LocJury is a privacy-preserving framework proposed for protecting location data in Internet of Connected Vehicles (IoCV) environments. It employs identity-based networking and clustering mechanisms to reduce the risk of location disclosure while maintaining service utility [7]. Although LocJury is designed for vehicular location privacy rather than social network anonymization, it is considered here for comparison due to its use of clustering-based privacy enforcement and constraint-driven data protection, which conceptually relate to the objectives of the proposed KMFC-GWO framework. LocJury, an Identity-Based Networking (IBN) location-privacy preservation framework for Internet of Connected Vehicles (IoCV), focuses primarily on protecting spatiotemporal trajectories, trust-based decisions, and contextual location disclosure. Unlike LocJury, which operates in a vehicular mobility environment, KMFC-GWO focuses on anonymization of graph-structured social networks while jointly satisfying K-anonymity, L-diversity, and T-closeness. KMFC-GWO performs optimization on both node attributes and structural edges, providing a multi-constraint anonymization method that addresses risks not covered in LocJury.

The characteristics of users within social networks commonly involve both graph properties, which pertain to the connections between users, and matrix properties, relating to personal attributes. Graph-based binary features indicate whether connections exist between users, while matrix-based features are typically classified into three categories [6]:

Primary Identifying Attributes: This category encompasses direct identifiers such as full name, surname, and national identification numbers, which can directly expose a user’s identity. Consequently, these attributes must be eliminated prior to any additional processing.

The majority of anonymization techniques rely on clustering-based KA, wherein users are clustered into separate groups, each containing at least K members [8]. Graph-based anonymization methods typically involve initial modifications to the network data, followed by propagation of the altered graph. Edge randomization techniques introduce random changes to the graph-based network structure by randomly adding/deleting edges or swapping pairs of edges. This randomization aids in safeguarding users against re-identification through probabilistic means. Graph structure modifications can occur random perturbation or with the goal of satisfying some predetermined constraints such as KA, LD, and TC [9].

Random unconstrained perturbation involves fundamental techniques that alter graph structures by randomly adding or removing edges. Casas-Roma [10] introduced a spectral approach to safeguard critical network edges while optimizing utility within specified privacy constraints. Although spectral methods enhance utility while minimizing information loss, they entail higher computational demands. Nguyen et al. [11] employed quadratic programming to create uncertain graphs, focusing on edge manipulation for anonymization. However, these techniques, despite their simplicity, overlook the diversity and distribution of sensitive attributes, making them vulnerable to attribute and similarity attacks. Moreover, they may not ensure the K-anonymity condition, as edge modification does not typically consider it.

Kumar and Kumar [12] proposed a K-degree technique based on the upper approximation method of rough sets, utilizing split-join operations to evaluate privacy preservation. They employed various metrics including variation of information, mutual information, and adjusted Rand index to assess utility. Kiabod et al. [13] introduced a time-efficient K-degree method for anonymizing network graphs without the need for rescanning the structure for different anonymity levels. This method constructs a connection tree by computing the degree sequence of the anonymized graph, ensuring it exceeds a specified threshold. While KA and the K-degree approach protect against identity attacks, they do not address threats such as attribute/link and similarity attacks.

Yazdanjue et al. [14] proposed a KA technique for social networks through integrating clustering with evolutionary algorithms. This technique initially clusters nodes in the social network based on similarities. Fitness evaluation follows, assessing cluster fitness considering intra-cluster similarity and inter-cluster dissimilarity. Employing an evolutionary process, clusters with low fitness undergo modification through genetic operators like mutation and crossover. Throughout this process, the algorithm ensures KA, maintaining that each cluster contains at least K indistinguishable individuals. Termination occurs upon finding a satisfactory k-anonymous solution or reaching a predefined stopping criterion.

Langari et al. [6] introduced a K-member Fuzzy Clustering and Firefly Algorithm (KFCFA) approach for privacy preservation in social networks by combining fuzzy clustering with the firefly algorithm. The method begins with fuzzy clustering to group nodes based on similarity, followed by the application of the firefly algorithm to optimize privacy parameters. Through an iterative optimization procedure, this algorithm aims to minimize information disclosure while maintaining network utility. Termination occurs upon reaching a satisfactory privacy-preserving solution.

Rajabzadeh et al. [15] proposed a graph-based modification technique for achieving KA by employing a genetic algorithm. The approach involves iteratively modifying the network’s structure to ensure that each generated subgraph satisfies the KA property. By employing a genetic algorithm, this method optimizes the required anonymity conditions and network utility. The algorithm terminates when a satisfactory level of KA is generated.

Clustering-based anonymization strategies commonly rely on K-anonymity as their fundamental privacy requirement, upon which additional diversity and distributional constraints can be enforced to mitigate attribute disclosure risks [4].

The existing methods for anonymizing social network data, while effective in addressing specific privacy concerns, exhibit notable limitations. Many clustering-based KA techniques, such as those employing random edge perturbations or K-degree strategies, fail to adequately address threats like attribute and similarity attacks. Methods like spectral approaches or quadratic programming-based uncertain graphs, though utility-focused, often neglect the diversity and distribution of sensitive attributes. Moreover, the high computational cost of these methods makes them impractical for large-scale networks. Evolutionary and metaheuristic-based approaches, such as genetic algorithms or firefly algorithms, offer improved optimization but tend to prioritize specific conditions, such as KA, at the expense of balanced utility and privacy across multiple dimensions like LD and TC. These limitations highlight a critical gap in developing comprehensive methods that achieve robust privacy protection while maintaining low information loss and ensuring balanced clustering.

Our proposed KMFC-GWO anonymization technique addresses the mentioned gaps by combining KMFC with GWO to form a hybrid method capable of meeting KA, LD, and TC conditions simultaneously. Unlike existing methods, KMFC-GWO incorporates a modified fuzzy c-means clustering algorithm to ensure well-balanced clusters, where each cluster contains at least K members, thereby guaranteeing the KA condition. Additionally, our method formulates privacy protection as an optimization problem, leveraging GWO to optimize LD and TC constraints. This dual-layer approach effectively mitigates identity, attribute, and similarity attacks while preserving the structural and attribute-based integrity of the data. Furthermore, KMFC-GWO demonstrates superior computational efficiency and scalability compared to existing metaheuristic approaches, making it a practical solution for large-scale social network datasets. By achieving a balanced trade-off between privacy and utility, KMFC-GWO represents a significant advancement over existing anonymization techniques.

As mentioned above, to satisfy the KA condition, we apply a KMFC technique on the FCM. Our KMFC method utilizes a customized version of the FCM algorithm to construct balanced clusters with at least K members, satisfying the KA condition. To achieve this purpose, first, we generate an initial clustering using FCM, and then, the generated clusters are balanced to satisfy the KA condition.

The FCM algorithm was initially introduced by Bezdek in 1981 [16]. Unlike traditional clustering techniques such as c-means and k-means, FCM assigns a degree of membership for each data point to every cluster. The objective of FCM is to minimize the total distance between the data points and the cluster centroids. Its primary aim is to partition N data points into C distinct clusters. Each data point can be characterized by two feature vectors: binary edges and personal attributes. More specifically, X={x1,x2,…xi,…xN}T is the input matrix of dimension N×F, where F=N+M is the number of clustering attributes, N is the number of all users, and M is the number of personal features. Also, s={s1,s2,…,sk,…,sC}T is the centroid of clusters (C×F). The objective of FCM can be formulated as follows [17]: (1)OFFCM=∑i=1N∑k=1CUikq×dxi,skwhere xi=[xi1,xi2,…,xif,…,xiF] represents sample i, sk=[sk1,sk2,…,skf,…,skF] is the center of cluster k, q (q > 1) is the exponential weight of FCM that adjusts the fuzziness degree, and Uik∈[0,1] is the membership of node i to cluster k which should fulfil the condition of Eq. (2). Also, dxi,sk measures the Euclidian distance between xi and sk, as formulated in Eq. (3). To optimize the objective function of the FCM algorithm described in Eq. (1), iterative partial derivations of OFFCM with respect to Uik and sk need to be computed as per Eqs. (4) and (5). (2)∑k=1CUik=1  ∀i=1,2,…,N (3)dxi,sk=(xi1−sk1)2+(xi2−sk2)2+…+(xiF−skF)2 (4)Uik=∑k′=1C(d(xi,sk)d(xi,sk′))21−q (5)sk=∑i=1N(Uikq×xi)∑i=1NUikq

The termination criterion for the FCM algorithm is determined by the condition Uik(t)−Uik(t−1)<ε, where t represents the current iteration count, and ε is a small positive value (we set ε = 0.001). Upon termination of the FCM algorithm, each data point (user) is allocated to the cluster with the highest fuzzy membership. Generally, the traditional KA-based clustering methods suffer from two drawbacks:

Number of clusters: Selecting the best value for the number of clusters C is a challenging issue.

Our proposed revision phase of the KMFC algorithm is provided in Algorithm 1. To handle the first problem, we consider the number of clusters in a such a way that minimizes the CAVG criterion [6], as formulated in Eq. (6). The CAVG metric (where CAVG ≥ 1) quantifies the degree of cluster balance, with values closer to 1 indicating higher balance among clusters. In the case of K-member clustering, CAVG tends to approximate 1, i.e., CAVG ≈ 1. (6)CAVG=NC×K

Since N and K are fixed parameters, we consider C ≈ N/K. To have somewhat a free level for clustering, we have set C = 0.9 × N/K in our simulations to have around 10% more samples on average in each cluster. Furthermore, to handle the second problem to satisfy the KA condition, we present a revision phase on the initial clustering results of the FCM algorithm.

The KMFC component ensures that clusters satisfy the minimum K-member requirement while preserving the inherent structural relationships within the graph. By incorporating both edge-based similarities and attribute-based distances, KMFC groups nodes with similar structural and semantic characteristics. The revision phase redistributes borderline nodes from overloaded clusters to underloaded ones, maintaining cluster balance and minimizing distortion. This ensures that the resulting clusters remain coherent and suitable for subsequent LD and TC optimization via the GWO module.

Once users are clustered into C balanced clusters (with at least K users in each cluster) to meet the KA requirement, GWO is used to further anonymize the social network, considering the requirements of LD and TC. This process is conducted simultaneously on both the graph matrix G and the attribute matrix A. At the graph level, anonymization involves randomly adding or removing edges between users, while at the attribute level, it entails randomly altering the values of user features.

Grey Wolf Optimizer (GWO) is a swarm intelligence–based metaheuristic algorithm designed to balance exploration and exploitation in continuous optimization problems. In this study, GWO is adopted as the optimization engine within the proposed framework due to its simple structure and effective search mechanism. The algorithm was originally introduced by Mirjalili et al. in 2014, inspired by the social hierarchy and hunting behavior of grey wolves [18]. Beyond its original formulation, GWO has been successfully applied to various real-world optimization problems, including adaptive fuzzy systems and healthcare-related applications, demonstrating its robustness and practical effectiveness [19]. Since its introduction, GWO has been widely employed in various optimization applications due to its balanced search mechanism and straightforward implementation [18]. As illustrated in Fig. 3, the algorithm commences by randomly initializing a population of grey wolves. In each iteration, the fitness of the current population is evaluated using a customized fitness function tailored to the particular application. Subsequently, the entire population undergoes adjustments through two phases: attacking prey (exploitation) and searching for prey (exploration), as described in the standard GWO procedure [18]. These main steps of GWO including random generation of initial population, objective function evaluation, and population updating, are described in the following:

Generation of initial population: As seen in Fig. 4, a feasible solution (SOL) can be represented as a matrix of N × (N + M) = N × F, where each row i represents the edge and attribute modifications in G and A matrices. In the case of edge modification, SOLi,j = 1 if the edge between nodes i and j is modified (adding a new edge between nodes i and j or removing a previously connection link between nodes i and j). Furthermore, for the attribute modification, SOLi, N + m = 1 if the value of the personal feature m is randomly changed for the data of user i.

• Objective function evaluation: There is a trade-off between the information loss (due to the modifications in G and A) and the anonymity level. The more changes in the G and A, the higher anonymity level, however by accepting more information loss. To minimize the information loss within the published social network data, we present an objective function to minimize the summation of the information loss of the modified G and A, denoted by IL_G and IL_A, while satisfying the constraints of KA, LD and TC. More specifically, the objective function of Grey Wolf Optimizer (OFGWO) is defined as follows: (7)OFGWO=minimize{(WG×ILG+WA×ILA)×2Penalty}subject to: (8)ILG=∑i=1N∑j=1NMDFi,jGN×N (9)ILA=∑i=1N∑m=1MMDFi,mAN×M (10)MDFi,jG={1ifedgei,jis modified0otherwise (11)MDFi,mA={1ifattributemofuseriis modified0otherwise (12)Penalty=∑k=1CPkLD+∑k=1CPkTCwhere PkLD = 1 and PkTC = 1 if the LD and TC conditions are not satisfied within cluster k, respectively. Furthermore, WG and WA are the relative weights of the information losses within the published graph and attributes, respectively. According to Eq. (12), Penalty is measured as the total number of unsatisfied LD and TC constraints. Considering the penalty function as 2^Penalty ensures that satisfying the constraints is of higher importance than the main objective function. It ensures obtaining the final solutions with no penalties.

Population updating: GWO models a hierarchical social structure consisting of four main levels, namely alpha, beta, delta, and omega wolves. The alpha wolf represents the best solution found so far, followed by beta and delta wolves, which guide the search process, while the remaining woves update their positions accordingly [17]. The alpha assumes the leadership role, with commands to be followed by all other wolves. The beta acts as an advisor to the alpha, supporting the alpha’s directives and offering feedback. The delta complies with the alpha and beta but holds dominance over the rest of the pack (omegas). During each iteration of the algorithm, the fitness function assesses the quality of all gray wolves, which are then sorted based on their fitness values, ranging from best to worst. The top three solutions are assigned as alpha (X_α), beta (X_β), and delta (X_δ), respectively. In each iteration t, the position of each gray wolf s is updated according to the following process:(13)Xst+1=Xsα+Xsβ+Xsδ3where Xsα, Xsβ, and Xsδ are respectively the factors of the encircling prey of the grey wolf s according to X_α, X_β, and X_δ, respectively, which can be calculated as follows: (14)Xsα=Xα−Asα.|Csα.Xα−Xst)| (15)Xsβ=Xβ−Asβ.|Csβ.Xβ−Xst|

(16)Xsδ=Xδ−Asδ.|Csδ.Xδ−Xst| where A = 2a.r1-a and C = 2.r2 are random vectors of the same dimension as X_s, where r₁ and r₂ are uniformly generated random vectors with elements within [0, 1], and α is decreased from 2 to 0 during the execution of the GWO.

After reaching the maximum iterations of GWO, the global best solution is considered as the final anonymized graph-based social network, which is ready to be sent to the data miners for further processing.

The computational complexity of the proposed KMFC-GWO framework is derived as follows: KMFC Phase: O(N × C × I_fcm) GWO Phase O(Pop × Iter × F) where F = N + M represents the dimensionality of the solution space. Total Complexity: O(NCI_fcm + Pop × Iter × (N + M)) This combined complexity shows that the framework scales linearly with the size of the network and the attribute dimensions.

The KMFC-GWO algorithm was effectively implemented using MATLAB R2022b on a Windows 10 PC equipped with a 2.6 GHz Core i7 CPU and 16 GB of RAM. Since GWO is a self-adjustable algorithm, we should only set the population size and iteration count of the algorithm to ensure enough function evaluation leading to a proper convergence in terms of OFGWO. In our simulations, we set the population size of GWO as 50 and maximum iterations as 200. The both weights W_G and W_A were set to 1 (W_G = W_A = 1), ensuring that the same weights are specified for the edge and attribute modifications. However, these weights can have any other relative values, based on the requirements specified by the system manager.

The datasets considered in this study are subsets of three large-scale social networks (Twitter, Facebook, and YouTube). Each dataset is represented in the form of a graph structure where nodes denote users and edges represent their relationships (e.g., friendships, subscriptions, or follow links). Alongside the graph matrices, we used an attribute matrix that contains numerical and categorical features such as age, gender, or activity-related metadata. While many social media users post images and videos, the publicly available benchmark datasets primarily provide graph and attribute information, not raw multimedia content. We therefore focus on graph-based anonymization, which remains highly relevant because the structural and attribute-based information alone is sufficient to re-identify users or disclose sensitive relations, as shown in prior privacy attacks. Due to the extensive size of these networks in terms of user count and edges, a subset of each network was selected for the simulations, as outlined in Table 1. Each dataset includes an attribute matrix A, where each row represents a feature vector containing the personal attributes of a user. Additionally, there is an edge graph matrix G, depicting the connection edges between users. While many social media users also share images and videos, the benchmark datasets used in this study primarily provide graph structures and attribute information, not raw multimedia content. Therefore, our evaluation focuses on graph-based anonymization, which remains highly relevant because structural and attribute-based data alone have been shown to enable user re-identification, attribute disclosure, or sensitive relationship inference in prior works. The KMFC-GWO algorithm was then employed to anonymize each social network, considering anonymity parameters set to K = 6, L = 4, and T = 0.5.

The results of the KMFC algorithm for different datasets are reported in Table 2. As mentioned above, we considered the number of clusters in such a way that minimizes the CAVG criterion as much as possible, which means generating more balanced clusters. By performing the KMFC algorithm on Twitter, Facebook, and YouTube datasets, CAVG has been obtained as 1.099, 1.112, and 1.103, respectively. It shows a proper generation of balanced clusters using the KMFC algorithm. Another point is that as expected, the KA condition has been satisfied in all datasets, however, the conditions of LD and TC are still not fulfilled, which need further anonymization using the GWO algorithm.

After satisfying the KA condition using the proposed KMFC algorithm, GWO is applied for further anonymization to fulfill the LD and TC conditions. The convergence of the GWO algorithm for the Twitter dataset is provided in Fig. 5. The figure shows that the algorithm started from an initial population with the average and best objective function values of 0.0491 and 0.0458, and after an iterative-based optimization procedure with 200 iterations, the algorithm converged finally to the global best solution with OFGWO = 0.0374.

To justify the performance of the GWO algorithm against other metaheuristic algorithms, we have considered the same optimization problem to be solved using Genetic Algorithm (GA) [20], and Greylag Goose Optimization (GGO) [21]. A comparison of the objective function value of the GWO algorithm with GA, AO, and GGO, on the different datasets is provided in Table 3. The results demonstrate that the AO and GGO have a slightly better performance than the GA. However, the GWO outperforms all compared metaheuristics in all datasets. To better understand the superiority of the GWO compared to other metaheuristics, the gain (improvement rate) of the GWO against GA, AO, and GGO is illustrated in Fig. 6 for different datasets.

To justify the proposed KMFC-GWO algorithm in terms of the anonymity results and clustering performance, the results of the proposed method are compared with KA, 3-Layer T-closeness L-diversity K-anonymity (3L-TLK), and KFCFA. Comparison of the obtained results in terms of the information loss of the attribute matrix A (ILA), information loss of the graph G (ILG), and the clustering balance metric (CAVG) are reported in Tables 4–6, respectively. The results presented in Tables 4–6 illustrate the effectiveness of the proposed KMFC-GWO technique in comparison to K-anonymity (KA), 3L-TLK, and KFCFA across three datasets: Twitter, Facebook, and YouTube. These tables evaluate the methods based on information loss in the attribute and graph matrices, as well as clustering balance.

When it comes to minimizing information loss in the attribute matrix (Table 4), KMFC-GWO consistently outperforms the alternative techniques. For instance, in the Facebook dataset, KMFC-GWO achieves an information loss value (ILA) of 0.0151, which is significantly lower than those reported for 3L-TLK (0.0324) and KFCFA (0.0239). Similar trends are observed in the Twitter and YouTube datasets, where KMFC-GWO consistently delivers the best performance, reflecting its ability to better preserve data utility.

A similar advantage is observed in minimizing information loss in the graph matrix (Table 5). KMFC-GWO exhibits superior results by achieving the lowest ILG values across all datasets. For example, in the Twitter dataset, KMFC-GWO reports an ILG of 0.0114, outperforming 3L-TLK (0.0135) and KFCFA (0.0191). This pattern of reduced information loss highlights the robustness of KMFC-GWO in preserving the structural integrity of graph-based data.

In terms of clustering balance (Table 6), KMFC-GWO demonstrates exceptional performance, achieving significantly lower clustering balance metric (CAVG) values compared to its counterparts. For example, in the YouTube dataset, KMFC-GWO achieves a CAVG of 1.103, which is notably better than KFCFA (1.44) and 3L-TLK (3.40). This indicates that KMFC-GWO not only protects privacy effectively but also ensures well-balanced clusters, which is critical for maintaining data utility.

The results show the superiority of the KMFC-GWO algorithm against the compared methods. Since KA considers just the clustering of users into distinguished groups, it does not generate any information loss neither within the attribute matrix nor graph matrix. However, as it does not consider the LD and TC conditions, it cannot protect the published data against attribute/link and similarity attacks. However, the other techniques have some distortions in the attribute and graph matrices, among them, the KMFC-GWO algorithm obtained the least information loss. Another remark is that the KMFC-GWO outperforms by far all techniques in terms of CAVG, by generating balanced clusters using the KMFC algorithm in the first phase of the proposed algorithm. These results underscore the advantages of KMFC-GWO as a powerful privacy-preserving approach that excels in reducing information loss in both attribute and graph matrices while achieving superior clustering balance, making it a reliable and efficient solution for publishing anonymized social network data.

Although KMFC-GWO is compared primarily with GA, AO, and GGO in the experimental results, further justification is provided for not adopting Particle Swarm Optimization (PSO) and Ant Colony Optimization (ACO) (Table 7). These methods either lack the exploitation intensity required to satisfy LD/TC constraints or impose computational costs unsuitable for high-dimensional graph anonymization.

To further illustrate the motivation and effectiveness of KMFC-GWO, we provide two examples. First, consider a Facebook subgraph where a small community of users share sensitive attributes (e.g., medical conditions). Without anonymization, even if the users voluntarily posted these attributes, attackers could easily re-identify them through structural overlaps with other public networks. Second, when applying our algorithm, the anonymized graph retains its utility for data mining tasks while substantially lowering the risk of identity and attribute disclosure compared to 3L-TLK and KFCFA. This demonstrates that KMFC-GWO not only improves clustering balance and reduces information loss but also delivers practical protection against real-world privacy threats in social media environments.

To evaluate the robustness of the anonymized graphs, we simulate three widely-used attack models that represent realistic adversarial behavior in social networks: degree-based re-identification, neighborhood-based structural attacks, and attribute inference attacks. These attacks were selected because they are among the most commonly used adversarial strategies in the social network anonymization literature and target different aspects of user privacy.

Degree-based re-identification exploits the uniqueness of node degrees. Attackers match a node in the anonymized graph to its unique or rare degree signature in the original graph.

For each attack, we implement a baseline adversary that has partial background knowledge of the original graph. The attacker attempts to map anonymized nodes back to their true identities using structural or attribute-based fingerprints. Table 8 reports the attack success rate before and after anonymization. As shown, KMFC-GWO reduces adversarial success by 63%–85% across all attack categories. This improvement results from both the KMFC clustering process, which ensures attribute diversity, and the GWO optimization mechanism, which disrupts structural uniqueness while preserving overall utility.

The information-loss evaluation examines how the anonymization process affects both structural and attribute components of the graph. Instead of relying solely on global IL scores, we separately analyze edge-modification rates, attribute-distortion rates, and their contribution to the final IL measure. A detailed breakdown demonstrates that KMFC-GWO achieves lower IL compared to baseline anonymization techniques due to its joint optimization of LD and TC constraints. The results confirm that KMFC-GWO introduces the minimum structural perturbation necessary for satisfying privacy requirements. Table 9 reports the execution time of KMFC and GWO for the three evaluated datasets.

To further examine utility, we evaluate how anonymization impacts downstream data-mining tasks, including community detection and link prediction. The modularity and AUC scores show that KMFC-GWO preserves the major community structure of the graph while introducing minimal distortion to link-formation patterns. This confirms that the structural adjustments introduced by GWO do not compromise the interpretability of the anonymized network. Table 10 reports the preservation of key graph–topology metrics before and after anonymization.

To further evaluate the utility of the anonymized graphs, several structural metrics were measured before and after applying KMFC-GWO. Table 10 reports the preservation of key topological properties including clustering coefficient, path length, diameter, modularity, and link-prediction performance.

We also analyze the scalability of KMFC-GWO by reporting execution times for KMFC clustering, GWO optimization, and the overall anonymization process across three datasets. The runtime grows approximately linearly with the number of nodes, consistent with the complexity analysis presented in Section 3. The results demonstrate that KMFC-GWO is suitable for medium-scale social networks and can be extended to larger datasets with parallelization. Although KMFC-GWO is designed for static social networks, it can be extended to dynamic environments where nodes and edges evolve over time. Incremental KMFC techniques can update cluster membership efficiently without full recalculation, while GWO can be re-initialized with partial populations to optimize only affected regions of the network. These extensions enable efficient anonymization of evolving graphs, consistent with dynamic anonymization approaches in recent literature.