Two-stage multi-objective evolutionary algorithm for overlapping community discovery

Description of the proposed algorithm

Figure 1 shows the optimization process for overlapping community discovery. First, a test network is given, and after the first stage of non-overlapping community discovery, a better non-overlapping community structure is obtained. Then, the obtained non-overlapping community structure is put through the second stage of overlapping community discovery, after which an accurate overlapping community structure can be obtained. Next, we will focus on the realization process of the two phases.

The first stage: non-overlapping community division

At this stage, to obtain higher-quality non-overlapping community structures, this article uses modularity and normalized mutual information as optimization objectives. We will introduce these two objectives in detail in the experimental section. Two objectives are optimized through a decomposition-based multi-objective optimization framework to obtain high-quality non-overlapping communities. For non-overlapping community discovery algorithms, it is mainly divided into an initialization part and an evolution part. The initialization part adopts the population initialization strategy based on the central node. In the evolution part, this article performs cross-mutation on individual genes and optimizes individuals by combining the boundary node occupancy allocation method (Cai, Zhou & Wang, 2023). Finally, by comparing all the offspring produced by parents, modules are selected from them. The individual with the highest degree is regarded as the optimal solution. The specific implementation is shown in Algorithm 1.

• (1) Initialization strategy based on central node

In the first stage of the population initialization strategy, this article uses an initialization strategy based on the central node to create initial individuals. First, in a real community, the node at the center of the community is likely to be more closely connected to other nodes. The node metric usually represents the number of connections between the node and other nodes. Therefore, the higher the node degree, it means that the node is in the network. With more connections, a node is more likely to become a community center node. Specific steps are as follows:

1. First, calculate the degree of each node in the network, and select the node with the largest degree as the initial central node.

2. Mark the initial central node as the central node of the initial community, and assign all nodes directly connected to the central node to the initial community to form the first initial community.

3. Among the remaining unassigned nodes, find the node with the largest degree as the center node of the next community.

4. For each newly selected central node, allocate its directly connected unallocated nodes to the community of the central node.

5. Repeat steps 3 and 4 until all nodes are assigned to the community of a certain central node.

Through this method, a well-constructed network community structure can be initially obtained. Each central node and its directly connected neighbor nodes constitute a community of the network. In addition, to improve the diversity of the initial population, only half of the individuals use the above-mentioned selection strategy of maximum node degree as the central node, and the remaining individuals are generated by randomly selecting nodes as the central node.

• (2) Multi-objective evolution strategy based on MOEA/D

To search efficiently and accurately to obtain high-quality community structures, this article uses a multi-objective evolutionary algorithm framework based on decomposition to optimize the two objectives of modularity and normalized mutual information. After using the two-way crossover mutation method of the genome matrix during the evolution process, reasonable communities can be searched more quickly and effectively, and the diversity of the population can be improved while avoiding falling into local optimality during the search process. In addition, the node correction strategy will be Section “Experiment” introduces it in detail.

Before performing the two-way crossover mutation operation, we convert the community structure into the form of an adjacency matrix. The intersection values of rows and columns in the adjacency matrix represent the relationship between two nodes. Among them, 1 indicates that there is a connection between the two nodes, 0 indicates that there is no connection between the two nodes, and −1 indicates that the two nodes are in the same community structure.

• 1) Two-way crossover mutation strategy

To form a new population, two individuals are randomly selected as parents and a crossover operation is performed to generate new individuals. The crossover mutation operation occurs directly in the individual’s genome, by randomly selecting different numbers of nodes in the parent generation and exchanging the genetic attribute values of the same nodes in different parents to generate new individuals. In addition, to prevent the generated individuals from falling into the local optimum, their genetic attributes are mutated with a certain probability. The mutation operation also randomly selects different numbers of nodes and inverts the genetic attributes of the nodes to generate new individuals and increase the population diversity. For example, as shown in Fig. 2, two individuals (a) and (b) are randomly selected for crossover operation. Assume that nodes 3 and 4 are randomly selected, and the genetic attribute values of nodes 3 and 4 in (b) are assigned to (a), the gene attribute values of the remaining nodes remain unchanged, and a new individual is generated (c). It can be found that (b) and (c) are the same, so mutation operations will continue to be performed on them with a certain probability to avoid duplication of the generated individuals. It is also assumed that nodes 3 and 4 are randomly selected for mutation operation, and the genetic attribute values of nodes 3 and 4 in (c) are inverted, and finally a new individual (d) is generated, thereby completing an evolutionary process.

• 2) Individual choice

Individual selection retains the optimal solution in a manner that eliminates poorer candidate solutions during the evolutionary iteration process. The Tchebycheff Approach method is used to calculate the weighted maximum deviation of modularity and normalized mutual information and retain the value with the smallest deviation to the neighborhood information to complete the individual update. After completing the evolution for the prescribed number of iterations, a group of communities with the same size and higher quality as the initial population can be obtained, making the initial population in the second stage closer to the real community.

The second stage: overlapping community discovery

The second stage is the process of finding overlapping nodes based on the non-overlapping communities in the first stage. On the decomposition-based multi-objective optimization framework MOEA/D, the population is optimized with the extended modularity EQ and the number of overlaps as optimization objectives. The initialization population in the second stage benefits from the results of the first stage. The central node and non-central node of each community are found in the divided non-overlapping communities as the population initialization in the second stage. Inspired by Tian, Yang & Zhang (2019), we used the fuzzy clustering idea to assign a fuzzy threshold to each node to control whether the node is overlapping. During the optimization process, we not only perform cross-mutation on the current individual to generate new individuals but also use the information feedback model (Wang & Tan, 2017) to guide the generation of subsequent individuals to make full use of the valuable information before iteration. The specific algorithm implementation is Algorithm 2.

• (1) Initialization strategy based on non-overlapping nodes

In the previous stage, this article divided network nodes into non-overlapping communities. To obtain better overlapping community division results at this stage, a good initialization population is essential. Therefore, in this stage of population initialization, this article randomly selects a node from each community in non-overlapping communities from the node set with a node degree greater than the average node degree as the central node of the population, and its remaining nodes are non-central. In this way, we get a better initial population of overlapping communities. In addition, to diversify the initial population and avoid a single initial population, this article adopts the above scheme for half of the initial population, and the remaining initial population randomly selects nodes from each community of non-overlapping communities as the initial central node.

• (2) Overlapping node discovery strategy based on fuzzy clustering strategy

Fuzzy clustering is a clustering method that permits a sample to be associated with multiple clusters, facilitating its integration into the process of identifying overlapping nodes. During this process, the sets of central and non-central nodes identified from the initial population are denoted as illustrated in Eq. (1).

(1) $$\begin{array}{rl}& CN=\{C{N}_1,C{N}_2,\dots ,C{N}_n\}\\ & NCN=\{NC{N}_1,NC{N}_2,\dots ,NC{N}_m\}\end{array}$$

Among them, the sum of n and m is the total number of nodes, CN represents the central node-set, and NCN represents the non-central node set. The node number position corresponding to the central node is 1, and the non-central node is 0.

To effectively identify overlapping nodes, it is necessary to calculate the membership matrix between the central node and the non-central node and the fuzzy threshold of each node. Calculating the distance between the central node and the non-central node according to Formula (2) (Bezdek, Ehrlich & Full, 1984) can obtain each element in the membership matrix.

(2) $$A_{ij}={\displaystyle \frac{1}{{{\sum }_{k=1}^{|CN|}{\displaystyle \frac{dis(NC{N}_i,C{N}_j)}{dis(NC{N}_i,C{N}_k)}}}^{\frac{2}{f-1}}}}$$

Among them, $A_{ij}$ represents the membership degree between $NC{N}_i$ and $C{N}_j$, $dis(NC{N}_i,C{N}_j)$ represents the distance between the two nodes, $i\in \left[1,n\right],j\in [1,m]$, and f is the parameter that controls fuzzy clustering. It is generally recommended to be 2. In the initial stage, each node will randomly generate a fuzzy threshold r. According to Formula (3), the appropriate fuzzy threshold ${r_i}^{\prime }$ can be found to divide overlapping nodes.

(3) $$\begin{array}{rl}& r=(r_1,r_2,\dots ,r_n),r_i\in [0,1]\\ & r^{\mathrm{\prime }}=min{A}_{il}+r_i\times \left(max{A}_{il}-min{A}_{il}\right)\end{array}$$where $r_i$ is the fuzzy threshold of the ith node, and $A_{il}$ represents all columns of the ith row of the membership matrix. By comparing the element values in the membership matrix with the fuzzy threshold, suitable overlapping nodes can be found. To better understand this process, a simple division example is given as shown in Fig. 3.

In the above legend, the central nodes are node 1 and node 6, represented by 1 in the vector, and the remaining nodes are non-central nodes, represented by 0; the fuzzy threshold r is randomly generated, and each threshold corresponds to a non-central node at the same position. central node. The membership matrix $A_{ij}$ and the fuzzy threshold ${r_i}^{\prime }$ are calculated according to Eqs. (2) and (3). By comparing the sizes of ${r_i}^{\prime }$ and $A_{il}$, the node will preferentially choose to join the community with a ${r_i}^{\prime }$ value less than or equal to $A_{ij}$. For example, ${r_2}^{\prime }=0.5417$, $A_{21}=0.7083$, $A_{22}=0.2917$, ${r_2}^{\prime }<A_{21}$, so node 2 will choose to join community 1, and all nodes will be compared in the same way. After all nodes are selected, they can be converted into overlapping communities.

• (3) Fuzzy threshold optimization based on information feedback model

The information feedback model uses previous information to guide the subsequent update process in a meta-heuristic algorithm. In the process of overlapping community discovery, more accurate overlapping nodes can be obtained by rationally using previous information to optimize the fuzzy threshold. In the process of selecting previous individual information, there are two methods: random selection and fixed selection. To increase the diversity of the population, this article uses the random selection method to select individuals. At the same time, to reduce the complexity of the algorithm, in this model only the previous historical information of the current update process is selected to guide the current update status. The specific formulas (Wang & Tan, 2017) are as follows Eqs. (4)–(6).

(4) $$r_i^{t+1}={\theta }_1{r}_{rand}^t+{\theta }_2{y}_i^{t+1}$$

(5) $${\theta }_1={\displaystyle \frac{F^{t+1}}{f_{rand}^t+F^{t+1}}}$$

(6) $${\theta }_2={\displaystyle \frac{f_{rand}^t}{f_{rand}^t+F^{t+1}}}$$

Among them, $r_{rand}^t$ is the randomly selected fuzzy threshold for the t iteration of an individual, and the corresponding fitness value is $f_{rand}^t$; $y_i^{t+1}$ is the individual obtained by the current algorithm iteration, and the corresponding fitness value is $F^{t+1}$.

• (4) Node correction strategy

Due to the randomness of the evolutionary algorithm, some nodes may be mistakenly assigned to unreasonable communities in the overlapping community population obtained according to the above method. To reduce the influence of algorithm randomness, this article proposes a correction strategy to reallocate overlapping nodes and non-overlapping nodes. Based on the idea of boundary node occupancy allocation, the occupancy of each node in all neighboring communities is calculated, and unreasonable nodes are allocated to communities with large occupancy. The specific occupancy formula is as shown in Eq. (7).

(7) $$f_C^k={\displaystyle \frac{d_C^k}{\sum _{i,j\in C}{e}_{ij}}}$$

Among them, $d_C^k$ represents the degree of the kth node in community C, $k\in [1,n]$, and $e_{ij}$ represents the edge connecting nodes in community C. If there is an edge between nodes, it is 1, otherwise it is 0.

In the correction strategy, the correction of nodes is generally divided into three situations. First, when the occupancy rate of the community where the node is located is greater than the occupancy rate of the node’s neighboring community, the node is correct; second, when the occupancy rate of the community where the node is located is equal to If the node is a non-overlapping node, then the node is converted into an overlapping node; thirdly, when the occupancy of the community where the node is located is less than the occupancy of the node’s neighborhood community, the node is corrected to the maximum share of the community.

Evaluation indicators

• (1) Extended modularity function EQ

The extended modularity function EQ (Ma et al., 2021) is an extension of traditional modularity and is used to evaluate the performance of overlapping community divisions. By maximizing the extended modularity, the algorithm can more accurately identify multiple overlapping community structures existing in the network, as shown in Eq. (8).

(8) $$EQ=\frac{1}{2m}{\displaystyle \sum }_{i=1}^t{\displaystyle \sum _{v\in C_i,w\in C_i}\frac{1}{O_v{O}_w}\left[A_{vw}-\frac{k_v{k}_w}{2m}\right]}$$

Among them, t represents the number of communities, $C_i$ represents the ith community, $O_v$ and $O_w$ represents the number of communities to which nodes v and w respectively belong. When both $O_v$ and $O_w$ only belong to one community, the extended modularity will return to the traditional modularity.

• (2) Generalized normalized mutual information

Generalized normalized mutual information (Lancichinetti, Fortunato & Kertész, 2009) refers to comparing the community structure detected by the algorithm with the real community of the ground truth, and can only be used in networks where real community divisions exist. Its range $\mathrm{g}NMI(A,B)\in [0,1]$, the larger the calculated value, indicates that the community structure detected by the algorithm is closer to the ground truth community structure. As shown in Eq. (9).

(9) $$\mathrm{g}NMI(A,B)={\displaystyle \frac{-2\sum { }_{\mathrm{i}=1}^{m^A}\sum { }_{j=1}^{m^B}{L}_{ij}\log (L_{ij}N/L_{i.}{L}_{.j})}{\sum { }_{\mathrm{i}=1}^{m^A}{L}_{i.}\log (L_{i.}/N)+\sum { }_{\mathrm{j}=1}^{m^B}{L}_{.j}\log (L_{.j}/N)}}$$where $m^A$ and $m^B$ represent the number of community A and community B respectively, $L_{ij}$ represents the same number of nodes in the ith community in A and the jth community in B, N represents the number of network nodes, and $L_{i.}(L_{.j})$ represents the ith row in L (elements in column j).

Synthetic and real network datasets

• 1) LFR network datasets

This article uses the LFR network model proposed by Lancichinetti, Fortunato & Radicchi (2008) to evaluate the performance of the proposed algorithm. The LFR model controls the generation of different network structures by adjusting different parameters. The parameter settings used in this article are shown in Table 2. Among them, the number of nodes is set to 100, 500, and 1,000 and represents three types of networks: LFR1, LFR2 and LFR3 respectively; $\mu$ controls the connectivity between communities and is set to 0.1 to 0.5; $O_n$ and $O_m$ represent the number of network overlapping nodes and the number of communities to which each overlapping node belongs; and represent the minimum number of nodes and the maximum number of nodes in each community respectively; $k$ and $k_{max}$ represent the average node degree and maximum node degree of the network respectively; ${\tau }_1$ and ${\tau }_2$ are the power law distribution indices.

• 2) Real network datasets

This article uses nine known real-world networks to evaluate algorithm performance. Six of these networks have real community structures. The specific information of the real network is shown in Table 3.

Evaluation of non-overlapping community discovery experiments

As shown in Table 4, the modularity Q values obtained by the first-stage non-overlapping community discovery algorithm proposed in this article and the other three proposed non-overlapping community algorithms in seven real-world network datasets are given. It can be seen that the maximum modularity value of the non-overlapping communities in the first stage of the algorithm proposed in this article is the highest than that of other algorithms, and the average modularity value of only one data is lower than DECS. It shows that the first-stage method proposed in this article has better accuracy in non-overlapping community division. However, the single-objective optimization method proposed by CNLLP based on core nodes and layer-by-layer label propagation only considers the value of modularity and ignores the relationship with the real community, thus only obtaining a poor Q value. It also shows that simultaneous optimization of multiple objectives can yield better results than optimization of a single objective.

When the LFR network is generated, two files will be obtained, including network data and the real community division structure of the network. Therefore, this article uses gNMI and EQ to evaluate the experimental results of the LFR network.

As shown in Figs. 4 to 6, the gNMI values obtained by 45 LFR networks on the four algorithms are shown. Taking LFR1 as an example, it shows the gNMI value detected when $\mu$ changes from 0.1 to 0.5 when node N is 100 and $O_m$ is set to 2, 4, and 6. It can be seen from the LFR1, LFR2, and LFR3 network diagrams that the gNMI values of all algorithms almost show a decreasing trend as the $\mu$ value increases. For the gNMI value corresponding to each $\mu$, most of the algorithms proposed in this article are higher than The performance of the remaining algorithms is particularly obvious on the LFR3 network. As the number of nodes increases, the community results obtained by the proposed algorithm have greater advantages than other algorithms. Therefore, the algorithm proposed in this article has better performance and can search for a network structure closer to the real one.

As shown in Figs. 7 to 9, the EQ values obtained by the LFR1, LFR2, and LFR3 networks on the four algorithms are shown. In the LFR1 network result EQ diagram, when $O_m$ is 2 and 4 and $\mu$ is between 0.1 and 0.3, the EQ value obtained by the algorithm proposed in this article is almost equal to that of EMOFM-DK and is higher than the other two algorithms. At the same time, the EQ value of this algorithm when $O_m$ = 6 is better than the other three algorithms. In addition, in the EQ diagram of the LFR2 and LFR3 network results, although the EQ value obtained by the algorithm proposed in this article is higher than the MRMOEA algorithm and the CNLLP algorithm, compared with the EMOFM-DK algorithm, it is either equal to or lower than this algorithm. This is because in the multi-objective evolutionary algorithm, we use modularity EQ and gNMI as optimization goals, and the optimal Pareto values obtained conflict with each other, so a better gNMI value is obtained in the algorithm proposed in this article Afterwards, the resulting EQ value may drop.

Real network experiment results

Table 5 and Fig. 10 shows the EQ values obtained by the algorithm proposed in this article and the other three algorithms in nine real-world network datasets. It can be seen that EMOFM-DK based on fuzzy clustering and the algorithm proposed in this article have obtained better modularity values on nine datasets, reflecting the better performance of the fuzzy clustering method in discovering overlapping nodes. CNLLP, which is based on core nodes and layer-by-layer label propagation, obtains poor results in dividing overlapping communities. The reason may be that it only uses the degree of belonging of the node to determine whether the node is overlapping. In addition, on the nine datasets, the number of maximum and average values of EQ obtained by the algorithm proposed in this article is seven, while the maximum and average number of EQ obtained by EMOFM-DK are five and three respectively, indicating that the two-stage overlapping community discovery method proposed in this article can show better performance on the test data set.

Table 6 and Fig. 11 shows the gNMI values obtained by the algorithm proposed in this article and the other three algorithms in six real-world network datasets. It can be seen that on the two datasets of Karate and Dolphin, the algorithm proposed in this article and the two algorithms MR-MOEA and EMOFM-DK can find the real network partition structure. On the Y2H dataset, MR-MOEA achieved the best results. On the remaining three datasets, the optimal results obtained by the algorithm proposed in this article are higher than the other three algorithms. Therefore, the information feedback model proposed in this article divides overlapping community structures and uses values from historical evolutionary periods to optimize the current moment and obtain results closer to real communities.