each community; End for End if temp = 0; NewCluster = 1; For kk = 1: k do Calculate CSE from Equation (10) for each community; If temp < CSE then temp = CSE; NewCluster = kk; Else Go to canonical mutation; End if End for End for End for IHJPAS. 2025,38(4) 449 Figure 3. Flowchart for the main steps of the Algorithm 2. 3. Results and Discussion In this section, we present and discuss the results that demonstrate the effectiveness of our improved MOEA/D algorithm compared to other state-of-the-art algorithms. We applied MOEA/D with canonical mutation and with the proposed mutation for these models (MOGA- NET, MODPSO, LMOEA, HMM-MODCD), all of which are mentioned in Section 3.1. Furthermore, Table 8 includes the equations used and the period to which they were applied. In each case, a table compares the different mutation strategies and describes the determinants and pathways used in each case. IHJPAS. 2025,38(4) 450 For fear comparison, we determined the population's size, or an excessive amount of potential solutions, to be 300. We additionally set each solution's size, or number of neighbors, to five. The mutation probability is 0.6, and the crossover probability is 0.8. We settled on 300 generations as the upper limit. The results were set to the mean of 10 simulated trials. The good non-dominated solution from each iteration saved with certain validity conditions into the Near Pareto Optimal Set (PS) archive. It was used to compute the average outcome for ten runs. We first illustrate the (MOEA/D) algorithm on 24 networks divided into two groups of networks (real networks and synthetic networks). 3.1 Dataset description. This section describes the dataset that was employed in this study. It consists of 24 datasets divided into two groups. The first group examined four real networks. Zachary's network was constructed by examining the friendship connections between 34 club members with 78 relationships within the network (32). The second network consists of 62 bottlenose dolphins from New Zealand and is called the Bottlenose Dolphins Network. This network contains 159 connections with two significant groups (33). Girvan and Newman (34) rated the game of American football (2000) as the third most famous network. The 115 teams that competed in the Football 2000 championship games make up this network. Teams were divided into 12 groups, each representing a different geographic location. The ultimate network consists of the compilation of Krebs' publications on American politics, authored by Krebs himself (35). The 105 books on politics that make up this network from the United States are available on Amazon.com. The second group consists of synthetic networks. Girvan and Newman (34) introduced a computer-generated benchmark that includes graphs of different levels of complexity. This benchmark was further developed by (36). The initial set of LFR benchmarks consists of a network with 128 nodes, built to accommodate four communities, each with 32 nodes. We evaluated the efficacy of the algorithms on networks produced between 0:05 and 0:5. The networks are referred to as LFR128. The next set of LFR benchmarks has been used to evaluate the four models using larger networks that closely match real-world networks. The benchmarks comprise 10 networks, each containing 1000 nodes. We refer to this set as LFR1000. According to (36), the size of the mixing parameter is between 0.5 and 0.05. 3.2. Experiment on Real Networks. This section employs four types of real networks to test our algorithm. We show the result and discuss it. Table 1 shows the results of applying the MOEA/D algorithm to the five models with canonical mutations. These results show how canonical mutations influence the algorithm's performance in different models and serve as a comparison point with the results of applying the proposed CSE strategy in Table 2. The results of the tables show that the CSE strategy improves the performance of the MOEA/D algorithm by reassigning nodes to a more appropriate community. Speed up the convergence of the algorithm to the optimum solution. Table 1. The maximum and average values of Normalized Mutual Information (NMI) and Modularity (Q) were evaluated for the five models. These models are tested on four real-world networks with canonical mutation. IHJPAS. 2025,38(4) 451 Table 1. The maximum and average values of Normalized Mutual Information (NMI) and Modularity (Q) were evaluated for the five models. These models are tested on four real-world networks with canonical mutation Dataset Measure MOCD MOGA-NET MODPSO LMOEA HMM-MODCD Zachary NMImax 0.8371 0.8371 0.8371 0.8371 0.8372 NMIav 0.8371 0.8371 0.8371 0.8371 0.8365 Qmax 0.4188 0.4188 0.4188 0.4188 0.3109 Qav 0.4188 0.4172 0.4188 0.4188 0.3079 Dolphin NMImax 0.9065 1 1 1 1 NMIav 0.8888 1 1 1 1 Qmax 0.5178 0.5052 0.5008 0.5246 0.3339 Qav 0.5032 0.4931 0.5008 0.4994 0.3339 Football 2000 NMImax 0.7571 0.6838 0.7598 0.7420 0.5493 NMIav 0.7315 0.6759 0.7292 0.7141 0.3487 Qmax 0.4727 0.4332 0.4713 0.4797 0.2799 Qav 0.4660 0.4231 0.4538 0.4573 0.2743 Krebs NMImax 0.6263 0.6285 0.6103 0.6921 0.5934 NMIav 0.6014 0.6174 0.6019 0.6264 0.5934 Qmax 0.5123 0.5104 0.5189 0.5247 0.3569 Qav 0.5118 0.5060 0.5135 0.5177 0.3569 Table 2. The maximum and average values of Normalized Mutual Information (NMI) and Modularity (Q) were evaluated for the five models. These models are tested on four real-world networks with community strength enhancement mutation. Dataset Measure MOCD MOGA-NET MODPSO LMOEA HMM-MODCD NMImax 1 1 1 1 0.8589 Zachary NMIav Qmax 0.8372 0.4188 1 0.4107 1 0.4198 1 0.4174 0.8365 0.3079 Qav 0.409 0.4033 0.4126 0.4174 0.3079 NMImax 1 1 1 1 1 Dolphin NMIav Qmax 1 0.5178 1 0.5144 1 0.5178 1 0.5163 1 0.3339 Qav 0.5032 0.507 0.5172 0.5163 0.3339 NMImax 0.9273 0.8599 0.9269 0.9309 0.7961 Football2000 NMIav Qmax 0.9273 0.6036 0.8559 0.5847 0.9177 0.6044 0.8946 0.6044 0.7961 0.3237 Qav 0.6036 0.5806 0.6044 0.5950 0.3237 NMImax 0.6772 0.6314 0.6299 0.6705 0.6313 Krebs NMIav Qmax 0.6174 0.5248 0.594 0.5249 0.6299 0.5178 0.6099 0.5253 0.5979 0.3539 Qav 0.5246 0.5211 0.5138 0.5145 0.3539 In the Zachary dataset, all nodes in the actual partition are strong nodes. Two nodes (3 and 10) connect equally within and between the communities. The NMI achieved by all models using traditional mutations was about the same (0.8371), except for the HMM-MODCD model with a slightly better result of 0.8372, indicating that very little diversity was achieved in solution exploration before application of heuristic mutations. Now, the results of the best mutated versions by heuristics are much better, all reaching NMI = 1, which means that the method improves the quality of the results concerning how well the technique achieves community IHJPAS. 2025,38(4) 452 detection. The HMM-MODCD with the traditional mutation method is not the best regarding heuristic mutation. It has now been improved only to 0.8589, possibly due to some inherent weaknesses in its functioning. In the dolphin dataset, all nodes in the actual partition of the dolphin dataset are strong nodes, except node (40), which has equal connections inside and between communities. All models achieved NMI = 1 for the Dolphin network except the MOCD model, which had 0.9065 with the traditional mutation. However, after applying our custom mutation, all models, including MOCD, reached NMI = 1, demonstrating their effectiveness in enhancing community detection precision. The football dataset has 15 weak nodes; some of them (12, 25, 51, 59, 60, 64,70, 81, 83, 98) have internal connections smaller than their external connections; on the other hand, nodes (22.93, 43, 91, 111) have internal connections of zero, but there are several external connections. In the football network, the results under traditional mutation produced notable variation, with the HMM-MODCD model having the lowest value of NMI at 0.5493, while MODPSO had the highest at 0.7598. These results show a lack of exploratory diversity in traditional mutation methods. However, after applying the proposed mutation, a significant improvement followed; LMOEA led with the highest value of 0.9309, while HMM-MODCD was 0.7961. This substantial improvement did not raise the NMI to one level, which reflects the complexity of network characteristics, the different responses of models to optimization processes, and the ultimate partitioning conditions that are sensitive and sometimes hard to attain. The Krebs dataset has 15 weak nodes. For these nodes (8, 19, 29, 51, 78, 86), the number of internal connections is smaller than the number of external connections; on the other hand, in nodes (30, 50), the internal connections of each node are equal to the number of external connections. Thus, the Krebs network was represented the highest by LMOEA with NMI = 0.6921 using a traditional mutation, while its lowest value was recorded at 0.5934 by HMM- MODCD. After its application, some models improved, whereby MOCD achieved the highest value of 0.6772. However, LMOEA dropped to 0.6705 and MODPSO had the least at 0.6299, further underscored by the variation in some models' response towards the optimization strategy. Figure 4. Zachary dataset, A- Correct partition and with CSE mutation, B- Canonical mutation. IHJPAS. 2025,38(4) 453 Figure 5. Dolphin dataset with and without Heuristic mutation. Figure 6. Football dataset, A- Correct partition, B- Canonical mutation, C- Heuristic mutation Figure 7. Krebs dataset (A) Canonical mutation (B) Heuristic mutation The visual representations (Figures 4, 5, 6 and 7) present precise comparisons of the performance of the method being considered in the Football and Krebs networks, respectively. The figures portrayed the correct partitions along with the results drawn from the application of the traditional mutation and the CSE strategy, highlighting their effects on quality in community detection. When NMI = 1, the CSE mutation perfectly fits the correct partition. IHJPAS. 2025,38(4) 454 Table 3. Real world dataset characteristics Dataset Nodes Edges Communities Zachary 34 78 2 Dolphin 62 159 2 Football 115 613 12 Krebs 105 440 3 3.3. Experiments on Synthetic Networks In this paper, we mainly perform experiments on the two groups of Synthetic Networks called the computer-generated benchmark network and the LFR benchmark network. We then displayed and discussed the result. Table 4 presents the Performance of the five models on synthetic LFR-128 networks without CSE mutation for maximum and average values of NMI n mo ul r ty Q, w c s s own to r u lly cr s s t r o t m x n p r m t r μ increases. Thus, it becomes more challenging to correctly recognize the existing community structures within the network as the mixing parameter increases. Throughout the variety of mo ls t st , un r ll μ l v ls, t s only t t low st m x n p r m t r v lu t t LMOEA outperforms HMM- MODCD in terms of NMI and Q; th v lu s 0.9498 t μ = 0.05. How v r, the relative ability of HMM-MODCD to other models is seen to decline significantly, particularly when the level of mixing is increased. It does not seem to perform well when the model has to be applied to very complex networks Table 5 presents comparison of the models with the suggested mutation strategy CSE. There is a significant improvement in the performance of models due to the incorporation of CSE for n rly ll t mo ls n l v ls o μ. Som mo ls, or x mpl , m nt n n NMI o 1.0 up to μ = 0.25, a much more visible enhancement over Table 4, whereas the average modularity values over the different runs are more coherent and higher than pre-update runs, showing stronger community quality. These provide evidence that the CSE strategy maintains the quality of the solution as the difficulty increases. Results of the models on larger-scale synthetic networks (LFR-1000) using canonical mutation are shown in Table 6. There is a notable decrease in NMI and modularity s n twork s z n μ increase, validating the scalability challenge faced by evolutionary models in community detection compared to the smaller networks. However, LMOEA maintains a relatively strong performance, compare to other models. The Q values drop s n c ntly w n μ xc s 0.3, showing more pain in preserving meaningful community partitions. Table 7 shows the impact of the CSE strategy on the same large-scale networks. A clear improvement is observed across most models, especially in NMI scores, where many models k p v lu s (clos to or qu l to 1.0) up to μ = 0.3. LMOEA, n p rt cul r, m nt ns ot high accuracy and modularity in challenging settings, thus confirming the fact that the strategy is scalable and robust. Moreover, the performance gap between the models is reduced, which means that CSE helps equalize their exploration-exploitation balance and make them work towards optimal solutions. IHJPAS. 2025,38(4) 455 Table 4. Performance metrics including maximum NMI, average NMI, maximum Q, and aver- age Q for synthetic networks were presented for all models, excluding CSE. NETWORK MEASURE MOCD MOGA-NET MODPSO LMOEA HMM-MODCD NMmax 1 1 1 1 1 LFR-128-0.05 Qmax NMlav 0.7012 1 0.7012 0.9874 0.7012 1 0.7012 0.9975 0.4131 1 Qav 0.7012 0.7012 0.6945 0.6999 0.4131 NMImax 0.9076 0.8868 0.9431 0.9498 0.9076 LFR-128-0.1 Qmax NMlav 0.6014 0.8837 0.5669 0.8478 0.618 0.9182 0.6307 0.8982 0.3662 0.8451 Qav 0.5837 0.5419 0.5987 0.5935 0.3528 NMImax 0.7862 0.729 0.873 0.8435 0.7565 LFR-128-0.15 Qmax NMlav 0.453 0.7587 0.4441 0.7104 0.4985 0.8125 0.5 0.7636 0.3037 0.5295 Qav 0.4431 0.4199 0.4778 0.4545 0.2887 NMImax 0.6502 0.6316 0.7139 0.6961 0.5356 LFR-128-0.2 Qmax NMlav 0.3634 0.6288 0.3281 0.5656 0.4012 0.6638 0.3827 0.6254 0.2929 0.4239 Qav 0.3521 0.3177 0.3658 0.3343 0.2657 NMImax 0.5629 0.462 0.5894 0.5682 0.4188 LFR-128-0.25 Qmax NMlav 0.2764 0.5239 0.2491 0,405 0.3019 0.5568 0.2875 0.523 0.25 0.354 Qav 0.2666 0.1965 0.2713 0.259 0.25 NMImax 0.5054 0.3653 0.5331 0.5189 0.3955 LFR-128-0.3 Qmax NMlav 0.2322 0.4641 0.1766 0.3428 0.2413 0.4558 0.2498 0.4448 0.25 0.3119 Qav 0.215 0.1398 0.2176 0.2068 0.25 NMImax 0.4514 0.087 0.4062 0.3949 0.29 LFR-128-0.35 Qmax NMlav 0.1846 0.3812 0.0229 0.0087 0.1949 0.3779 0.1852 0.3677 0.25 0.239 Qav 0.1759 0.0023 0.17010 0.1654 0.25 NMImax 0.3593 0.087 0.3444 0.4063 0.2525 LFR-128-0.4 Qmax NMlav 0.1636 0.3151 0.021 0.0087 0.1613 0.32 0.1613 0.3167 0.25 0.2065 Qav 0.1462 0.0021 0.1446 0.1432 0.25 NMImax 0.2878 0.03 0.3198 0.307 0.208 LFR-128-0.45 Qmax NMlav 0.1339 0.264 0.0015 0.02 0.1333 0.2344 0.1324 0.2415 0.25 0.171 Qav 0.1287 0.001 0.1274 0.1245 0.25 NMImax 0.2585 0.03 0.26 0.2418 0.2445 LFR-128-0.5 Qmax NMlav 0.1349 0.2214 0.0015 0.02 0.1262 0.2132 0.135 0.1891 0.25 0.161 Qav 0.1215 0.001 0.1165 0.1169 0.25 IHJPAS. 2025,38(4) 456 Table 5. Performance metrics including maximum NMI, average NMI, maximum Q, and aver- age Q for synthetic networks were presented for all models, including the CSE heuristic. NETWORK MEASURE MOCD MOGA-NET MODPSO LMOEA HMM-MODCD NMmax 1 1 1 1 1 LFR-128-0.05 Qmax NMlav 0.7012 1 0.7012 1 0.7012 1 0.7012 1 0.1431 1 Qav 0.7012 0.7012 0.7012 0.7012 0.1431 NMmax 1 1 1 1 1 LFR-128-0.1 Qmax NMlav 0.6533 1 0.6533 1 0.6533 1 0.6533 1 0.3892 1 Qav 0.6533 0.6533 0.6533 0.6533 0.3892 NMmax 1 1 1 1 0.8679 LFR-128-0.15 Qmax NMlav 0.5995 1 0.5995 1 0.5995 1 0.5996 0.9707 0.3447 0.8571 Qav 0.5995 0.5931 0.5995 0.583 0.3447 NMmax 1 1 1 0.9748 1 LFR-128-0.2 Qmax NMlav 0.5508 0.9666 0.5508 0.95 0.5508 1 0.5389 0.8622 0.3379 0.8622 Qav 0.5355 0.5292 0.5508 0.4726 0.3252 NMmax 1 0.9325 1 0.9247 0.8571 LFR-128-0.25 Qmax NMlav 0.5 0.922 0.4728 0.8974 0.5 1 0.4713 0.7601 0.3027 0.651 Qav 0.4669 0.4574 0.5 0.3851 0.2871 NMmax 0.9452 0.9247 1 0.7999 0.5911 LFR-128-0.3 Qmax NMlav 0.4149 0.8408 0.4247 0.8791 0.4502 1 0.3719 0.5704 0.2705 0.4947 Qav 0.3738 0.4046 0.4502 0.2567 0.2594 NMmax 0.9248 0.8682 1 0.5819 0.5002 LFR-128-0.35 Qmax NMlav 0.3212 0.6842 0.3556 0.7584 0.4023 0.9422 0.2554 0.4691 0.25 0.3238 Qav 0.2883 0.3129 0.382 0.2049 0.25 NMmax 0.5531 0.686 0.6667 0.436 0.3017 LFR-128-0.4 Qmax NMlav 0.2118 0.4323 0.2638 0.5665 0.2441 0.6085 0.1843 0.338 0.25 0.2319 Qav 0.1777 0.2227 0.2227 0.1537 0.25 NMmax 0.42 0.3 0.6667 0.3475 0.2375 LFR-128-0.45 Qmax NMlav 0.1615 0.2939 0.0015 0 0.1973 0.395 0.1531 0.2582 0.25 0.1734 Qav 0.1396 0 0.1703 0.1319 0.25 NMmax 0.2585 0.03 0.3868 0.2978 0.1636 LFR-128-0.5 Qmax NMlav 0.1707 0.2133 0.0015 0 0.1692 0.2609 0.1255 0.1891 0.25 0.13281 Qav 0.1327 0 0.1474 0.1169 0.25 IHJPAS. 2025,38(4) 457 Table 6. Performance metrics including maximum NMI, average NMI, maximum Q, and average Q for synthetic networks were presented for all models, with canonical mutation. NETWORK MEASURE MOCD MOGA-NET MODPSO LMOEA HMM-MODCD NMmax 0.9321 0.9259 0.931 0.9367 0.9369 LFR-1000-0.05 Qmax NMlav 0.7353 0.7353 0.7501 0.7501 0.7709 0.7709 0.7605 0.7605 0.3907 0.3907 Qav 0.9221 0.9147 0.9274 0.936 0.6261 NMmax 0.8624 0.8506 0.8726 0.8635 0.8411 LFR-1000-0.1 Qmax NMlav 0.6353 0.8577 0.5897 0.8402 0.64 0.8657 0.6237 0.8635 0.3288 0.8307 Qav 0.6257 0.5875 0.6277 0.6175 0.3288 NMmax 0.7915 0.7613 0.78 0.7709 0.7203 LFR-1000-0.15 Qmax NMlav 0.5 0.7736 0.4828 0.7509 0.4909 0.7771 0.4813 0.7672 0.2554 0.645 Qav 0.4846 0.4675 0.4726 0.4726 0.2497 NMmax 0.709 0.692 0.7033 0.7017 0.6221 LFR-1000-0.2 Qmax NMlav 0.3966 0.7018 0.384 0.6829 0.3831 0,7012 0.3865 0.7017 0.2186 0.5486 Qav 0.3909 0.3712 0.38 0.3865 0.214 NMmax 0.6314 0.6053 0.646 0.6316 0.4753 LFR-1000-0.25 Qmax NMlav 0.3293 0.6265 0.3046 0.5981 0.3209 0.6408 0.3181 0.6316 0.2056 0.39 Qav 0.3213 0.2928 0.3135 0.3181 0.2049 NMmax 0.5801 0.5738 0.593 0.5769 0.4332 LFR-1000-0.3 Qmax NMlav 0.2617 0.5747 0.2562 0.5397 0.2608 0.5839 0.2553 0.5638 0.2051 0.3577 Qav 0.2614 0.2418 0.2566 0.2529 0.2051 NMmax 0.5211 0.4945 0.5313 0.5151 0.3464 LFR-1000-0.35 Qmax NMlav 0.2182 0.5132 0.1997 0.4815 0.2055 0.519 0.206 0.5136 0.2072 0.3105 Qav 0.2122 0.1907 0.2018 0.2029 0.2072 NMmax 0.474 0.4336 0.464 0.467 0.3048 LFR-1000-0.4 Qmax NMlav 0.1716 0.4585 0.1581 0.3955 0.1714 0.4653 0.1673 0.467 0.2174 0.2344 Qav 0.1705 0.1389 0.1684 0.1665 0.2125 NMmax 0.4185 0.3772 0.4296 0.4285 0.2442 LFR-1000-0.45 Qmax NMlav 0.1475 0.4073 0.1233 0.3219 0.143 0.4147 0.1375 0.3973 0.2204 0.1852 Qav 0.1468 0.0982 0.1408 0.1466 0.2204 NMmax 0.4003 0.2858 0.3857 0.3703 0.2443 LFR-1000-0.5 Qmax NMlav 0.1305 0.3747 0.0709 0.2141 0.1281 0.3787 0.1285 0.3727 0.2287 0.1426 Qav 0.1285 0.0428 0.1235 0.1205 0.2287 IHJPAS. 2025,38(4) 458 Table 7. Performance metrics including maximum NMI, average NMI, maximum Q, and aver- age Q for synthetic networks were presented for all models, with the CSE heuristic. NETWORK MEASURE MOCD MOGA-NET MODPSO LMOEA HMM-MODCD NMmax 1 0.9839 1 1 0.9942 LFR-1000-0.05 Qmax NMlav 0.8602 0.9992 0.8942 0.9784 0.9101 1 0.9101 0.9977 0.4645 0.9934 Qav 0.8587 0.8896 0.9101 0.9098 0.4645 NMmax 1 0.9392 1 0.9857 0.9832 LFR-1000-0.1 Qmax NMlav 0.8602 0.9992 0.8602 0.938 0.8602 1 0.8542 0.9823 0.4382 0.9832 Qav 0.8602 0.8022 0.8602 0.8533 0.4382 NMmax 1 0.8953 1 0.9727 0.9878 LFR-1000-0.15 Qmax NMlav 0.8099 1 0.7234 0.8892 0.8098 1 0.7959 0.9727 0.4138 0.9878 Qav 0.8098 0.7098 0.8098 0.7959 0.4138 NMmax 1 0.8519 1 0.9652 0.9522 LFR-1000-0.2 Qmax NMlav 0.7624 1 0.6486 0.8448 0.7624 1 0.7419 0.9652 0.387 0.9522 Qav 0.7624 0.63337 0.7624 0.7419 0.387 NMmax 1 0.8364 1 0.9202 0.9359 LFR-1000-0.25 Qmax NMlav 0.7131 1 0.5768 0.8364 0.7131 1 0.687 0.9202 0.3618 0.9359 Qav 0.7131 0.5706 0.7131 0.687 0.3618 NMmax 0.992 0.8137 0.9843 0.9259 0.9145 LFR-1000-0.3 Qmax NMlav 0.6661 0.9901 0.5327 0.7969 0.6661 0.9917 0.6194 0.9119 0.3371 0.8716 Qav 0.6661 0.5258 0.6661 0.6031 0.3358 NMmax 1 0.7466 1 0.9802 0.4994 LFR-1000-0.35 Qmax NMlav 0.6154 0.9988 0.4582 0.7281 0.6153 0.9939 0.6089 0.9802 0.2812 0.4994 Qav 0.6153 0.4421 0.6153 0.6089 0.2812 NMmax 0.9699 0.699 0.9723 0.8246 0.292 LFR-1000-0.4 Qmax NMlav 0.5619 0.9661 0.414 0.6861 0.5614 0.9675 0.4919 0.8021 0.2548 0.1568 Qav 0.5614 0.4051 0.5614 0.4619 0.25483 NMmax 0.9962 0.6705 0.9955 0.6474 0.2809 LFR-1000-0.45 Qmax NMlav 0.5151 0.9933 0.348 0.6611 0.5152 0.9836 0.3498 0.6474 0.253 0.2809 Qav 0.5149 0.344 0.5142 0.3498 0.253 NMmax 0.9855 0.6294 0.9626 0.492 0.2206 LFR-1000-0.5 Qmax NMlav 0.467 0.95 0.3117 0.5891 0.4671 0.967 0.251 0.492 0.25 0.1108 Qav 0.4665 0.3007 0.4665 0.251 0.25 Table 8. A comprehensive overview of the applied model: its name, formula and when it is applied Name Model functions References used the model MOCD CS Eq1 and CF Eq2 (14, 15, 16) MOGA-Net Intra Eq3 and Inter Eq4 (28, 18) MODPSO KKM Eq5 and RC Eq6 (19, 22, 21) LMOEA RC 6 and NRA 7 (23, 24, 25) HMM-MODCD fintra 8 and finter 9 (7) IHJPAS. 2025,38(4) 459 Table 9. Comparative table summarizing the difference between state-of-the-art mutation strategies mentioned in Section l and our CSE proposed mutation strategy. Reference Name of mutation strategy The role of the proposed strategy to improve the evolutionary algorithm The limitation of the mutation method (11, 30) Migration mutation operator This method shows that a weak node joins the community to which most of its neighbors belong. 1. Focusing on binary relationships only may lead to ignoring network complexities and an imbalanced distribution of information. 2. Difficulty in handling highly interconnected networks, thereby reducing the accuracy of community detection. (31) Community Based mutation operator This method modifies a whole community. One community is selected at random and its cohesion is assessed. If it is weak, a small group is transferred to the highly cohesive neighboring community as identified by Weighted Roulette Selection. 1. Unexpected changes in community structure influence solution stability. 2. May lead to lower diversity and lower-quality outcomes. 3. Strengthening a community is not a guarantee of the best solution for all scenarios. (25) Mutation strategy This Mutation occurs where nodes at the boundary of different communities have connectivity with multiple communities, irrespective of the number of internal connections. They are assigned to the majority community among their neighbors to improve clustering accuracy in complex networks 1. Based on boundary node identification and mutation, it increases the complexity of the algorithm. 2. Reclassification of nodes can be done wrongly in cases where the conditions for mutation are not accurately defined. Our mutation strategy Community strength enhancement (CSE) Our strategy is to measure the proximity of each node to different communities within the network. This proximity is defined by assessing the structural relationships between nodes, such as common neighbors, and the strength of their connections. Therefore, each node gets signed to the community for which such relatedness is at its maximum, that is, optimally assigning communities, based on relational proximity. 1. The increase in the number of communities makes identification more difficult. 2. The complexity of the process comes from the comparison of proximity between communities. IHJPAS. 2025,38(4) 460 (a) LFR-128-maximum NMI (b) LFR-128-average NMI (c) LFR-1000-maximum NMI (d) LFR-1000-average NMI Figure 8. Overall performance of all models based on their maximum and average NMI. The results with the community strength enhancement (CSE) are shown by dashed lines, whereas the results without it are represented by solid lines. 4. Conclusion This work presents a new mutation operation called community strength-based enhancement (CSE) to be incorporated into the MOEA/D algorithm to improve its efficiency, stability, and convergence rate during evolution. This strategy uses information about community structure and explores neighboring node-internal and node-external links, guiding mutation operations with more structural awareness. Results show that the modified MOEA/D algorithm with CSE performs better regarding normalized mutual information (NMI) and modularity (Q) than the unmodified MOEA/D algorithm and other state-of-the-art community detection methods. These results justify the promise of enriching advanced multi-objective evolutionary optimization algorithms with the present strategy. 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