The study of community detection algorithms in complex networks has been very active in the past several years. In this paper, a Hybrid Selfadaptive Community Detection Algorithm (HSCDA) based on modularity is put forward first. In HSCDA, three different crossover and two different mutation operators for community detection are designed and then combined to form a strategy pool, in which the strategies will be selected probabilistically based on statistical selfadaptive learning framework. Then, by adopting the best evolving strategy in HSCDA, a Multiobjective Community Detection Algorithm (MCDA) based on kernel
Since many complex systems, such as the Internet, social networks, and biological networks, can be modeled as complex networks, the study of complex networks is essential to better understand and analyze such systems. In complex networks, community structure [
Fortunato [
Smallscale communities cannot be detected from some large complex networks by optimizing the modularity, which is the resolution limit [
The above work shows that the modularitybased intelligent optimization algorithms for community detection attract much attention of researchers. In order to further improve the performance of intelligent optimization algorithms for community detection, the paper proposes a new framework including hybrid evolving strategies and adaptive learning mechanism based on evolutionary algorithm. The work includes two parts. In the first part, the modularity
The rest of this paper is organized as follows: Section
Assume that a network
Community structure is a universal property of many complex networks in real world. The community is the node subset, which has a relatively tight connection between the inner nodes and a relatively sparse connection between the external nodes [
The modularity
In order to solve problem of limit of modularity resolution [
The smaller the KKM value is, the closer the internal group will be, and the smaller the RC is, the sparser the links between nodes of internal and external community will be. Therefore, community detection problem can also be modeled to a multiobjective optimization problem by minimizing KKM and RC.
In this section, the detailed information of HSCDA and MCDA is depicted.
In order to further improve the solution quality of intelligent optimization algorithms for community detection problems based on modularity, HSCDA is proposed based on evolutionary algorithm. In HSCDA, three different crossover and two different mutation operators for community detection are designed and then combined to form a strategy pool, in which the strategies will be selected probabilistically by roulette wheel selection based on statistical selfadaptive learning framework. The flow of HSCDA is shown in Algorithm
Input: Adjacent matrix
Parameters: population size (
mutation probability (
Output: Optimal solution of the current iteration
(1.1) Initialize each individual by label propagation mechanism (see Algorithm
(1.2) Calculate objective function
For each individual, select a strategy from hybrid evolutionary strategy pool (see Section
using roulette wheel selection according to the selected probability,
then update the selected probability of each strategy by selfadaptive learning framework (see Section
Apply hillclimbing search (see Section
Once a better individual is generated, the new individual will replace the chosen one.
Repeat until no more better individual is get or the number of search reaches the maximum,
then the individual is the current best solution of the population.
If (iterations < gen), iterations ++, and go to Step
Input: Population with each node divided into different communities, that is,
Output: Population after initialization
A partition
To both reduce the searching space and promote diversity, the paper adopts initialization mechanism based on label propagation [
Assume that the neighbor set of a node
In order to enhance the capability of evolution of the algorithm and thus to improve the quality of solution, six different strategies for community detection are designed to make up the hybrid evolutionary strategy pool. Every evolutionary strategy includes crossover and mutation operators. Individual chooses different strategies adaptively and then gradually improves its solution structure.
Given two individuals
Combine the above three crossover and two mutation operators mutually and thus generate the following six evolutionary strategies to form the hybrid evolutionary strategy pool:
Strategy 1: Crossover 1 + Mutation 1.
Strategy 2: Crossover 1 + Mutation 2.
Strategy 3: Crossover 2 + Mutation 1.
Strategy 4: Crossover 2 + Mutation 2.
Strategy 5: Crossover 3 + Mutation 1.
Strategy 6: Crossover 3 + Mutation 2.
Based on strategy pool, a statistical selfadaptive learning framework is introduced into HSCDA. The individual adaptively chooses the appropriate strategy in different stages of the algorithm depending on the evolution effect of the strategy. In the selfadaptive learning framework, each strategy is given the corresponding probability of being selected. Individual selects evolution strategy by roulette wheel selection.
In particular, each individual
The difference of the individual before and after evolving by a strategy is used to measure the evolution effect of that strategy, which is defined as follows:
However for other strategies, the selective probability
Individuals in the next generation will make a choice of the evolving strategies according to the updated selective probabilities. Therefore, HSCDA can make the individual adaptively choose the appropriate strategies at different stages.
In order to improve convergence speed and alleviate trapping into local optima, the hillclimbing method suggested in [
Experiments (see Sections
Input: Adjacent matrix
Parameters: population size (
Output: Pareto front solutions.
(1.1) Initialize the population with population initialization algorithm based on label propagation mechanism (Algorithm
(1.2) Calculate individual objective functions KKM and RC with formula (
(1.3) Calculate the rank of each individual
If at least one objective value of individual
and all objects of
and the rank of all nondominant individuals is defined as 1,
and the other individual’s rank plus 1 with the number of individuals who control it.
(1.4) Calculate crowding distance
Calculate the distance between one individual and other individual in the same rank by
the crowding distance calculation method refers to [
The rank of all individuals is calculated first,
then select the individuals whose rank is 1 to construct dominant solutions of the current generation.
(4.1) Combine the dominant solutions with the present population to form a new population
(4.2) Calculate the rank of each individual and sort them from small to large.
(4.3) Select
If (iterations < gen), iterations ++ and go to Step
Normalized Mutual Information (NMI) [
The parameters of HSCDA are set as follows: population size is 100, crossover probability is 0.8, mutation probability is 0.2, the initial selection probabilities of evolving strategies in strategy pool for each individual are set as
Zachary’s Karate Club network [
Characteristics of four real world networks.
Network  Nodes  Edges  True clustering results 

Karate Club  34  78  2 
Dolphin  62  159  2 
Football  115  613  12 
Polbooks  105  441  3 
HSCDA is applied to four real networks, respectively; the average of optimal solutions of HSCDA after running 30 times is recorded. Table
NMI of HSCDA, GN, FN, and BGLL in four real networks.
Algorithm  Karate Club  Dolphin  Football  Polbooks 

GN  0.58  0.55  0.88  0.56 
FN  0.69  0.57  0.76  0.53 
BGLL  0.59  0.52 


HSCDA 




Algorithm  Karate Club  Dolphin  Football  Polbooks 

GN  0.4013  0.5194  0.5996  0.5168 
FN  0.3801  0.4897  0.5773  0.5020 
BGLL  0.4188  0.5188  0.6021  0.4986 
HSCDA 




Clustering results on four real networks by HSCDA.
Clustering results on Karate Club network by HSCDA. (A) is the structure with highest NMI value; (B) is the structure with highest
Clustering results on Dolphin network by HSCDA. (A) is the structure with highest NMI value; (B) is the structure with highest
Clustering results on Football network by HSCDA. (A) is the structure with highest NMI value; (B) is the structure with highest
Clustering results on Polbooks network by HSCDA. (A) is the structure with highest NMI value; (B) is the structure with highest
To analyze the actual evolution effect of evolving strategy in hybrid strategy pool, the selected count of each evolving strategy of the optimal solutions (run 30 times independently) is recorded and shown in Figure
Selected proportion of strategies of the optimal solutions.
From the results in Tables
The parameters of MCDA are set as follows: population size is 100, crossover probability is 0.9, mutation probability is 0.1, and maximum number of iterations is 100. MCDA and three multiobjective algorithms (MOGAnet [
In order to compare with other community detection algorithms based on multiobjective optimization, we do experiments on artificial synthetic benchmark network proposed by Lancichinetti et al. [
By adjusting values of mixing parameter
Max NMI values averaged over 30 runs for different algorithms.
In Figure
MCDA is applied to four real world networks mentioned above. Cluster results with max
Cluster results of MCDA in Karate Club network.
Cluster results of MCDA in Dolphin network.
Cluster results of MCDA in Football network.
Clustering results of MCDA in Polbooks network.
From Figure
Figure
Some nodes in Football network are not connected with nodes in the same community, while the connection between nodes of this community and nodes of other communities is more close. When the network is in the real clustering, the modularity
Similar to Football network, Books on US politics network itself shows high complexity. From the comparison of Figures
To further improve the solution quality of intelligent optimization algorithms for community detection, HSCDA and MCDA are proposed based on evolutionary algorithm, respectively. In HSCDA,
The authors declare that there is no conflict of interests regarding the publication of this paper.
This paper was supported by China Postdoctoral Science Foundation funded project (2015M571790) and NUPTSF (Grant nos. NY213047, NY213050, NY214102, and NY214098).