Keywords

1 Introduction

A complex network is a web containing a collection of nodes connected through edges, for instance, the world wide web, technological networks, biological networks, brain networks, collaboration networks, online social networks, etc. Community detection (CD) problem deals with finding groups of nodes that have strong intracommunity connections and weak intercommunity connectivity. Investigating important nodes in such networks through community detection can provide better insights to analyze the quality of interconnections between different nodes. Community detection problem is considered to be NP-Hard due to the high complexity of the network structure [1]. It has numerous applications in social networks, healthcare, modeling of epidemic spreading on networks, business, fraud detection, communication networks, biological networks, etc. [1]. Previous studies done on community detection problems found in the literature have worked upon graph partitioning methods, hierarchical clustering approaches, genetic algorithms, and many other evolutionary algorithms and swarm intelligence-based techniques. As shown in Fig. 1 community detection problems can be studied for disjoint communities (no nodes common in 2 or more different communities) and overlapping community detection (nodes common in two or more communities). Many approaches used for detecting community structure in the literature have focused on static communities satisfying the Modularity fitness function for assessing the quality of partitions. Later on, algorithms for detecting dynamic communities concerning the temporal smoothness along with community partitions have also been examined [2, 3]. CD is considered as an optimization problem, so approaches used for community detection have used either single-objective function for optimization or multiobjective functions. Some of the metaheuristic approaches which employs random searching algorithm have been implemented with great efficiency resulting in global optimal solutions which are discussed in this paper as those using genetic algorithms and PSO. These metaheuristic approaches with heuristic operators are used by many researchers for detecting communities [4, 5].

Fig. 1
An organizational chart of classification of community detection algorithms. It is branched into disjoint and overlapping community detection algorithms; disjoint community detection algorithm is further branched into traditional, modularity based and dynamic algorithms, and the specifics of each are listed further.

Classification of Community Detection algorithms [7]

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1.1 Classification of Various Types of Algorithms Used in Literature for Community Detection Problem

Community Detection is an important direction of research in multidisciplinary areas. So many algorithms are classified and proposed in the literature by the scientists and researchers according to the dimension of the work chosen as enlisted in Fig. 1. The algorithms implemented for community detection can be broadly categorized into Graph partitioning, Clustering, and Genetic algorithms for disjoint communities and clique-based algorithms for overlapping community detection.

1.2 Classification of Various Methods of Community Detection Based on Social Network

As social networks are so vast and widespread according to the applications, the detection of community structure in different types of social networks needs to exploit different algorithms for the analysis. The community detection problem is intended to identify the highly interrelated nodes or vertices in a network within a group which is strongly communicating with each other.

Initially, most of the networks were static but later on with the widespread use of social networking, the networks became more dynamic in nature. The different types of social networks to be investigated in research work are static, signed, positive, dynamic, directed, and heterogeneous [6]. So, the aforementioned task is to:

  • Detect communities in static networks

  • Community finding in signed networks

  • Community detection in dynamic networks

  • Detection of community in positive networks

  • Detecting communities in heterogeneous networks

  • Community detection in a directed network.

The sequence flow of the paper is as follows “Literature Survey” discusses the background or the related work, “Contribution” describes the novel work done by previous researchers in the latest approaches used, and the “Conclusion” the final viewpoint concluding the review done in the paper.

2 Literature Survey

Traditional methods used for community detection inspired by clustering methods are those using hierarchical and partitioning methods [8, 9]. But these methods require high computational time and are inefficient to generate optimal solutions in a reasonable time. Also not found efficient for implementation in large-scale networks. The aforementioned issues are addressed very well by evolutionary algorithms using heuristic search. The primary methods used for discovering communities are those by optimizing single-objective function (SO) and using modularity(Q) as an evaluation function which computes intracommunity edges [10]. But using only one objective function may direct the evolving population to form a particular type of community structure or result in a resolution limit problem [11]. Thus, this issue can be addressed by using a multiobjective function (MO) for optimization. MO methods find an optimal solution by establishing a trade-off between different objectives [12]. The concept of Pareto optimality is employed by many evolutionary methods utilizing MO functions. Here in the case of community detection problem MO can find communities with dense intracommunity links and sparse intercommunity connections by optimizing two objective functions simultaneously.

A short glimpse of the previous work done on community detection reveals its societal impact using social networks. The valuable knowledge which can be drawn from studying community structures has led many researchers to investigate the literature behind it. Many comprehensive surveys focus on detecting community structures in multilayer networks [13].

The work presented by (Che et al., 2021) focused on community detection in two modes (bipartite graphs). Their work proposal includes an algorithm which is known as IABC-BN (artificial bee colony algorithm) for detecting communities in bipartite graphs. The experimental results have proved the ABC method to be an excellent algorithm for the discovery of clusters in two-mode graphs. The main contribution of this new algorithm seen is cluster partition for bipartite graphs [14].

Yin et al., (2020) approached the real problem occurring in dynamic networks. The proposed method used DYN-MODPSO for dynamic community detection is an improved evolutionary clustering framework. The multiobjective method is devised for large-scale dynamic networks using PSO. The basic idea is to detect the evolving community structures based on temporal intervals [15].

Reference [16, 17] proposed the use of genetic algorithms with multiobjective criteria to detect communities in complex networks using the algorithm MOGA-Net [18]. His work contributed to the first proposal of using multiobjective GA to discover communities. This algorithm used two objective functions which were optimized to identify partitions in the network structure. The first one uses a community score to evaluate meaningful partitions in the network called communities. A high value of community score corresponds to dense clustering. Another objective function called community fitness is used to analyze the fitness of the nodes confined to a certain group. Further, they extended their work for the application in dynamic networks using DYNMOGA optimizing modularity and Normalized Mutual Information(NMI) as fitness functions [19].

A particle swarm intelligence-based algorithm called MOPSO-Net was proposed by the authors [20]. Kernel k-means (KKM) and Ratio Cut (RC) are the objective functions to be minimized here. In each iteration, the swarm moved in the direction to achieve the global best solution using the NMI criterion. A Locus-based encoding scheme is used for representation and effective exploration of the solution space. In [21], the authors proposed a many objective(MaOPs) approach for community detection to address the challenges faced by multiobjective methods(using only 2 or 3 quality metrics) in community detection in multi-structural networks. Each quality measure has its specific property for detection thus ignoring other important features to be detected. For example, using only modularity as a quality metric, small communities are left unseen. This issue is addressed by using at least four or larger number of objective functions for identifying community structures.

2.1 Datasets Description

The datasets often used by many researchers for conducting experimental studies in research work for community detection can be categorized as real datasets (Zachary’s karate club, Political blogs, Less Miserables, American college football, Books about US politics, Internet, Coauthorships in Network Science) as well as artificial datasets. They are also known as the benchmark datasets (Lancichinetti et al., Girvan and Newman). These network datasets are in GML format which can be interpreted by many network analysis packages like NetworkX, Cytoscape, etc. (link to download http://www-personal.umich.edu/~mejn/netdata/).

2.2 Network Analysis Packages and Tools Used for Identification of Communities

Some of the popular social network analysis frameworks and tools used for analyzing social network data and graphs are Igraph, Cytoscape, SocNetV, Stanford Network analysis platform (SNAP), Network workbench, NetMiner, NetworkX, Gephi, Graphviz, Neo4j, etc. These social network analysis tools accept network data as GraphML, CSV, GML, and Graphviz file formats and can analyze any type of network data and files. Also, they analyze social networks and outputs important network statistics such as link strength, node density, node strength, visual representation of data, etc. The output file of analyzed network data or graph can be saved or exported in the form of GraphML, GML, BMP, PNG, etc.

2.3 Community Detection (CD) as an Optimization Task

In most of the research papers, CD is formulated as an optimization task solved using either a single-objective function or multiobjective function. For instance, reference [12] used two objective functions Negative Ratio Association(NRA) and Ratio cut(RC)(sum of the density of intercommunity links) to be minimized. NRA corresponds to negative RA (sum of internal edge densities of the communities identified). Some of the papers have used modularity as single-objective function [22,23,24,25] and many of them used more than one objective function like modularity and NMI(when ground-truth communities are known in advance) [19, 26, 27]. Reference [28] used two objective functions Kernel k-means and Ratio cut with PSO algorithms. Kernel k-means finds solutions with maximum intracommunity edges density and Ratio Cut tries to approach solutions with minimized intercommunity links. The authors [29] have used different variants of objective functions like (Kernel k-means, Ratio Cut, Modularity) as the first variant and (community score, community fitness and modularity) as the second variant with a non-dominated sorting genetic algorithm(NSGA-III). Reference [21] used many objective quality functions such as modularity, NMI, Community Score, Normalized Cut, Conductance, Purity and Rand Index for evaluating the structural properties and quality of the detected communities.

2.4 Representation of the Solution

The success of any algorithm depends on the encoding scheme used for representing a solution in the computational search space. Some of the most widely used solution representation schemes used for addressing the community detection problems are discussed below [30].

Label-based encoding—Label-based encoding scheme represents the population in the computational space as an integer vector of size(position) n. Here n stands for the number(genotype) of nodes. Each location in this vector 1 ≤ 1 ≤ n. Suppose if k is the number of communities in {1,2……, k}, the ith position(gene) corresponds to the ith node. Provided that a genotype has k number of communities then each gene has a value in the set (1……, k} which is actually the label identifying the community to which the node i belongs to, thus known as label-based representation. The network in Fig. 2 is partitioned into 3 individual communities as ({1,2,3}, {4,5,6}, {7,8,9}}. Figure 3 below shows the label-representation scheme for Fig. 2.

Fig. 2
An illustration of a network depicts 9 nodes, 17 edges, and 3 communities with labels 1 to 9.

A network of 9 nodes, 17 edges and 3 communities

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Fig. 3
An illustration depicts the label based presentation of a network. It has 2 rows labeled, position, and label.

Label-based representation of network of Fig. 2

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Locus-based representation—This type of solution representation scheme employs an individual g consisting of n number of genes g1, g2, g3, ……, gn and each gene gi can be mapped to take any adjacent connected node of any node i as shown in Fig. 4. Thus, in this graph-based representation, a value j which is assigned to the ith gene can further be used as a link between node i and j in the resultant division of the nodes as communities or partitions of the network. It can be concluded that nodes i and j belong to the same community. When this representation is decoded all the connected components of the network are identified. The nodes present in one connected component are assigned to one community. The decoding step here helps in finding connected components of the graph. The nodes which form these connected components are assigned to the desired community. This type of representation exhibits redundancy. Label-based representation scheme reduces the complexity of the search space from nn (in case of) to \(\prod^n_{i = 1} k_i\), ki = degree of node i.

Fig. 4
An illustration depicts the locus based representation. It has 2 rows labelled, position, and neighbor.

An example of Locus-based representation

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Medoid- based representation – It’s a prototype-based representation. Here, an n-dimensional array is used with input elements as the number of communities. For example, from Fig. 2 the partitioned communities are {1,2,3}, {4,5,6}, {7,8,9}. Here 1 is the element of the array indicating the prototype of community likewise. This is the medoid-based representation for Fig. 2. These community prototypes coincide with elements of the array. This type of representation scheme shows efficiency for space complexity. However, it has many drawbacks like it is redundant in nature because medoid can be any element of a particular community and also prior knowledge of k is required.

Label-based and Locus-based solution representation schemes are the most widely used ones in the literature. The above-described representation schemes refrain a node from becoming a member of more than one community. To overcome this drawback a new representation scheme for overlapping communities was introduced by [31].

2.5 Crossover Operator

Although one-point or two-point crossover fits well with label-based representation still it has two main drawbacks. The first drawback is that a community may contain disconnected subgroups of the node means nodes having no connections are placed in the same community. To allay this problem, the idea of one-way crossover was proposed by [32]. But it produces only one child from two parents. Another drawback observed is that the children doesn’t receive the genetic characteristics of the parent nodes fully. This issue was encountered by [33]. However, according to the author's observation and view point this crossover enhances the global search fitness of their method but they didn’t throw any light on the increase in computational time. While medoid-based representation works with one-point crossover and standard uniform crossover is exploited by locus-based representation. Standard uniform crossover is used by the locus-based representation scheme in which the off springs fully inherit the genetic properties of their parents [16].

3 Contribution

Evolutionary Algorithms (EA)

The category of EA algorithms particularly the genetic algorithms (GA) work on the concept of random population generation. These individuals in the population refer to chromosomes in the case of GA. The structure of chromosomes is organized according to the type of problem GA addresses. An objective function quantifies the quality of chromosomes in the population. This objective function evaluates the fitness value of the chromosomes and a percentage of high-fitness valued chromosomes are selected for the next iteration. Crossover and mutation operators on the chromosomes generate an improved population of individuals until the termination condition is achieved. An optimal solution is produced at the last step of the algorithm. These are the widely embraced techniques to solve NP-complete problems related to optimization due to their robustness in contrast to other traditional methods. GAs that can use different representation schemes are good for solving dynamic problems [34].

Particle swarm optimization (PSO)

PSO is used as a population-based stochastic searching algorithm for the community detection problem. It solves the optimization problems simulating the bird flocking behavior which are randomly searching for food in an area. The exact location of the food particle is not known to them. So, they apply the strategy of following those birds which are in close proximity to the food particle. To address any problem, a population or swarm of particles(solutions) is randomly generated initially. These particles search for the optimal solutions in the state space of possible solutions by updating generations. Each particle is associated with a position vector (Xi) and a velocity vector (Vi). At each iteration, every particle is attracted towards its personal best position (Pbestid) and best position of all particles (Gbestid) while moving randomly at the same time. [35, 36].

$$v_i^{t + 1} = v_i^t + c_1 z_1 (Gbest_{id} - x_i^t ) + c_2 z_2 (Pbest_{id} - x_i^t )$$
(1)
$$x_i^{t + 1} = x_i^t + v_i^{t + 1}$$
(2)

where, c1, c2 stands for acceleration parameters known as cognitive and social components r1, r2 are random numbers between {0.1}.

Bat Algorithm (BA)

Bat Algorithm is also a metaheuristic algorithm that mimics the prey hunting behavior of bats using an echolocation strategy to sense distance and velocity with static variations and loudness frequency. Bat algorithm address the CD as an optimization task where each ‘bat’ represents an individual in the population. It adapts the features of both particle swarm optimization (PSO) and simulated annealing. These combined features make Bat algorithm an outstanding one to achieve global search capability and strong convergence capability. BA simulates the emission rates, loudness and frequency variations of bats when they go for prey hunting. Bats transform their wavelength according to pulse frequency variations to locate the target. The updation rules for position and velocity for BA are similar to those of PSO algorithms. Continuous process of frequency and loudness adjustment maintains a balance between the intensification and diversification operations of the algorithm. BA overcome the drawback of PSO by generating a random solution using random flight behavior to avoid sinking into local optimum [37, 38, 27, 37]. The main equation for updating the bat location based on frequency and velocity is shown below:

$$f_i = f_{\min } + (f_{\max } - f_{\min } )\beta$$
(3)
$$v_i^t = v_i^{t - 1} + (x_{i }^{t - 1} - x^\ast )f_ i \\$$
(4)
$$x_i^t = x_i^{t - 1} + v_i^t$$
(5)

where, \(f_{\min }\) is minimum frequency, \(f_{\max }\) is a maximum frequency, \(\beta\) is a random number which takes a value between 0 and 1. \(x_i^t\) is the current location of the ith bat, \(x_{i }^{t - 1}\) is the previous location of the ith bat, \(v_i^t\) is the new velocity, \(v_i^{t - 1}\) is the previous velocity of the ith bat.

Differential Evolution (DE)

DE is a new population-based stochastic search evolutionary algorithm. As compared to the traditional GA algorithm, DE algorithm exhibits some merits: fast convergence, identifies optimized solutions regardless of initial parameters, requiring only a few control parameters. DE initiates the search procedure with a population of NP individuals randomly sampled where each individual signifies the target vector is selected from the population used to generate the mutant vector using the mutation operator. DE’s performance depends on the setting of control parameters like the size of the population, crossover, scale factor and the mutation scheme. These parameters should be set properly for the efficient solution of the problem. The mutation scheme of the DE algorithm exploits the genetic information of several individuals to utilize the distributed population characteristics and improve the search ability [39, 40]. Some mutation strategies often used with DE are: DE/rand/1(known as classical mutation scheme in DE), DE/ best/1, DE/best/2, and DE/rand-to-best/1. DE/rand/1 is the most popular mutation strategy used with DE in community detection is as follows:

$$v_{i,m} = x_{r1,m} + F \ast (x_{r2,m} - x_{r3,m} )$$
(6)

where i = {1, 2, ………., NP}, r1, r2 and r3 are randomly selected integer values from 1, 2……, NP, satisfying r1 ≠ r2 ≠ r3 ≠ I, scaling factor F is a real number between {0,1}.

Memetic Algorithms (MAs)

Memetic Algorithms (MA) are considered as the hybridization of previous GA based evolutionary algorithms. It is also a population-based approach with separate individual learning or optimization intersperse sing the recombination of high-quality solutions. They consider evolution as a baseline principle of working. It relies on the local search improvement procedures for problem search thus reducing the premature convergence. The word “memetic” is stirred by the Dawkin’s notion of the word ‘meme’, an element of social development resulting in local refinement [41]. The meme used in MAs represents a distinct learning procedure which can exhibit local refinements. MA uses the combination of GA and local search procedure to solve the optimization problem. MA outperforms existing genetic algorithms for specific applications of community detection [42,43,44].

Ant Colony Optimization

It is a metaheuristic optimization algorithm, basically a simulation of the ants foraging behavior independently communicating with each other through pheromone. It is also considered as a distributed multi-agent system where the search for food begins from different locations at the same time [45]. The population of ants construct solutions iteratively by finding the shortest path using pheromone and leaving the heuristic information behind them by crossing the paths. ACO algorithms are used in finding the community structure in the network. The positive feedback mechanism is used to find optimal solutions. The quality of the solution achieved by each artificial ant is assessed by its modularity. The probability of selecting a route by the ants from vertex x to y is given by the following formula below:

$$p_{xy} = \frac{{\mu_{xy}^\alpha h_{xy}^\beta }}{{\sum {^n_{x,y = 1} } \mu_{xy}^\alpha h_{xy}^\beta }}$$
(7)

where,

\(\mu_{xy}\) is the pheromone concentration of the path between x, y

\(h_{xy}\) is a heuristic function with a likelihood to select an edge from point x to y.

\(\alpha ,\beta\) determines relative influence of trail information and visibility.

Firefly Algorithm

It’s a population-grounded algorithm where each firefly represents a feasible solution. This algorithm imitates the flashing patterns and activities of the fireflies [46]. The main principle for the sparkle of fireflies is to attract other fireflies. This algorithm was proposed with a few assumptions like a firefly is attracted towards another firefly according to the brightness intensity. With the increase in the distance the brightness of the firefly decreases. The movement of fireflies towards the brightest firefly is to achieve a global optimal solution. FA algorithm depends on the parameters like random movement and attractiveness as performance measures. Community detection problems can be solved using the FA algorithm as an optimization algorithm by maximizing the modularity function. The main update formula [47] for any pair of two fireflies xi and xj is

$$x^{t + 1}_i = x^t_i + \beta_0^{ - {\gamma} r_{ij^2 } } (x_j^t - x^t_i ) + \alpha_t e_i^t$$
(8)

where, \(x^t_i\) represents the ih solution(firefly) at iteration t.

\(\beta_0\) is brightness at source.

A solution \(x_i\) will be attracted towards a brighter firefly \(x_{_j }\), means \(x_i\) moves towards \(x_{_j }\), \(\alpha\) randomization parameter, \(e_i^t\) vector of random variables.

4 Conclusion

The aim of this comprehensive review is to encompass various evolutionary and swarm intelligent-based algorithms for community detection that have encouraged a flurry of research. The widespread use of the aforementioned algorithms has shown an outstanding performance in detecting communities in static, dynamic, complex or multi-structural networks. Classification of different types of methods and algorithms used in addressing the community detection problem on the basis of social networks is also discussed here. The discussion of evolutionary and nature-inspired (NIA) algorithms based on single-objective or multiobjective has also been covered along with the most commonly adopted evaluation metrics like Modularity and NMI. A detailed description of the most widely used EA and NIA algorithms is statistically broken down and summarized in the tabular form according to the common key components used. These statistics provides a direction to the readers and researchers to select the characteristics of the algorithms like population initialization methods, perturbation operators and types of objective functions. It is observed that most of the research papers have shown a research gap for community detection in overlapping communities, multilayer networks and large-scale networks, implementing the algorithms independent of the increasing network size and substantial improvement in speed and accuracy.

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