Multi-objective optimization for community detection in multilayer networks

A multilayer network is abstracted to $G = (\{G_1,\ldots,G_l\}, R)$, $G_l=(V_l,E_l)$ stands for a network at the l-th layer where V_l represents nodes and E_l the inter-layer links. $R = \{E_{ij} \subset V_i \times V_j, i, j \in{1,\ldots,l}, i \ne j \}$ indicates connections among nodes of layer G_i and G_j . The elements of R represent inter-layer connections or crossed layer connections. That is, l subnetworks and the connections among these l networks make up a multilayer network together.

There are few indexes to directly assess the quality of the multilayer networks compound community. At present, the composite modularity $(Q_c^{\prime})$ [17], redundancy index (R_c ) [18], normalized mutual information (NMI) [19] and adjusted rand index (ARI) [20] are commonly used as the evaluation indicators.

As defined in eq. (1), the composite modular $Q_c^{\prime}$ is used to estimate the community index. The greater $Q_c^{\prime}$ is, the better the quality of the multilayer network compound community performs. M denotes the amount of communication links and n indicates dimensions or communication layers of a network. A denotes the corresponding adjacent. d infers to the degree of a node. X_i indicates that node i belongs to community X. When $X_i=X_j$, $\delta(X_i, X_j)=1$, 0 otherwise:

$$\begin{equation} Q_c^{\prime} = \frac{1}{2M}\sum_{i,j}^{n}\left(A_{ij}^{\prime} - \frac{d_i \times d_j}{2M}\right)\delta(X_i,X_j). \end{equation} \tag{ 1 }$$

The redundancy index R_c  [18] denotes the calculated ratio of redundant connections of multilayer networks. The larger R_c is, the better the quality of the compound community is. Intuitively, a compound community should have links across multiple layers. $\|p\|$ represents the amount of communities. $S_i^{\prime}$ denotes a couple $\{c_1,c_2\}$ that can be connected at the lowest layer of the G community. If $\{c_1,c_2\} \in S_i^{\prime}$, then $\beta(c_1,c_2,E_l)=1$, otherwise $B(c_1,c_2,E_l)=0$. The redundancy index is formulated in the following equation:

$$\begin{equation} R_c = \frac{1}{t \times \|p\|}\sum_{G_\delta \in G} \sum_{\{c_1,c_2\} \in S_i^{\prime}}\beta(c_1,c_2,E_l). \end{equation} \tag{ 2 }$$

The normalized mutual information (NMI) can assess the comparability of the optimized community and the original one [19]. $F^{\prime}$ denotes a confusion matrix. $F_x^{\prime} (F_y^{\prime})$ denotes the amount of elements in the x-th row (or the y-th column) in $F^{\prime}$. $r_{c_1}~(r_{c_2})$ signifies the total amount of clustering in a partition c₁ (c₂). The range of NMI value is [0, 1]. The greater the NMI is, the higher the similarity between optimized and original networks is. Suppose that c₁ and c₂ are two partitions in a network, then the following equation is applied to calculate the value of $\textit{NMI}(c_1,c_2)$:

$$\begin{eqnarray} & \textit{NMI}(c_1,c_2)=\nonumber\\ & \frac{-2\sum_{x=1}^{r_{c_1}}\sum_{y=1}^{r_{c_2}}F_{xy}^{\prime}\log(F_{xy}^{\prime}N/F_{x.}^{\prime}F_{.y}^{\prime})}{\sum_{x=1}^{r_{c_1}}F_{x.}^{\prime}\log(F_{x.}^{\prime}/N)+\sum_{y=1}^{r_{c_2}}F_{.y}^{\prime}\log(F_{.y}^{\prime}/N)}. \end{eqnarray} \tag{ 3 }$$

The adjusted Rand index $(\textit{ARI})$ is used to calculate the similarity between real communities and clustering ones. e and h denote the number of node pairs. The former one denotes that located in the same community in the real partition (c₁) and the obtained one (c₂), the other in a disparate community in c₁ and c₂. f and g are for the quantity of node pairs. f denotes which located in the same community in c₁ and different community in c₂. g represents that located in the same community in c₂ and different community in c₁. The more similar the real partition and the obtained ones are, the greater the value of $\textit{ARI}$ is. $\textit{ARI}$ can be defined as follows [20]:

$$\begin{equation} \textit{ARI}=\frac{2(eh-fg)}{(e+f)(f+h)+(e+g)(g+h)}. \end{equation} \tag{ 4 }$$

In this paper, the modularity is used as the optimization function, and NMI, R_c , ARI are used as evaluation indexes due to their capability of finding high-quality solutions in multilayer networks [11,21–24].

Existing community detection algorithms for multiplex networks fall into four categories, i.e., matrix-decomposition–based method, spectral-based method, information-based method, and modularity-based method.

Matrix-decomposition–based methods extract the community division by decomposing the matrix such as NMF [ [16,22], SNMF [11,25] and so on. Spectral-based methods compute the community division by employing eigendecomposition to Laplace matrices, such as MIMOSA [26], and SC-ML [27]. The performance of such kinds of algorithm stands out because they capture the global information across different layers. Information-diffusion–based methods integrate different layers of the multilayer network by employing the diffusion of networks [28]. For example, similarity network fusion (SNF) computes the fused matrix of all layers through a parallel interchanging diffusion process, and then explores the community division by employing the spectral clustering method to the fused matrix [21]. The generalized Louvain (GL), one of the most efficient modularity-based algorithms, achieves a great efficiency by optimizing their generalized modularity function [23]. However, GL suffers from great difficulties in mining the consensus community division of all layers.

As for the methods mentioned above, they can handle almost all multilayer networks accurately and efficiently, but some drawbacks are unavoidable [11]. For example, the spectral-clustering–based method is good at extracting small-scale and tight communities, which may lead to some crucial information such as SC-ML being ignored. Many modularity-based algorithms are extended from the single-layer method that do not address the information across different layers. To overcome these problems, a novel multi-objective optimization algorithm is proposed to balance the community structure of every layer.

An appropriate code scheme of the solution can effectively reduce the computation and expedite the algorithm convergence. The label-based and locus-based representation play an important role in encoding methods for community detection. However, the label-based code scheme is redundant, which means that if there are t labels in the pattern, then t! different chromosomes might be mapped to the same division [29]. In order to get the utmost output of the information contained in the pattern, this paper employs the locus-based adjacency presentation scheme. Suppose the solution of chromosomes in the population is set to $S=\{s_1, s_2,\ldots,s_N\}$. The length of the gene is N and each gene i can be arbitrary integer between 1 and N, namely, 1 < i < N. The i-th gene value could be j, if i, j are linked at the lowest one layer in a network.

The uniform crossover is applied for genetic operation in this paper. In fact, the uniform crossover pertains to multi-point crossover category, and becomes an efficient operator in evolutionary algorithms. First, the binary mask chromosomes with equal number of nodes are generated at random. According to the mask, the corresponding gene bits are selected from two parent chromosomes to form new chromosome as offspring chromosomes. To be specific, if the mask is 0, the operator selects the gene of the first parent chromosome; otherwise, the operator selects the second parent chromosome.

The mutation operator in chromosomes integrates the correlative information of the layer nodes neighbourhood, which makes random mutations in the form of the probability. The i-th gene in a chromosome is selected at random in the form of predefined probability, and then this gene mutates into the j-th neighbour of the i-th gene.

The NSGAMOF algorithm includes local search operations, namely the hill-climbing (HC) method. The first step is to define the neighbours of a chromosome. Given a chromosome $S_k = \{ S_k^1, S_k^2, \ldots, S_k^n \}$, node $S_k^i$ is selected from chromosome S_k at random. Then, the gene $S_k^i$ is a substitute for other neighbour nodes $S_k^j$ of the location i, and $S_k^j \neq S_k^i$, denoted by $S_k^{\prime}$. The newly generated chromosome $S_k^{\prime}$ is used as a neighbour of chromosome S_k . In local search operations, a chromosome is chosen to be refined at random. Meanwhile, all possible neighbour chromosomes are identified. If the newly generated chromosome is better than the original one, we replace it with the new chromosome.

In MOOP (multi-objective optimization), a set of optimal solution sets will be obtained, which signify the optimal trade-off among optimization objectives. In order to obtain the final solution of PS in MOOP, we use the disparate optimal solution selection policy. There are three types of multi-criterion decision making: 1) a posteriori-based, 2) a priori-based, and 3) interactive-based [30].

The a priori-based policy leverages the value maximization of the objective function (i.e., the modularity maximum) which is the optimal solution in the Pareto set. This policy using prior knowledge is called NSGAMOF-prik.

Different from the a priori-based policy, the a posteriori-based strategy takes the overall information of the network into the consideration. It calculates the average modularity of each layer and the chromosome with the highest average value is the optimal solution. In this case, the method is called NSGAMOF-postk.

The last one is the interactive-based strategy, which mainly uses the k-means clustering method to classify the collection of data got by disparate models and enhance the quality of community detection. Such a strategy which selects the fittest solution by using the clustering method in the Pareto solution is called NSGAMOF-clu. Figure 2 shows an example of NSGAMOF-clu with three-layer networks and six nodes per layer.

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The optimal selection strategy mentioned above can adapt to different network structures, which ensures a high performance. For example, the prik-based method adapts to networks which have uneven information distribution for each layer (i.e., most of the information exists in one layer). However, for the network with the uniform information distribution, the prik-based approach may lose some information but the postk-based strategy can address the problem. In general, the proposed strategies can improve the accuracy for multilayer networks with various structures.

The synthetic network m-LFR128 function has 128 nodes in each layer [31]. The actual partition structure is known. To change parameters allows to control the total edges among communities and the difference of node degrees among layers. Therefore, 2, 3 and 4 layers of the network are generated respectively and these networks have different network topologies. The mixture parameter u signifies the part of the connection between one node and every other node in a community. The quality of community detecting usually becomes more dissatisfying as u gets larger. The degree of each node in different network layers is determined by Dc, that is, the degree change chance. The greater the parameter Dc is, the more diverse the nodes of various layers might be. Moreover, eight real-world network datasets are used in this paper, i.e., Kapferer Tailor Shop, KTS ¹ , Cs-Aarhus Social, CAS [32], Bibliographic Data, CoRA [33], Mobile Phone Data, MPD [34], Worm Brain Networks, WBN ² , Word Trade Networks, WTN [35] and Social Network Nata, SND including two different ground truth divisions, i.e., SND(o) and SND(s) [36] (see table 1).

Figure 3 compares three strategies of NSGAMOF with the MLMaOP algorithm (i.e., proj, cspa, mf) [14] in terms of NMI based on different parameters (i.e., d, Dc and u) in 12 network structures from mLFR dataset. Results show that each strategy the NSGAMOF algorithm performs better than all strategies in MLMaOP algorithm. Moreover, the increasing number of layers scarcely influences the performance of NSGAMOF algorithms. Although the parameter Dc affects the network architecture, NSGAMOF can still find out the optimal division beneath diverse network architectures. We can conclude that NSGAMOF algorithms (i.e., NSGAMOF-prik, NSGAMOF-clu, NSGAMOF-postk) perform better than MLMaOP algorithms (i.e., proj, cspa, mf) in the diverse optimal selection policy, network architecture and network layers.

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Comparative results of NSGAMOF-prik, NSGAMOF-clu, NSGAMOF-postk, BGLL [37], the MOEA-MultiNet algorithms [11] in the KAPFERER TAILOR SHOP and CS-AARHUS SOCIAL multilayer networks are shown in table 2. More specifically, L₁, L₂, L₃ and L₄ of the one-layer strategies show that the simplified composite modular $Q_c^{\prime}$ and redundancy Rc are calculated by utilizing each community returned by BGLL algorithms in each layer of the networks. MOEA-MultiNet and NSGAMOF of multilayer strategies calculate the corresponding simplified composite modular $Q_c^{\prime}$ and redundancy Rc.

Table 2:. The $Q_c^{\prime}$ and R_c metric comparative results.

Dataset	Strategy	Algorithm	$Q_c^{\prime}$	Rc
KAPFERER TAILOR SHOP NETWORK	One-layer	BGLL/L1	0.2179	0.3964
		BGLL/L2	0.2006	0.4717
		BGLL/L3	0.1380	0.2657
		BGLL/L4	0.0932	0.4094
	Multilayer	MOEA-MultiNet	0.2094	0.4735
		NSGAMOF-prik	0.4343	0.3705
		NSGAMOF-clu	0.4698	0.3511
		NSGAMOF-postk	0.4810	0.3134
CS AARHUS NETWORK	One-layer	BGLL/L1	0.4685	0.2852
		BGLL/L2	0.1672	0.0472
		BGLL/L3	0.0832	0.1205
		BGLL/L4	0.2893	0.1611
		BGLL/L5	0.4115	0.2715
	Multilayer	MOEA-MultiNet	0.4010	0.3186
		NSGAMOF-prik	0.2316	0.3703
		NSGAMOF-clu	0.2315	0.3617
		NSGAMOF-postk	0.2287	0.3847

Results show that the multilayer strategy is superior to the one-layer strategy in solving multilayer network community detection. And the proposed NSGAMOF-postk algorithms of multilayer strategy outperforms obviously the several other algorithms, especially in simplified composite modular metric. In general, we can conclude that the compound community structure acquired by the proposed algorithms NSGAMOF-prik, NSGAMOF-clu, NSGAMOF-postk) have better performance than the BGLLs [37] and the MOEA-MultiNet algorithm [11] in the single layer.

Tables 3 and 4 further compare the proposed strategies of NSGAMOF with other algorithms (i.e., CSNMF, CPNMF, CSNMTF, CSsNMTF [11], SNF [21], SC-ML [27], CGC [22], GL [23] and Infomap [38]) on five different real multilayer datasets in terms of NMI and ARI, respectively. Obviously, at any rate one of the presented methods is superior to other algorithms, especially in NMI metrics in most real-world datasets. And the ARI index value of the proposed algorithm has significant advantages over other algorithms. In addition, to validate the performance of the proposed algorithm roundly, the results on the HBN network, which contains 13281 nodes and 8 layers, are shown in table 5. The results denote that the proposed algorithm achieves an acceptable result.

Although the ground truth represents consensus division of a kind, the information and importance of each layer are different. To identify the varying frequency of the different importance of each layer, three strategies are proposed to better adapt to different network structures. Here, two datasets (i.e., MPD and SND(o)) are used to verify the validity of the proposed strategies. Figure 4 plots the NMI and ARI of each layer.

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For the results shown in tables 3 and 4, the interactive-based strategy (postk) performs well when run on the SND(o) dataset and the a priori-based strategy acquires a better solution when run on the MPD dataset. For SND(o), the best solution is extracted by choosing the maximum average of all layers since complementary information can improve the accuracy, but for MPD, the information of other layers reduces the accuracy so the prik strategy get the best solution. The experimental result demonstrates that the proposed strategies can improve the performance by adapting to various data structures.

To demonstrate the convergence performance of NSGAMOF, NMI and conductance indices are calculated to observe the convergence of the NSGAMOF algorithm. As shown in fig. 5, the NSGAMOF algorithm converges slowly at the beginning, and then the convergence starts to accelerate after 55 iterations. Finally, the NSGAMOF algorithm stably converges within 90 times. In the whole operation process of the NSGAMOF algorithm, the curve of the NMI index generally rises and tends to be stable after some reasonable fluctuation, which proves that NSGAMOF algorithm has strong convergence.

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Figure 6 shows the result of scalability as the network nodes and dimension increase. The experiments employ 10 diverse datasets with 2 dimensions. In fig. 6(a), the curve exhibits the characteristics of a quadratic equation as the networks increase in size. Overall, it is reasonable for the execution time of the NSGAMOF-prik algorithm. Figure 6(b) depicts the worst running time of the proposed algorithm as the network dimension increases. According to this result, the algorithm presents the scalability for the high-dimensional network.

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