Detecting composite communities in multiplex networks: A multilevel memetic algorithm

Swarm and Evolutionary Computation 39 (2018) 177–191

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Swarm and Evolutionary Computation journal homepage: www.elsevier.com/locate/swevo

Detecting composite communities in multiplex networks: A multilevel memetic algorithm Lijia Ma *, Maoguo Gong, Jianan Yan, Wenfeng Liu, Shanfeng Wang Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education, Xidian University, Xi’an 710071, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Composite community detection Multiplex networks Memetic algorithm Multilevel local search

Nowadays, many systems can be well represented by multiplex networks, in which entities can communicate with each other on multiple layers. A multiplex network under each layer has its own communities (i.e., a higher-order organization with a group of similar nodes) while it has a composite structure which is most likely to describe its community structures at all layers. Many algorithms have been proposed to detect communities in unweighted single-layered networks, but most of them cannot be well applied to detect composite communities in multiplex networks. The aim of this paper is to detect composite communities in weighted multiplex networks using a multilevel memetic algorithm. First, a simplified multiplex modularity is adopted for evaluating the fitness of composite communities, and then the community detection problem in multiplex networks is modeled as a combinational optimization problem. Second, we devise a multilevel memetic algorithm that combines a networkspecific genetic algorithm with problem-specific multilevel local search operators. In the presented algorithm, the network-specific knowledge (i.e., the layer neighborhood and the consensus neighborhood) and the problemspecific information (i.e., the fast computation of multiplex modularity under each local refinement) are adopted to guide its search processes. Last, extensive experiments are performed on eight real-world networks ranging from social, transport, financial to genetic areas, and the results demonstrate that our algorithm discoveries composite communities in multiplex networks more accurately than the state-of-the-art.

1. Introduction Complex network has demonstrated to be a simple but effective model reflecting the fundamental structures and behavior attributes of real systems in the domains of sociology, finance, transportation, ecology, genetics, and etc [3,34]. Nodes in complex networks interpret the agents of real systems while the edges among nodes reflect the interconnections of entities. Most existing studies mainly model real systems to single-layered networks with homogeneous links, i.e., interconnections between entities are only reflected by a single type of link and their weights are set to the same value. However, many real systems are heterogeneous, i.e., their communications exist in multiple platforms and they are different in types and weights [36,47,8]. For instance, in social systems, individuals can communicate with each other through the following platforms: Facebook, Telephone, Webchat and Email. In transportation systems, travellers can choose bikes, buses, taxies, trains, boats and airplanes as transportation way, and the travel cost under each transportation way are different. These systems can be properly represented by multiplex networks with a set of nodes and

multiple layers of weighted links. Links in different layers interpret the different types of activities of agents in the systems and their weighted values represent the communication degree of pair of agents [33,45]. Community structure, a macroscopic structure of networks, reflects the higher-order organization and functionality which are hidden in the low-order nodes and communications of the networks [36]. A community in a single-layered network is composed of a set of similar nodes [17, 20]. However, it has no uniform definition in a multiplex network as the network under each layer has its own community structures which cannot well reflect its composite community structure (i.e., the uniform communities under all layers) [36]. The detection of communities of networks are important to understand the structural functionality (e.g., motif structure, small world, scale-free structure, robustness, synchronization, and controllability) and individual behaviors (e.g., spreading, competition, balance, cooperation and evolution) of complex systems [6, 29,27,22,52,42,14]. Recently, many community detection algorithms have been presented for single-layered networks [25]. However, they cannot be well applied to multiplex networks as their results only reflect the community property of the networks at one layer.

* Corresponding author. E-mail address: [email protected] (L. Ma). https://doi.org/10.1016/j.swevo.2017.09.012 Received 12 December 2016; Received in revised form 12 August 2017; Accepted 26 September 2017 Available online 28 September 2017 2210-6502/© 2017 Elsevier B.V. All rights reserved.

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Compared with a single-layered network, a multiplex network has more complex communities and topology structures. The community detection in multiplex networks is not an easy problem and it has to solve the following problems: i) the definition of composite community and ii) the utilization of comprehensive topology structures in multiplex networks. Existing work can be divided into two classes: the community definition based optimization approach and the layer aggregation and the consensus clustering approaches. Generally, the former approaches firstly give a definition of composite community, and then use a heuristic or meta-heuristic algorithm to detect the defined communities. For instance, Mucha et al. [36] presented a multiplex modularity to define the composite community in multiplex networks, and then used a heuristic clustering division and grouping algorithm for the community detection. Amelio and Pizuti et al. [1] used modularity [39] and normalized mutual information [23] to evaluate the quality of communities, and proposed a network-specific multi-objective optimization algorithm to detect the communities. Hmimida et al. [21] given the definition of local communities, and then presented a send-centric algorithm to find the local communities. Demenico et al. [11] used the modular flow redefine the community structure, and proposed a network flow compression method to detect the modular flows. The latter approaches take into account the comprehensive topology structure of multiplex networks, and they use the layer reduction method to detect communities. Specifically, the layer aggregation approach firstly aggregates the topological connections of multiplex networks into a weighed single-layered network, and then uses existing community detection algorithms to detect composite communities [48,53]. The consensus clustering approach firstly detects communities of multiplex networks under each layer, and then uses a consensus clustering technique to detect composite communities [24]. Recent studies have demonstrated that the layer reduction in multiplex networks can improve the community detection [12,50]. Note that, the community definition based optimization approaches [36,1,21,11] neglect the comprehensive topological connection of multiplex networks, and thus they are easy to get into local optimal community divisions. Moreover, these approaches ignore the overall layer interconnections, and therefore they are hard to be applied to high-dimensional multiplex networks. The layer aggregation and the consensus clustering approaches [48,24] may result in the missing and the increasing of topology connection, and therefore the detected community structure cannot well reflect the composite community structure of original multiplex networks. The composite community detection in multiplex networks is a discrete NP-hard problem. Recent years, Memetic algorithm (MA) has become a simple but effective way for solving many discrete NP-hard problems [13,31,2], including capacitated art routing problem [32], VLSI floorplanning [49], minimum energy broadcast [56], resource allocation and scheduling [54,51], multi-robot path planning [10], vertex coloring [35], influence maximization [19] and etc. MAs comprehend the explorations of a population based global search (e.g., a genetic algorithm) and the exploitations of one or more individuals based local search (e.g., problem-specific individual learning) [41,37,56,30]. Actually, some network-specific MAs have been presented for the identification of communities in unweighted single-layered networks. For instance, Gong et al. [18] and Shang et al. [44] combined a network-specific genetic algorithm (GA) with a hill climbing and a simulated annealing to detect communities in small-scale networks, respectively. We [30] combined a GA with a neighborhood based learning search to detect communities in large-scale networks. Wu et al. [55] presented a MA for multiobjective community detection. However, little attention has been paid to the utilization of MAs for composite community detection in multiplex networks. In this paper, we propose a community definition based multilevel memetic optimization algorithm for composite community detection (MMCD for short) in multiplex networks. The main contributions of this paper are summarized as follows.

1. We adopt a simplified multiplex modularity Qe to evaluate the quality of composite community structure in multiplex networks, and model the composite community detection problem as a Qe based constrained optimization problem. The Qe index comprehensively considers i) the composite community definition in a global view, ii) the community structures over all network layers, iii) the weighted communication topology and iv) the community sensitivity to different composite community structures. 2. We propose a knowledge-based and problem-specific MMCD algorithm for solving the modeled constrained optimization problem. MMCD combines a GA with a multilevel local search, in which the GA combines its operation with the network knowledge while the multilevel local search operators combine their update and search rules with the network knowledge and the optimization problem information at the node level, cluster level and partition level, respectively. 3. We reveal two types of network-specific knowledge from multiplex networks: the layer neighborhood and the consensus neighborhood. The utilization of layer neighborhood information can promote that two entities with more types of communications are easier to be the same community, while the utilization of consensus neighborhood can help the optimization algorithm discover composite communities with high Qe value. We also reveal the problem-specific information from the optimization model (i.e., the fast computation of multiplex modularity under each local refinement). The utilization of this problem-specific information can largely reduce the time consuming during the optimization processes. 4. We conduct extensive experiments on eight real multiplex networks that come from social, transport, financial and genetic areas. Moreover, we use three criteria to comprehensively compare the performance of comparison algorithms on the detected composite communities. The experiments demonstrate that MMCD is superior to the state-of-the-art in terms of the accuracy and redundancy of detected composite communities. The rest of the paper is organized as follows. Related work are given in Section 2. The problem formation and our solution are given in Sections 3 and 4, respectively. Section 5 makes systematic experiments. Finally, we give our conclusions and future work in Section 6. 2. Related work 2.1. Multiplex networks Multilayer network. A multilayer network [8] is a pair GM ¼ ðfG1 ; G2 ; …; Gα g; CÞ where Gi ¼ fV ½i ; E ½i g is a single-layered network with nodes V ½i and intra-layer links E ½i , α is the dimension of the multilayer network, ½ij

and C ¼ fepq ⊆V ½i  V ½j ; vp 2 V ½i ; vq 2 V ½j ; i; j 2 1; 2; …; α; i 6¼ jg is the set of inter-layer links between nodes of networks Gi and Gj at different layers. Here, nodes V ½i may be different with nodes V ½j when i 6¼ j. Multiplex network. A multiplex network is a special type of multilayer network in which the nodes at all layers are the same, all nodes are only internally linked to their counterpart nodes in the rest of layers, i.e., C ¼ ½ij

fepq ⊆V ½i  V ½j ; vp 2 V ½i ;vq 2 V ½j ;p ¼ q; i;j 2 1;2;…; α;i 6¼ jg, and all these inter-layer links have no explicit cost associated [5,40]. In other words, a multiplex network consists of a set of nodes V and multiple types of edges fE ½1 ; E½2 ; …; E½α g, where α is the number of communication types, and it can be represented by a multi-layered network GM. GM ¼ fG1 ; G2 ; …; Gα g;

(1)

where Gi ¼ ðV; E ½i Þ, 8i 2 f1; 2; …; αg, represent i-th layer network of the multiplex network. For the network Gi, its communication E ½i can be represented by a weighted adjacency matrix A½i (i.e., each element A½i pq  0 denotes the communication weight between nodes vp and vq in i-th 178

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Fig. 2. Schematic illustration of the framework of layer aggregation approach on the toy multiplex network G1M .

Fig. 1. Schematic illustration of the representation of a toy multiplex network G1M . The multiplex network is composed of 8 nodes and 3 types of edges (i.e., the communications in Facebook, Telephone and Email).

layer network). Fig. 1 gives a schematic illustration of the multi-layered network representation of a toy multiplex network G1M with 8 nodes and 3 types of edges. As shown in Fig. 1, the networks under different layers have the same nodes and different types of communications (i.e., the communications in Facebook, Telephone and Email). Fig. 3. A schematic illustration of the framework of consensus clustering approach on the toy multiplex network G1M .

2.2. Classical community detection algorithms in multiplex networks

algorithm combines the community grouping technique (i.e., BGLL [7]) with the community division technique (i.e., Kernighan-Lin [38]), and it can effectively detect communities with high quality. In [1], Alessia and Clara modeled the detection of communities in multiplex networks as a multiobjective optimization problem with two objectives: modularity Q and normalized mutual information (NMI). In the optimization processes, the Q criterion is used to evaluate the quality of detected communities in one layer while the NMI criterion is adopted to evaluate the community similarity between two neighboring layers of networks. They proposed a network-specific multiobjective evolutionary algorithm MultiMOGA to optimize the two objectives simultaneously. Note that, MultiMOGA neglects the consensus communications and the community correlations of two nonadjacent networks in multiplex networks, and therefore it is hard to be applied to high-dimensional multiplex networks. In [21], Hmimida and Kanawati given a definition of community structure (i.e., the extension of seeds in networks), and proposed mux-Licod to detect the communities in multiplex networks. The mux-Licod algorithm firstly identifies a set of special nodes (called as seeds) in networks, and then uses a local community computation method to find the communities around the identified seeds. However, it neglects the global network information and the global community definition, and thus it is hard to detect high-quality communities in high-dimensional multiplex networks.

In multiplex networks, there are two common challenges for composite community detection: i) the definition of composite community and ii) the utilization of comprehensive topology structures in multiplex networks. 2.2.1. Community definition based optimization approach The community definition based optimization approach firstly gives an index to evaluate the quality of composite communities in multiplex networks, and then uses a heuristic or meta-heuristic algorithm to reveal the defined communities by optimizing the index. Multiplex modularity [36] is widely used to evaluate the quality of community structure of arbitrary multiplex networks, and it is computed as follows. QM ¼

1 X 2u pqij

" A½i pq  λi

! # ½i k ½i p kq δði; jÞ þ δðp; qÞ δðxp ; xq Þ; 2 m½i

(2)

where the normalization factor u represents the number of links (i.e., communication links and coupling links), k½i p is the degree of node vp in ith layer network, xp denotes the cluster label of node vp, and δðp; qÞ is a binary delta function (i.e., if p ¼ q, δðp; qÞ ¼ 1; and δðp; qÞ ¼ 0, otherwise). λi is a resolution parameter and it is set to 1 for community detection of multiplex networks in most cases [36]. QM is the multiplex modularity, and it is composed of two parts in Eq. (2). The left part evaluates the quality of composite community structure of networks under all layers while the right part evaluates the number of coupling links within composite communities. For a multiplex network with N nodes, the right term is a constant value αðα  1ÞN. Mucha et al. [36] adopted QM to define the quality of composite communities, and proposed a heuristic algorithm (called as GenLouvin) to detect the communities of multiplex networks. The GenLouvin

2.2.2. Layer aggregation approach and the consensus clustering approach Layer aggregation approach. This approach firstly integrates a multilayered network GM ¼ fV; A½1 ; A½2 ; …A½α g into a weighed singleP layered network G ¼ fV; 1α αi¼1 A½i g, and then uses existing community detection algorithms (e.g., Infomap [43] and LPAm [4]) to detect the communities of the integrated network [28]. Fig. 2 gives a schematic

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illustration of the framework of layer aggregation approach on the toy network G1M . Consensus clustering approach. This approach firstly uses existing community detection algorithms to discover communities of networks under all layers, and then finds a consensus community clustering from the detected communities. One of the most notable work is CBGLL [24]. It adopts a classical community detection algorithm (i.e., BGLL [7]) to detect communities under all layers, and then constructs a consensus community matrix based on the detected communities. Next, it uses BGLL to reveal the composite clusters in the consensus community matrix. Fig. 3 gives a schematic illustration of the framework of consensus clustering approach on the toy network G1M . Note that, the layer aggregation and the consensus clustering approaches neglect the interactions of communications of multiplex networks, which result in that the constructed consensus communication information cannot well reflect the real communications of multiplex networks. Moreover, both of them neglect the global definition of composite communities in multiplex networks.

then design a problem-specific multilevel MA which is an extension of our previous work [18,30]. 3. Problem definition and formulation Given a multiplex network GM and a community index, our problem is to detect composite community structures of the network under nonoverlapping constraint (i.e., each node of the network cannot be allocated to two or more communities), so as to the community index is maximized. Here, a simplified multiplex modularity Qe is adopted as the community index, and it is computed as follows. Qe ¼

! ! ½i α X k½i 1 X p kq A½i δðx  ; x Þ ; p q pq 2 m½i 2 m i¼1 pq

(3)

where m is the number of communication links and α denotes the number of communication layers or dimensions of the network. The simplified multiplex modularity computes the average difference between the real fraction and the expected fraction of edges that fall within the detected composite communities of multiplex networks. A community division with high Qe value corresponds to a high-quality composite community structure. The Qe index is a simplified multiplex modularity that neglects the coupling factor (i.e., the second term in Eq. (2)). The reasons why we adopt the simplified multiplex modularity rather than the multiplex modularity to evaluate the quality of composite communities in multiplex networks are as follows. Firstly, the inter-layer links between counterpart nodes in multiplex networks have no explicit cost associated [5,40]. Secondly, for a large-scale or high-dimension multiplex network, the multiplex modularity with the coupling factor would be insensitive to the detected composite community structures. This is because the number of coupling inter-layer links is far more than that of intra-layer links in large-scale or high-dimension multiplex networks. In addition, for a given multiplex network, the number of coupling links is a constant value. Finally, the simplified multiplex modularity is also an extended version of classical community index (i.e., modularity [39]) in single-layered networks. It is the average modularity of community structures over all network layers, which satisfies the intuition of composite community structures that are most likely to describe the potential community structures of the networks at all layers. If we use Qe index to evaluate the quality of composite community structures in multiplex networks, our composite community detection problem can be formulated as follows.

2.3. Memetic algorithms for community detection MAs are hybrid global-local search algorithms. Generally, a global search algorithm (e.g., GA) simultaneously optimizes a population of individuals and it facilitates explorations. And a local search refines an individual via a problem-specific information based local refinement, and it benefits exploitations. A MA combines a population-based global search with an individual-based local search, and it can facilitate both explorations and exploitations [13,31,2]. Recent years, MAs have been widely applied to real applications in the fields of physics, bioinformatics, engineering, computer science, social computing, and etc. MAs have been used to detect communities in unweighted singlelayered networks. The most notable work are [18,44,30,55,15]. Their main differences are in i) the purposes, ii) the operations of GAs and local searches, and iii) the adopted network-specific knowledge and problem-specific information. Their operations are summarized as follows. Step 1) Representation: Each possible community division is represented by a solution (or called as an individual). Classical representation includes the locus-based representation and the string-based representation. Step 2) Initialization: A population of initial individuals (i.e., each individual represents a community division) are generated. Classical initialization includes the random initialization, the label propagation, and the network-specific initialization. Step 3) Selection: A subpopulation of individuals are selected as parent individuals to the mating pool. Classical selection includes the random selection and the roulette selection. Step 4) Genetic operation: The crossover and mutation operations work on parent individuals, and then generate children individuals. Classical crossover operators include the one-point crossover, the twopoint crossover and the two-way crossover while the common mutation operators are the random node mutation, the random cluster mutation and the network-specific mutation. Step 5) Local Search: One of children individuals is selected to be refined. Classical local searches are the hill climbing, the simulated annealing, the tabu search, the node learning and the cluster learning. Step 6) Update: The individuals in the population are updated based on the generated children individuals. Termination: If the termination condition is satisfied, then the algorithm ends. Otherwise, go to Step 3). Note that, existing work [18,44,30,55,15] only use the communication information of unweighted single-layered networks, and they cannot be applied to weighted multiplex networks well. In order to detect the composite communities of multiplex networks, in this paper, we use an index to define the composite community in multiplex networks, and

8 ! ! ½i α X > k½i 1 X > p kq ½i > > Apq  δðxp ; xq Þ max Qe ¼ > > 2 m i¼1 2 m½i > pq < ; n X > > > yjp ¼ 1; 8vp 2 V s:t:; > > > j¼1 > :

(4)

where yjp is a binary value denoting whether node vp is allocated to the jth community, and n is the number of communities. Theorem 1. The Qe based composite community detection problem in a multiplex network is NP-hard. Proof. We consider a special case of the problem: α ¼ 1 (i.e., the number of communication layer is 1). The special case is equivalent to the modularity based community detection problem in single-layered networks. The study [9] has proved that the modularity based community detection problem can be reduced to a well-known NP-hard problem: 3-PARTITION [16]. Therefore, the modularity based community detection problem is NP-hard. Our Qe based composite community detection problem is also NP-hard because one of its special case is NP-hard.□

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4. Multilevel memetic algorithm for composite community detection in multiplex networks In this section, we design a multilevel MA, which is an extension of our previous work, for the Qe based composite community detection problem (which is NP-hard). Firstly, the main framework and the details of the algorithm are given. Secondly, the adopted network-specific multilevel neighborhoods and the problem-specific information are introduced. Thirdly, the computational complexity of the algorithm is analyzed. Finally, comparisons between the presented method and related methods are made. 4.1. MMCD Fig. 4. Schematic illustration of the decoding of the locus based representation and the string based representation. In the locus based representation, the network is divided into a community whereas in the string based representation the network is divided into two communities. Nodes within an ellipse are in the same community.

MMCD is composed of a global search GA and a multilevel local search. The global search GA incorporates network-specific knowledge (i.e., the layer node neighborhood) to help the search of promising solution regions, and the multilevel local search incorporates networkspecific knowledge (i.e., the consensus neighborhood) and problemspecific information (i.e., the computation of ΔQe ) to accelerate the search convergency to optimal solutions. The details of network-specific knowledge and problem-specific information are given in Section 4.2. In order to describe MMCD clearly, we give some notations in a GA and our local search as follows. Definition 1: evolving population. The evolving population X is composed of a population of Np chromosomes (i.e., a chromosome represents a composite community division) in the evolution processes. Definition 2: parent population. The parent population Xp is composed of a set of No chromosomes that are chosen from the evolving population, and they are used for genetic operations (i.e., crossover and mutation). Definition 3: offspring population. The offspring population Xo is composed of a set of No chromosomes that are the production of genetic operations on the parent population. Definition 4: local chromosome. The local chromosome xr is the individual in the offspring population with the highest Qe value, and it chosen to be executed the multilevel local search. Definition 5: global chromosome. The global chromosome xg is the best solution (i.e., the solution with the largest Qe value) in the evolving population, and it is used to guide the refinement of local chromosome. Given the optimization problem in Eq. (4) and the introduction of MAs in Section 2.3, the presented MMCD algorithm is implemented as Algorithm 1. Firstly, a population of Np solutions are initially generated based on the function Initialization(). Secondly, the initial solutions are iteratively evolved based on the functions Genetic-Operation() and Multilevel-Local-Search(). Here, the Initialization(), Genetic-Operation() and Multilevel-Local-Search() are the devised network-specific initialization, genetic operation and multilevel local search, respectively, and they are detailed in the following.

of N genes x ¼ fx1 ;x2 ;…;xN g, where N denotes the number of vertices in the network, and each gene xj, 1  j  N, can be take an integer value q within ½1 N. The difference between the two representations lies in the decoding step. Fig. 4 gives an illustration of the two representations on the toy network G1M . As shown in Fig. 4, in the decoding step, nodes connected by the decoding links are assigned to the same community in the locus based representation, while nodes with same gene values are assigned to the same community in the string based representation. Both of the two representations can be used as our encoding way. In MMCD, the string based representation is adopted because it uses less information than the locus based representation to describe a community division. Function 1. Initialization() 1: 2: 3: 4: 5: 6: 7: 8: 9:

Input: Population size: NP; a multiplex network: GM. Output: X ¼ fx1 ; x2 ; …; xNP g for i¼1 to NP do for j¼1 to N do xij ←j. end for fs1 ; s2 ; …; sN g ← Generate a random sequence. for j¼1 to N do

10:

vq ←Choose a node vq from Γ sj based on the communication weight and the roulette selection. xisj ←xiq . end for end for

½g

Γ sj ← Choose one type of layer node neighborhood from GM randomly, e.g., g-layer.

11: 12: 13:

½g

Network-specific initialization. The Initialization() (shown in Function Algorithm 1. Framework of MMCD 1: 2: 3: 4: 5: 6: 7: 8: 9:

Input: population size: Np; parent population size: No; crossover probability: pc; mutation probability: pm; maximum number of generations: gmax. X← Initialization ðNp Þ do. For i¼1 to gmax Xp ← choose No solutions from X randomly. xg ← find the global chromosome in X. ðXo ; xr Þ← Genetic-Operation ðXp ; pc ; pm Þ. xr ← Multilevel-Local-Search ðxr ; xg Þ. X← Choose Np solutions with high Qe value from X, Xo and xr . end for

4.1.1. Representation and network-specific initialization Representation. The locus based representation and the string based representation are two main encoding ways of a community division [30]. In the two encoding ways, a chromosome or solution x is composed

Fig. 5. Schematic illustration of the initialization on the node v1 of the multiple network G1M . 181

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(continued ) 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20:

xbq ←xap ; f8q j xap ¼ xaq g. xak ←xbp ; f8k j xbp ¼ xbk g. end if fs1 ; s2 ; …; sN g ← Generate a random sequence. for j ¼ 1 to N if (A random value is smaller than pm Þ then ½g

Γ sj ←Choose one type of layer node neighborhood from GM randomly, e.g., glayer. ½g

vq ←Choose a node vq from Γ sj based on the communication weight and the roulette selection. xasj ←xaq and xbsj ←xbq . end if end for Xo ←add xa and xb into offspring population. end for xr ←choose the solution with the largest Qe value in Xo .

Crossover. The two-way crossover work on two parent chromosomes xa and xb with probability pc. The two-way crossover selects a node (e.g., vp) randomly, and then finds the cluster of xa (xb ) that has the same cluster label as xap. Next, the found cluster of xa (xb ) is assigned to the chromosome xb (xa ). Fig. 6 gives an illustration of the two-way crossover (node v1 is chosen) on two solutions xa and xb . After crossover, the cluster plotted with circle in xa is assigned to xb . Meanwhile, the cluster plotted with triangle in xb is assigned to xa .

Fig. 6. Schematic illustration of the crossover on parent solutions xa and xb . Here, v1 is chosen. Nodes in different dotted circles are in different clusters.

1) is to generate an initial population of network partitions. It incorporates the layer node neighborhood defined in Section 4.2, and it is devised here based on the following principles: in weighted multiplex networks: i) all types of links are important and they are independent of each other; ii) two linked nodes are easier to be assigned to the same cluster than two unconnected nodes; and iii) tightly linked nodes are easier to be allocated to the same cluster than two loosely linked nodes. The Initialization() work on each chromosome with xij ¼ j, i ¼ 1; 2; …; Np and j ¼ 1; 2; …; N, and it is described as follows. Firstly, a random sequence is generated fs1 ; s2 ; …; sN g. Secondly, for each sequential node vsj , one of types of layer node neighborhood is chosen randomly, and then the node vsj is assigned to the cluster of its layer node neighborhood based on the communication weight and the roulette selection. Fig. 5 gives an illustration of our initialization on the node v1 of the multiple network G1M .

Mutation. The mutation work on the chromosomes generated by the crossover and it incorporates the layer node neighborhood knowledge. For each chromosome, each node in a randomly generated sequence is assigned to the cluster of one of nodes in its layer node neighborhood with a predefined mutation probability pm. And the assignment processes of the mutation are similar to those of the initialization. The combination of the chromosome mutation with the layer neighborhood knowledge makes it easier for two nodes with more types of communications to be the same community, which can generate mutated composite community partitions with good redundancy property. Moreover, this network-specific mutation avoids improper community formations in multiplex networks. Function 3. Multilevel-Local-Search()

The combination of the population initiation with the layer neighborhood knowledge makes it easier for two nodes with more types of communications to be the same community, which can generate initial composite community partitions with good redundancy property. Moreover, the combination of the population initiation with the weighted communication information makes it easier for two nodes with a higher communication weight to be the same community. In addition, this initialization avoids improper community formations in multiplex networks, i.e., two entities with no communications in all layers are initially assigned to the same community. 4.1.2. Network-specific genetic operation The function Genetic-Operation() (shown in Function 2) is composed of two steps: crossover (lines 4–9) and mutation (lines 10–17). Here, a widely used crossover (i.e., a two-way crossover [30]) is adopted to execute the crossover operation, and a multiplex based mutation is devised which incorporates the layer node neighborhood of multiplex networks.

1: 2: 3: 4: 5: 6: 7: 8:

Input: local chromosome: xr ; global chromosome: xg ; a multiplex network: GM. Output: local chromosome: xr . % Low-level node local search: lines 4–10 Γ← Construct consensus node neighborhood. fs1 ; s2 ; …; sN g ← Generate a random sequence. While xr is not changed do forj ¼ 1 to N xrsj ←argmaxxrq ;q2Γs ΔQe ðvsj ; xrq Þ.

9: 10: 11: 12: 13: 14: 15: 16:

end for end while % Medium-level cluster local search: lines 12–18 χ ← Construct consensus cluster neighborhood. fs1 ; s2 ; …; sNc g ← Generate a random sequence. while xr is not changed do for j ¼ 1 to Nc do xrcsj ←argmaxxrc ;cb 2χ cs ΔQe ðcsj ; xrcb Þ.

17: 18: 19: 20: 21:

end for end while % High-level partition local search: lines 20–21 xc ← Construct a consensus cluster partition between xr and xg . xr ← The operations in lines 4–18 are executed on xc to find a good solution xr .

j

b

j

Function 2. Genetic-Operation() 1: 2: 3: 4: 5: 6:

Input: Parent population: Xp ; a multiplex network: GM; parent population size: No; crossover probability: pc; mutation probability: pm. Output: Offspring population: Xo ; local chromosome xr . for i ¼ 1 to No =2 do fxa ; xb g← two chromosomes from Xp are chosen randomly. vp ←a node is chosen randomly; if (A random value is smaller than pc Þ then

4.1.3. Problem-specific multilevel local search The problem-specific multilevel local search (shown in Function 3) is to refine the local chromosome xr , and it incorporates network-specific knowledge (i.e., the consensus node neighborhood, the consensus cluster neighborhood, and the partition neighborhood) and problem-specific information (i.e., the fast computation of multiplex modularity under each refinement ΔQe ) into the search processes, which are detailed in

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node vp in i-th layer consists of all nodes that have links with the node in

Section 4.2. The multilevel local search is composed of three phases: i) low-level node local search, ii) medium-level cluster local search and iii) high-level partition local search. Low-level node local search. It is to refine the local chromosome xr by tuning the cluster label of nodes, and it works as Function 3 in lines 4–10. Firstly, a random sequence is generated (s1 ; s2 ; …; sN ), where N is the number of nodes. Then, each sequential node is assigned to the cluster of its consensus node neighborhood for which the ΔQe > 0 is the largest. The two steps iteratively operate when none of nodes changes its cluster label. Medium-level cluster local search. It is to refine the local chromosome xr by tuning the cluster label of clusters, and it works as Function 3 in lines 12–18. Firstly, a random sequence is generated (s1 ;s2 ;…;sNc ), where Nc is the number of clusters in xr . Then, each sequential cluster is assigned to the cluster of its consensus cluster neighborhood for which the ΔQe is the largest. The two steps iteratively operate when none of clusters changes its cluster label. High-level partition local search. It is to refine the local chromosome xr by tuning the cluster label of partitions, and it works as Function 3 in lines 20–21. Firstly, a consensus partition xc between xr and xg is generated (i.e., if they are in the same cluster of both xr and xg , the nodes will be assigned to the same cluster in xc . Otherwise, they will be assigned to different clusters). Then, the node local search and the cluster local search are applied to xc to find a good solution xr . The consensus partition xc is the partition neighborhood of both xr and xg . The node local search and the cluster local search are cluster grouping techniques while the partition local search is a combination of cluster division and cluster grouping. The combination of these cluster grouping and cluster division techniques enables MMCD to find good composite communities in multiplex networks with high multiplex modularity and redundancy values.

½i

this layer network (i.e., Γ p ¼fvq jA½i pq > 0g). Definition 7: consensus node neighborhood. The consensus neighborhood Γ p of a node vp is composed of all layer node neighborhoods (i.e., ½i

Γ p ¼ fvq j9i; vq 2 Γ p g). ½i

Definition 8: layer cluster neighborhood. The layer neighborhood χ a of a cluster ca in i-th layer consists of all clusters that have edges with the ½i

cluster in this layer network (i.e., χ a ¼fcb j9vp 2 ca ; 9vq 2 cb ; A½i pq > 0g). Definition 9: consensus cluster neighborhood. The consensus neighborhood χ a of a cluster ca is composed of all layer cluster neighborhoods (i.e.,

χ a ¼ fcb j9i; cb 2 χ ½i a g). Definition 10: partition neighborhood. The neighborhood of a partition x is composed of a set of partitions that have division and grouping relationships with the partition. Problem-specific information. In our local search, each refinement operation (i.e., the migration of an isolated node vp to a cluster cb) will result in the change of Qe (i.e., ΔQe ðvp ; cb Þ). From Eq. (4), we can know P that the complexity of computing Qe is Oð αi¼1 m½i Þ, where α and m½i denote the number of layers and the number of links at i-th layer, respectively. For a large-scale or high-dimension network, it takes a lot of computation resources for each refinement. Here, in order to reduce the computation complexity, we rewrite Qe as follows. Qe ¼

" # 2 α X 1 X ðK ½i c Þ 2l½i ;  c 2 m½i 2 m i¼1 c

(5)

where l½i c is the number of links within cluster c in i-th layer network, and ½i

K c denotes the degree of community c in i-th layer network. Moreover, based on Eq. (5), we compute ΔQe ðvp ; cb Þ as follows.

2 13 0 " # " # 2 ½i 2 ½i ½i ½i 2 α α ðK ½i ðk ½i k ½i k ½i 1 X 1 X 1 X 6 ½i B ½i ðK cb Þ p Þ C7 p K cb p K cb cb þ k p Þ ½i ½i ½i 2l ¼ 2l ; ¼ ΔQe ðvp ; cb Þ ¼      42ðlcb þ lðp;cb Þ Þ  A 5 @2l cb ðp;cb Þ ðp;cb Þ 2 m i¼1 2 m i¼1 2 m ½i 2 m½i 2 m½i 2 m½i m½i m½i

(6)

i;Γ p 6¼½ 

The consensus neighborhoods of multiplex networks interpret the topological communications of nodes in a global view. The combination of the multilevel local searches with the consensus neighborhoods of multiplex networks can promote the search of all promising solution regions, which makes it possible to find composite communities with high multiplex modularity value.

½i

where lðp;cb Þ is the number of links between node vp and cluster cb in i-th layer network. Then, the complexity of the computation of ΔQe ðvp ; cb Þ is P ½i ½i Oð αi¼1 k Þ, where k is the average degree of nodes in i-th layer ½i

network. Generally, k  m½i . 4.3. Computational complexity of MMCD

4.2. Network-specific knowledge and problem-specific information

MMCD is an iterative algorithm, and it is mainly composed of the genetic operations and the multilevel local search in iterations. From function Genetic-Operation() (shown in Function 2), we can see that the For loop in lines 3–19 is executed No =2 times. Within the For loop, i) the operations in lines 4–10 are executed 2N times at most and ii) the For loop in lines 11–17 is executed N times. Within the For loop in lines

Network-specific knowledge. The entities and their communications are two basic elements of multiplex networks [8]. Here, the entities can be represented by the low-level nodes, the medium-level clusters and the high-level partitions, and the communications can be represented by the low-level links and the high-level division and grouping relationships. Following these representations, we devise the low-level node neighborhood, the medium-level community neighborhood and the high-level partition neighborhood information of multiplex networks, and use them to guide the research. Moreover, based on the complex link structures of multiplex networks, we construct two types of neighborhoods: the layer neighborhood and the consensus neighborhood. Given a community partition x ¼ fx1 ; x2 ; …; xN g and a multiplex network GM ¼ fV; A½1 ; A½2 ; …; A½α g, we give related definitions as follows.

½i

11–17, it takes Oð1Þ, Oðk Þ, and Oð1Þ to execute the operations in line 13, ½i

line 14 and line 15, respectively, where k is the average node degree of the network at layer i. Moreover, the fitness computation of each indiP vidual has a computational complexity Oð αi¼1 m½i Þ. Therefore, the geP ½i netic operations has a computational complexity OðNo ðNk þ αi¼1 m½i ÞÞ, where α is the number of types of links. From Function 3, we can see that the Multilevel-Local-Search() consists of three parts. The low-level node local search in lines 4–10 has a While loop. Within the While loop, the For

½i

Definition 6: layer node neighborhood. The layer neighborhood Γ p of a 183

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P ½i loop in lines 6–10 is executed N times, and it takes Oð αi¼1 k Þ for each round within the For loop. Generally, the While loop is executed logN times. Therefore, the computational complexity of the node local search P ½i is OðNlog N αi¼1 k Þ. Similarly, the cluster and the partition local P ½i searches have a computational complexity OðNðlog NÞ2 αi¼1 k Þ. For a Pα ½i Pα ½i sparse network, i¼1 k N  i¼1 m , and therefore MMCD has a P computational complexity Oðgmax ððlogNÞ2 þ No Þ αi¼1 m½i Þ, where gmax is the maximum number of generations.

Table 1 Parameter settings of MMCD. Parameter

Definition

Empirical value

Np No pc pm gmax

Population size Parent population size Crossover rate Mutation rate Number of generations

200 20 0.9 0.15 200

methods are used to tackle unweighted single-layered networks whereas our MMCD is adopted to handle weighted multiplex networks. Compared with unweighted single-layered networks, the weighted multiplex networks have more complicated communication topology, communication weight information and potential composite community structures, which results in that the methods in [18,44,30,15] cannot be directly apply to detect the composite community structure of weighted multiplex networks. In the network-specific knowledge. The methods in [18,44,30,15] consider the layer neighborhood knowledge of single-layered networks. MMCD takes into account both the layer neighborhood knowledge, the consensus neighborhood knowledge and the communication weight knowledge of multiplex networks. The comprehensive utilization of these knowledge can effectively enable MMCD to detect composite communities of weighted multiplex networks with high both redundancy and multiplex modularity properties. In the problem-specific information. MMCD has different problemspecific information from the methods in [18,44,30,15]. More specifically, MMCD uses the simplified multiplex modularity criterion and the fast computation of local refinement to guide the search processes. The existing memetic community detection methods adopt the community criteria of single-layered networks, i.e., modularity [39] and modularity density [26], to guide their search processes. Moreover, some of them (e.g., our previous method [30]) also use a fast local refinement (a modularity based label propagation refinement) to accelerate the search processes.

4.4. Comparisons between MMCD and previous methods Comparisons between MMCD and the composite community definition based methods [36,1,21]. They firstly give a definition of composite community, and then use heuristic algorithms to discover the communities. However, they are totally different in the following aspects. In the community definitions. GenLouvin [36] and MMCD use a multiplex modularity which takes into account the composite community structures over all network layers to define composite communities in multiplex networks. Compared with GenLouvin, MMCD adopts a simplified version of multiplex modularity, which increases the sensitivity of the community criterion to different composite community structures. MultiMOGA [1] uses the modularity and the normalized mutual information criteria to define composite communities simultaneously. The modularity is used to define the community structure of multiplex networks in one layer, while the normalized mutual information is adopted to define the similarities of communities in any two neighboring layers of networks. mux-Licod [21] uses the seeds of networks and their neighboring nodes to represent the composite communities of multiplex networks. In the optimization frameworks. GenLouvin and mux-Licod use heuristic greedy optimization frameworks which are local optimizers. MultiMOGA adopts a heuristic evolutionary optimization framework which is a global optimizer. MMCD uses a memetic optimization framework which is a hybrid global-local optimizer. This global-optimal optimizer comprehends the good explorations of global optimizers and the fast exploitations of local optimizers. In the network-specific knowledge. GenLouvin uses the consensus neighborhood knowledge to guide the merging of two small neighboring communities to a large community, which makes it possible for find composite communities with high multiplex modularity value. muxLicod uses the consensus node degree knowledge to guide the search of seeds of communities, and MultiMOGA uses the consensus node neighborhood knowledge to guide the genetic operations. MMCD uses both the consensus neighborhood knowledge, the layer neighborhood knowledge and the communication weight knowledge to guide the genetic operations and the multilevel local searches, which enables a comprehensive understanding of composite community structures in terms of their redundancy property and multiplex modularity property. Comparisons between MMCD and the memetic community detection methods [18,44,30,15]. They use the memetic optimization techniques to identify the community structure of networks, and incorporate the network-specific knowledge and problem-specific information. In the motivations. The existing memetic community detection methods [18,44,30,15] are to either detect the multiresolution community structures of small-scale single-layered networks or discover the community structures of large-scale single-layered networks. They have good performances on the detection of community structure in single-layered networks. However, they cannot be used to detect the composite community structures of multiplex networks. MMCD is proposed to i) solve the limitations of existing composite community detection algorithms and ii) extend the existing memetic community detection methods, especially for our previous work [30], to discover more complicated community structures in multiplex networks. In the tackled objects. The existing memetic community detection

5. Experimental analyses In this section, eight real multiplex networks are chosen as our experimental networks, and three criteria (i.e., Qe, redundancy Re [46] and modularity Q [39]) are comprehensively used to test the quality of the detected composite communities. Meanwhile, comparisons between MMCD and five classical community detection algorithms are made to demonstrate the superior performance of MMCD. 5.1. Simulation settings 5.1.1. Parameter settings and comparison algorithms Parameter settings. In MAs, there are five common parameters, i.e., maximum number of generations gmax, population size: Np; parent population size: No; crossover probability: pc and mutation probability: pm. All parameters of MMCD are set to empirical values that are widely chosen in other MA based community detection algorithms [18,44,30,55, 15], and their settings are shown in Table 1. Although they are not optimal, these parameter settings can effectively converge MMCD to good composite community divisions. Comparison Algorithms. Our comparison algorithms include four classical composite community detection algorithms (described in Section 2.2), including a community definition based optimization algorithm: GenLouvin [36], two layer aggregation algorithms: CLPAm [4,48] and Cinfomap [43,48], and a consensus clustering algorithm: CBGLL [24]. Comparisons between MMCD and the four algorithms are made to demonstrate the superior performance of MMCD on the quality of detected composite communities. To make fair comparisons between MMCD and GenLouvin, we set the objective function of GenLouvin as the 184

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Table 2 Detailed properties of the tested multiplex networks. N, α and m denote the number of nodes, the number of types or layers of links, and the number of communications, respectively. Multiplex Networks

Node

Edge

N

α

m

Weighted ?

Directed?

Field

London Transport CKM Physicians Innovation EU-Air Transportation FAO Trade Plasmodium GPI Celegans GPI Arabidopsis GPI MUS GPI

Train Station physicians City Country Plasmodium Elegan Elegan Elegan

Routing Social relation Airline routing Import/export trade Genetic connection Genetic connection Genetic connection Genetic connection

369 246 450 183 1203 3879 6980 7747

3 3 37 339 3 6 7 7

441 1551 3588 16,048 2521 8181 18,654 19,842

Yes No No Yes No No No No

No Yes No Yes Yes Yes Yes Yes

Transport Social Transport Financial Genetic Genetic Genetic Genetic

simplified multiplex modularity, which has no effect on its optimization processes. Moreover, a simple version of MMCD (called as GACD), which neglects the multilevel local search, is also chosen to compare with MMCD, which is to demonstrate that the multilevel local search can effectively improve the effectiveness and accelerate the convergence of GACD.

The Plasmodium GPI multiplex network, collected from Interaction Datasets (BioGRID, thebiogrid.org), represents the genetic connections of plasmodium falciparums in the Biological General Repository. It consists of three layers of interactions: direct interaction, physical interaction and association. The Celegans GPI multiplex network, collected from BioGRID, denotes the genetic connections of Caenorhabditis elegans. It consists of six layers of interactions: direct interaction, physical interaction and inequality based additive interaction, inequality based suppressive interaction, association and delocalization. The Arabidopsis GPI multiplex network, collected from BioGRID, considers the genetic connections of Bos Linnaeus. It consists of seven layers of communications: direct interaction, physical association, inequality additive genetic communication, inequality suppressive genetic communication, inequality synthetic genetic communication, association and colocalization. The MUS GPI multiplex network, collected from BioGRID, denotes the genetic connections of Bos Linnaeus Mus musculus. It consists of seven layers of communications: physical association, association, direct communication, colocalization, inequality synthetic genetic communication,inequality additive genetic communication and inequality suppressive genetic communication. All multiplex networks considered here are their undirected and weighted version, and their weights at each layer are computed as follows.

5.1.2. Real-world multiplex networks Eight real multiplex networks (shown in Table 2) are used to test the performance of MMCD, and they can be downloaded from http://deim. urv.cat/~manlio.dedomenico/data.php. The experimental networks are chosen from multiple fields, including society, transportation, economy and genetic engineering. The tested networks are composed of two small-scale networks (i.e., London Transport and CKM Physicians Innovation), two high-dimensional networks (i.e., EU-Air Transportation and FAO Trade), two medium-scale networks (i.e., Plasmodium GPI and Celegans GPI) and two large-scale networks (i.e., Arabidopsis GPI and MUS GPI). In the following, their detailed descriptions are given. The London Transport multiplex network, collected from the official website of Transport (i.e., https://www.tfl.gov.uk/), represents routing relations of train stations in London. The multiplex relations of train stations can be further described by the routing connection in the following three layers: underground, overground and DLR. The CKM Physicians Innovation multiplex network reflects the influence of social relations on the physicians’ adoption of a new drug or an innovative technique. Three layers of social relations were generated based on the social questions about the information sources, the weekly discussion and the physicians’s name. The EU-Air Transportation multiplex network denotes the airline routing in the cities of Europe. It has 37 layers or types of connections, and each type corresponds to a different airline operating. The FAO Trade multiplex network, collected from Food and Agriculture Organization of the United Nations, reflects the import/export trade relationships among countries. It has 183 types of edges, and each type of edges correspond to import/export relationships of a different food product among countries.

A½i pq ¼

8 > <

½i A½i pq þ Aqp ½i ½i > ðApq þ Aqp Þ=2 : 0

½i if A½i pq > 0 or Aqp > 0; ½i if Apq > 0 and A½i qp > 0; otherwise:

(7)

5.1.3. Metrics As the real communities of tested multiplex networks are unknown, classical supervised indexes (i.e., Normalized mutual information) cannot be applied to evaluate the quality of their composite communities. Here, three unsupervised indexes, i.e., the simplified multiplex modularity Qe, the redundancy Re, and the single-layered modularity Q, are

Table 3 a s Experimental performance Qe of comparison algorithms on real multiplex networks. Qm e , Qe and Qe denote the maximum value, the average value and the standard deviation of Qe in 30 independent trials, respectively. The best values are highlighted in bold. Algorithm

Qe

London Transport

CKM Physicians Innovation

EU-Air Transportation

FAO Trade

Plasmodium GPI

Celegans GPI

Arabidopsis GPI

MUS GPI

Cinfomap

Qm e/ Qae Qm e/ Qae Qm e/ Qae Qm e Qae Qse Qm e Qae Qse

0.7456

0.6579

0.012

0.3113

0.4601

0.4905

0.6576

0.6108

0.7352

0.6833

0.0479

0.2378

0.4734

0.4732

0.6471

0.6273

0.7892

0.6944

0.1154

0.3174

0.5183

0.5445

0.7067

0.6794

0.5496 0.5352 0.0064 0.7956 0.7932 0.0012

0.6169 0.5702 0.0284 0.7058 0.7056 0.0008

0.1122 0.1033 0.0045 0.1235 0.1219 0.0009

0.3152 0.3124 0.0010 0.3178 0.3174 0.0002

0.3456 0.3406 0.0023 0.5310 0.5237 0.0045

0.4099 0.3998 0.0043 0.5571 0.5524 0.0037

0.6052 0.5700 0.0193 0.7199 0.7165 0.0041

0.5208 0.5106 0.0052 0.6875 0.6840 0.0020

CBGLL GenLouvin CLPAm

MMCD

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Fig. 7. Composite community division of (a) the London Transport multiplex network and (b) the CKM Physicians Innovation multiplex network. Nodes in different communities are plotted with different colors. Here, isolated nodes are omitted.

used to evaluate the obtained composite community divisions of multiplex networks. Qe evaluates the fraction of composite communications of multiplex networks that fall in the detected communities, and its computation is shown as Section 3. The redundancy Re evaluates the fraction of redundant links of intracommunities in multiplex networks. The intuition behind this criterion is that the connected entities within the same community should have connections in multiple layers. Re is computed as follows [46,21]. P 1 X ðp;qÞ2Ec Re ¼ jcj c

layered community information of multiplex networks. The modularity Q(i) for i-th layer network is computed as follows [39] QðiÞ ¼

c

α

i¼1

l½i c

;

c m½i

 

K ½i c 2 m½i

2  :

(9)

A network partition with high Q(i) value corresponds to a good composite community division in i-th layer network.

jfjj9A½j pq > 0gj α P

X l½i

5.2. Experimental results (8) Table 3 records the statistical results Qe (i.e., maximum, mean and standard deviation of Qe) of comparison algorithms over 30 independent trials. From Table 3, we can see that i) Cinfomap and CBGLL have low a values of Qm e and Qe for the high-dimensional EU-Air Transportation and FAO Trade multiplex networks. This is because the layer aggregation approach would result in the missing of heterogeneous link properties while the ensemble clustering technique would lead to the production of spurious link information; ii) CLPAm cannot work well for all tested multiplex networks except for the FAO trade network. CLPAm neglects the separation of merged clusters and the global link properties of

where jcj is the number of communities, Ec represents all links in the community c, and ðp; qÞ denotes a link between vp and vq in one layer of multiplex networks. A partition of multiplex networks with high Re value corresponds to a good composite community division. The modularity Q is widely used to evaluate the community quality in single-layered networks. Here, the criterion Q is adopted to demonstrate whether the detected composite communities can well reflect the single-

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Fig. 8. Comparisons between MMCD, Cinfomap, CBGLL, GenLouvin and CLPAm on real multiplex networks in terms of Re. (a) London Transport. (b) CKM Physicians Innovation. (c) EU-Air Transportation. (d) FAO Trade. (e) Plamodium GPI. (f) Celegans GPI. (g) Arabidopsis GPI. (h) MUS GPI.

networks, and thus it is easy to get into a solution with low-quality composite communities; iii) MMCD and GenLouvin have high values of a Qm e and Qe for all tested multiplex networks. MMCD and GenLouvin use heterogeneous link properties to define the composite community, and they incorporate these network-specific information into their search strategies (i.e., GA and local search in MMCD, and community grouping and community division in GenLouvin). Therefore, MMCD and GenLouvin can find composite community divisions with high Qe values; And a iv) of comparison algorithms, MMCD obtains the largest Qm e and Qe values for all tested multiplex networks, which demonstrates its superior performance on the detection of composite communities. Table 3 also illustrates that CBGLL, GenLouvin and Cinfomap are deterministic algorithms whereas CLPAm and MMCD are nondeterministic algorithms. CBGLL and GenLouvin adopt a fixed optimization order (i.e., from the first node/cluster to the last node/cluster) in the optimization processes while Cinfomap adopts a certain rule based search strategy (i.e., random walker). Therefore, CBGLL, GenLouvin and Cinfomap can obtain the same community division under different independent trials. In order to enhance avoid trapping into a certain search region, CLPAm adopts a random optimization order strategy in the community grouping processes while MMCD uses genetic operations (i.e., crossover and mutation) and a random optimization order strategy in the local search. Therefore, CLPAm and MMCD obtain non-

deterministic solutions under different independent trials. Note that, MMCD can obtain robust solutions (i.e., its Qse values are small for all networks). Moreover, the Qse values of MMCD are far less than those of CLPAm for most of networks, which demonstrates that MMCD is more robust than CLPAm. MMCD is a robust algorithm because it adopts a hybrid global-local search framework and incorporates network-specific knowledge (i.e., the layer neighborhood and the consensus neighborhood) into the optimization processes. The hybrid search framework uses a global search (i.e., network-specific GA) to search a population of promising community division regions, and it adopts a local search (i.e., problem-specific multilevel local search) to accelerate the convergency of the algorithm to optimal community partitions around these regions. The incorporated network-specific information can effectively help MMCD avoid useless search processes (e.g., the grouping of unconnected nodes to a community and the division of linked nodes sparsely into different communities). In order to illustrate the detected composite communities clearly, we plot the corresponding community division of our best solution on two small-scale multiplex networks (i.e., London Transport and CKM Physicians Innovation) in Fig. 7. From Fig. 7(a), we can see that i) our solution with Qe ¼ 0:7956 corresponds to a composite community division of London Transport network with 21 communities; ii) the communities of the network at each layer (e.g., Underground, Overground or DLR)

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Table 4 Experimental performance Qe &Re of comparison algorithms on real multiplex networks. Each term in Qe &Re is labeled as ‘good’ (‘bad’) if the term value is larger (smaller) than the mean value of the term values obtained by all comparison algorithms. The best values are highlighted in bold. Algorithm

London Transport

CKM Physicians Innovation

EU-Air Transportation

FAO Trade

Plasmodium GPI

Celegans GPI

Arabidopsis GPI

MUS GPI

Cinfomap CBGLL GenLouvin CLPAm MMCD

good&bad good&bad good&good good&bad good&good

bad&bad good&bad good&bad bad&good good&bad

bad&good bad&bad good&bad good&bad good&good

good&bad bad&good good&good good&bad good&good

bad&good good&bad good&bad bad&bad good&good

bad&good bad&bad good&bad bad&bad good&good

bad&good bad&bad good&good bad&good good&bad

bad&good good&bad good&bad bad&good good&bad

Fig. 9. Comparisons between MMCD, Cinfomap, CBGLL, GenLouvin and CLPAm on real multiplex networks in terms of modularity under each layer. (a) London Transport. (b) CKM Physicians Innovation. (c) EU-Air Transportation. (d) FAO Trade. (e) Plamodium GPI. (f) Celegans GPI. (g) Arabidopsis GPI. (h) MUS GPI.

cannot completely reflect the composite communities of the multiplex network (i.e., the communities at aggregated layer); and iii) the communities of the network at different layers are different. Therefore, it is necessary to find the composite communities that are most likely to describe the community structures of the London Transport network at all layers. From Fig. 7(b), we can see that the CKM Physicians Innovation multiplex network is mainly divided into 8 communities with Qe ¼ 0:7058. At each layer, the network has high-quality communities because nodes with the same color are connected densely whereas these with different colors have no communications. Our composite community division can effectively reflect the high-quality communities of the

network at different layers. Fig. 8 records the averaged Re of community divisions detected by comparison algorithms over 30 independent trials. The results illustrate that i) of comparison algorithms, MMCD obtains the largest Re values for the London Transport, Plamodium GPI and Celegans GPI multiplex networks; ii) MMCD only obtains smaller Re values than CLPAm for the Physicians Innovation network, Cinfomap for the EU-Air Transforation network and CBGLL for the FAO Trade multiplex networks, respectively. Note that, CLPAm, Cinfomap and CBGLL obtain smaller Qe values than MMCD for all tested networks; iii) GenLouvin obtains smaller Re values than MMCD for most of networks, especially for the London Transport, 188

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Fig. 10. Comparisons between MMCD and GACD on real multiplex networks in terms of Qe VS. number of generations. (a) London Transport. (b) CKM Physicians Innovation. (c) EU-Air Transportation. (d) FAO Trade. (e) Plamodium GPI. (f) Celegans GPI. (g) Arabidopsis GPI. (h) MUS GPI.

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algorithm (i.e., the missing of heterogeneous link properties), the CBGLL (i.e., the production of spurious linking information) and the Cinfomap algorithm (i.e., neglecting the separation of merged clusters and the global link properties of networks). The layer aggregation approach and the consensus approach have high Qe values for some comparison networks. This is because their operations make it easier for two nodes with more types of communications to be the same community. Fig. 10 shows the comparisons between MMCD and GACD on real multiplex networks in terms of Qe VS. number of generations. As shown in Fig. 10, GACD is hard to converge to optimal solutions within 200 generations whereas MMCD can quickly converge to optimal solutions. This demonstrates that i) MMCD has better convergency than GACD; and ii) MMCD can converge certain composite community divisions for all tested multiplex networks using the parameter setting in Table 1. Table 5 shows the comparative results of Qe between GACD and MMCD on real multiplex networks. The results illustrate that MMCD has higher average values of Qe than GACD, which indicates that MMCD has better performances than GACD for the detection of composite communities in multiplex networks. Due to the multiplex information about heterogeneous links, GACD cannot work well for any tested multiplex networks, even for the small-scale London Transport and CKM Physicians Innovation networks. MMCD uses a problem-specific local search to accelerate the search of GACD to promising solutions, and it can effectively find composite community structures with high quality. The results also demonstrate that MMCD has smaller standard deviation values of Qe than GACD, which indicates that MMCD is more robust than GACD. In the optimization of Qe, the solution space increases with the increasing of the scale and the dimension of multiplex networks. MMCD uses a local search strategy (i.e., multilevel local search) to help GA to get into optimal solutions around the found solution regions, and thus it has better stability than GACD.

Table 5 Comparisons between GACD and MMCD on real multiplex networks in terms of Qe (i.e., mean standard deviation). All results are recorded in 30 independent trials. The best values are highlighted in bold. Networks

GACD

MACD

London Transport CKM Physicians Innovation EU-Air Transportation FAO Trade Plasmodium GPI Celegans GPI Arabidopsis GPI MUS GPI

0.5729 0.0100 0.5513 0.0464 0.0022 0.0025 0.2506 0.0093 0.2993 0.0041 0.2572 0.0050 0.2270 0.0045 0.2328 0.0049

0.7932 0.0012 0.7056 0.0008 0.1219 0.0009 0.3174 0.0002 0.5237 0.0040 0.5524 0.0037 0.7165 0.0041 0.6840 0.0020

EU-Air Transforation Plamodium GPI and Celegans GPI multiplex networks. This is because compared with GenLouvin, MMCD takes into account both the consensus neighborhood and the layer neighborhood information in the optimization processes; and iv) in large-scale networks, MMCD has lower Re values than Cinfomap and CLPAm. This is because the solution regions of problems are increased with increasing the scale and the genetic operations of MMCD are hard to find composite communities with good redundancy property in a given generation for large-scale networks. Fig. 8 and Table 3 show that there is no clear correlation between Qe and Re. It is difficult for most of comparison algorithms to detect composite communities with high both Qe and Re values. Table 4 gives the comprehensive performances of comparison algorithms on the detection of composite communities. In Table 4, each term (Qe &Re ) is composed of two parts (i.e, Qe and Re). Each part is labeled as ‘good’ (‘bad’) if the corresponding criterion value is larger (smaller) than the averaged value of the criterion obtained by all comparison algorithms. The results in Table 4 demonstrate that i) CBGLL, Cinfomap and CLPAm have no ‘good’&‘good’ results for tested networks; ii) our algorithm MMCD can obtain ‘good’&‘good’ results for most of networks; iii) although it can get ‘good’ & ‘good’ results for the London Transport and FAO Trade networks, GenLouvin has smaller both Qe and Re values than MMCD. Comprehensive comparisons in Table 4 further demonstrate the superior performance of MMCD on the detection of composite communities compared to the state-of-the-art. Fig. 9 records the Q values of the communities of multiplex networks under each layer. Figs. 9(a), 9(b) and 9(e) illustrate that MMCD has larger modularity values than Cinfomap, CBGLL, GenLouvin and CLPAm for all layers of multiplex networks. Figs. 9(c) and 9(d) show that MMCD and GenLouvin obtain higher Q values than CBGLL, Cinfomap and CLPAm for most layers of multiplex networks. Figs. 9(f), 9(g) and 9(h) demonstrate that for the Celegans GPI network, the Arabidopsis and the MUS GPI networks, MMCD only obtains smaller Q values than GenLouvin in most layers of networks. Note that, MMCD has a larger Qe value than GenLouvin for these networks. This demonstrates that the communities in most layers of multiplex networks cannot represent the composite communities of multiplex networks. Generally speaking, for most experimental networks, MMCD has larger both Qe and Re values than GenLouvin. This is because i) MMCD uses a global search (i.e., network-specific GA) to search a population of promising community division regions, and it adopts a local search (i.e., problem-specific multilevel local search) to accelerate the convergency of the algorithm to optimal community partitions around these regions. The incorporated network-specific information can effectively help our algorithm avoid useless search processes (e.g., the grouping of unconnected nodes to a community and the division of linked nodes sparsely into different communities); and ii) MMCD takes into account both the consensus neighborhood and the layer neighborhood information in the optimization processes. For most experimental networks, MMCD has larger both Qe and Q values under each layer than Cinfomap, CBGLL and CLPAm. This is because MMCD is a community definition based optimization method, which avoids the limitations of the Cinfomap

6. Conclusions With the development of information technique, many real systems enable entities to communicate with each other through multiple platforms. These heterogeneous communication information make it harder for existing community detection algorithms to reveal the composite communities of multiplex systems. In this paper, we proposed a community definition based community detection algorithm (i.e., multilevel MA) for the revelation of composite communities in weighted multiplex networks. The proposed algorithm used a simplified multiplex modularity to evaluate the quality of composite community structures, and combined a network-specific GA with a problem-specific multilevel local search to find high-quality composite communities in weighted multiplex networks. Moreover, based on the lower-order connectivity patterns of multiplex networks, we constructed higher-order node, cluster and partition neighborhoods. The proposed algorithm incorporated the link weight, the layer neighborhood and the consensus neighborhood into GA and the local search, and it introduced the problem specific information (i.e., the computation of ΔQe ) into the multilevel local search. The problem-specific information and network-specific information can effectively promote the proposed algorithm to find promising solution regions and accelerate the convergency to high-quality composite community divisions. Systematic experiments on eight real multiplex networks demonstrated that i) our algorithm has better performances than the state-of-the-art in the detection of composite communities; ii) the multilevel local search can effectively improve the effectiveness, stability and convergency of the network-specific GA; and iii) the community definition based methods work well for the composite community detection in all tested multiplex networks whereas the consensus clustering and the layer aggregation techniques are failed to find the composite communities of high-dimensional multiplex networks. Here, we mainly find the composite community structures of weighted multiplex networks with thousands of nodes and hundreds of layers. However, in real-world systems, e.g., online communication, 190

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interactive online game and recommendation systems, there are millions of users and thousands of layers. Moreover, most of these systems show sparsity, dynamicity, spatiality, temporality, evolution and cooperation properties. It is still an open challenge to identify the potential functionality in these systems. Existing algorithms and MMCD cannot be well implemented at these systems. Recently, the parallel computation and network sampling techniques have widely been used to tackle real largescale optimization problems. In the nearly future, we aim to combine our multilevel MA with the parallel computation and network sampling to identify composite communities in large-scale and high-dimensional multiplex networks.

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