Multi-objective optimization of community detection using discrete teaching–learning-based optimization with decomposition
Introduction
Real-world networks involve many important and complex interactions among individuals [3], for example, social networks [46], biological networks [6], biochemical networks [13], and communication networks [22]. These networks can be naturally represented as complex networks. In mathematics and computer science, complex networks are often modeled as graphs. More specifically, a complex network consists of nodes (or vertices) and edges (or links), which represent the individual members and their relationships. Generally, the cluster or community is defined as a subset of vertices that are closely knit in a relatively sparse neighborhood. Hence, the cluster or community is composed of individuals with common properties inherent to complex networks [48]. Community detection or network clustering is essential in complex networks; it is one of the most relevant features of networks that represent real systems and a key aspect of network analysis, the purpose of which is to discover complex network topology structures and understand complex network functions [16]. Community detection or the network clustering problem is typically viewed as the division of networks into subsets of vertices according to their similarity, other characteristics related to the nodes and edges of the graph, or both. Naturally, it can be considered an optimization problem [2], [38].
The teaching–learning-based optimization (TLBO) algorithm [43] is a population-based algorithm. The algorithm requires common control parameters rather than algorithm-specific control parameters. Because TLBO is characterized by simple computation and rapid convergence, it is used in many optimization fields. Although the TLBO algorithm and its variants have achieved effective results in continuous optimization, little progress has been made for discrete TLBO variants. To solve job shop scheduling problems (JSSP), Keesari and Rao used a learner in the TLBO algorithm to represent an operation scheduling list in JSSP, where each dimension of a learner represents one operation of a job [27]. Xia et al presented a simplified teaching–learning-based optimization (STLBO) algorithm for solving disassembly sequence planning problems (DSP) with discrete solution spaces [50]. In this method, a Feasible Solution Generator (FSG) is used to generate a feasible disassembly sequence, and a Teaching Phase Operator (TPO) and a Learning Phase Operator (LPO) are used to learn and evolve the solutions towards better solutions by applying the method of precedence preservation crossover operation. Baykasoglu et al analyze the performance of the TLBO algorithm with respect to flow shop (FSSP) and job shop scheduling problems (JSSP) [5]. In the proposed method, a random key-based approach with the largest position value rule is used to obtain a job permutation for FSSP and JSSP, and G&T algorithm is used for the construction of active schedules within the TLBO search. Xu et al. proposed an effective teaching–learning-based optimization algorithm for the flexible job-shop scheduling problem with fuzzy processing time (FJSPF) [51]. In this method, a solution is expressed by an operations sequence vector and a machine assignment vector, and a decoding method is employed to transfer a solution to a feasible schedule in the fuzzy sense. Then, a bi-phase crossover scheme based on the teaching–learning mechanism and special local search operators are incorporated into the search framework of the TLBO to balance the exploration and exploitation capabilities. Dede utilized TLBO for the discrete optimization of truss structures [12]. Li et al. proposed a discrete teaching-learning-based optimization for solving the flow shop scheduling problem (FSSP) [29]. In the proposed DTLBO algorithm, four discretization operators are introduced into the teacher phase and learner phase and a modified iterated greedy (IG)-based local search is embedded to enhance the search ability of the proposed algorithm.
In essence, community detection in a complex network is a clustering optimization problem. For this reason, evolutionary computation and swarm intelligence can be used for the community detection or network clustering problem (Seen in Section 2.1). Compared with traditional algorithms, intelligent optimization algorithms can effectively find a proper, high-quality solution within a reasonable period of time. This paper attempts to apply the multi-objective discrete teaching–learning-based optimization algorithm based on decomposition (MODTLBO/D) to address communities in complex networks. First, unlike the above method, we proposed a novel discrete TLBO for community detection problems. In the proposed DTLBO, the problem-specific locus-based adjacency representation strategy is used to initialize a population with N learners in which each learner represents the cluster values of all nodes; the difference between the Teacher (the best learner in the current generation) and NMean (the mean solution of the local learners) of the learner's neighbors in the Teacher Phase and the difference between the learner and another learner, which is randomly selected from the class in the Learner Phase, are updated using the XOR operator; the niching mechanism is applied to update the new learner according to the learner and the difference; and the mutation operation is implemented after both the Teacher Phase and Learner Phase at each iteration instead of the duplicate elimination in the original TLBO. The features of complex networks are taken into account, and DTLBO is extended to MODTLBO/D, which is a multi-objective optimization variant with a decomposition strategy. Experimental results and relevant comparative analyses show that the proposed algorithm achieves high-quality detection results.
The remainder of this paper is organized as follows. Related works are introduced in Section 2. Section 3 describes the proposed method for community detection. Section 4 presents tests performed on real-world complex networks, and the experiments are conducted along with statistical tests. Conclusions are given in Section 5.
Section snippets
Literature review
In recent years, community detection in complex networks has attracted an increasing amount of attention, and several popular algorithms for community detection have been presented. Most community mining algorithms can be classified into several categories:
Graph partitioning-based methods: In these methods, the network is divided the vertices into a number of groups with pre-defined size so that the number of edges between groups is minimized. The Kernighan-Lin algorithm [14] and Spectral
Teaching–learning-based optimization
Rao et al. [43] first proposed a novel teaching–learning-based optimization (TLBO) inspired from the philosophy of teaching and learning. TLBO mimics the teaching and learning process of a typical class. In optimization algorithms, the population consists of different design variables. For the TLBO algorithm, a group of learners or a class of learners is considered the population. The teacher and learners are vital components of the TLBO algorithm. The output of the TLBO algorithm is considered
The DTLBO and MODTLBO/D algorithm
In this section, the proposed MODTLBO/D method for community detection problems is described. First, the motivation for proposing the discrete TLBO (DTLBO) algorithm for complex community detection problems is given. Next, the DTLBO algorithm is described in detail. Then, based on the proposed discrete DTLBO variant, a novel multi-objective discrete method (MODTLBO/D) is proposed. The following sections describe the updating strategy and implementation process of the proposed algorithm.
Application of MODTLBO/D to community detection
We used real-world datasets to test the efficiency of MODTLBO/D. To evaluate the performance of MODTLBO/D, we also simulated the representative single-objective algorithms (GA [44], FM [9], BGLL [7]) and multi-objective algorithms (MOCD [44], MOGA-net [40], MOEA/D-net [21], MODPSO [19]) and then presented a comparison of their results.
Conclusion
One of the simplest and most efficient techniques, TLBO has been empirically shown to perform well on many optimization problems. In this paper, a discrete variant of the teaching–learning-based optimization algorithm (DTLBO) is first proposed. In this discrete TLBO framework, the learner vectors and their updating operators have been redefined. Moreover, based on the proposed discrete variant DTLBO, a novel multi-objective discrete optimization algorithm is first presented to solve the
Acknowledgment
This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 61572224, 61304082) and the National Science Fund for Distinguished Young Scholars (Grants No.61425009). This work is also partially supported by the Major Project of Natural Science Research in Anhui Province (Grant No.KJ2015ZD36) and the Natural Science Foundation in colleges and universities of Anhui Province (Grant No.KJ2016A639).
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