Community structure is a key component in complex network systems. This paper aims to improve the effectiveness of community detection and community discovery in complex network systems by providing directions for the reconstruction and optimization of community structures to expand the application of intelligent optimization algorithms in community structures. First, deep learning algorithms and ant colony algorithms are used to elaborate the community detection and community discovery in complex networks. Next, we introduce the technology of transfer learning and propose an algorithm of deep self-encoder modeling based on transfer learning (DSEM-TL). The DSEM-TL algorithm’s indicators include normalized mutual information and modularity. Finally, an algorithm that combines the ant colony optimization (ACO) algorithm and the quantum update strategy, called QACO, is proposed. The proposed community structure reconstruction scheme is compared with other methods using the accuracy rate as the indicator. The results show that the DSEM-TL algorithm exhibits the optimal detection rate, better applicability, and higher effectiveness in real networks. Under the given the condition that the number of edges between communities Z_{out} is >6, DSEM-TL shows better performance on the Girvan–Newman benchmark network than several other community discovery algorithms. Furthermore, under the given condition that the mixed parameter μ is >0.65, the DSEM-TL algorithm outperforms several other algorithms on the Lancichinetti–Fortunato–Radicchi benchmark network. When given μ < 0.4, the QACO algorithm can determine the proper division of the corresponding network. When the case is μ > 0.45, the division result corresponding to the QACO algorithm is closer to the real community division, which has a faster convergence speed and better convergence performances. Consequently, the proposed community structure reconstruction scheme has higher accuracy. The proposed two intelligent optimization algorithms have potential application in the reconstruction and optimization of community structure.

As a branch of network science, the complex network can be regarded as a collection formed by some individuals, which shows the interconnection between individuals [

Therefore, to explore the reconstruction and optimization of the community structure in the complex network systems, the optimized deep learning algorithm and quantum ant colony optimization (QACO) algorithm are applied to community detection and community discovery. The algorithms aim to improve the effectiveness of community detection, discovery, and mining. Hopefully, the results can provide some references for the reconstruction and optimization of community structure.

Community detection is a critical characteristic of complex networks. The analysis and mining of community structures in complex networks are of great significance for discovering the inherent development rules of complex networks and the correlation of relevant indicators within these networks [

In the real network structure, the community structure is widespread. From the perspective of network nodes, it can be divided into a collection of many nodes. If there is a close connection within the network node collection, and this connection between them also shows a relatively loose state, it will be determined that the corresponding complex network has a community structure. The accurate identification of community structure has developed into a crucial stage of obtaining sufficient information [

where:

Because deep learning algorithms have excellent performance in feature extraction, deep learning has been successfully applied in the fields of image classification and semantic recognition [

The proposed deep self-encoder modeling based on transfer learning (DSEM-TL) consists of the data preprocessing module, the feature extraction module, and the optimization module. The data preprocessing module is primarily responsible for the preprocessing of the original adjacency matrix in the complex network. Based on the topology of the complex network, the community discovery problem is to divide complex network systems into nodes as different groups or communities. The problem can be expressed as:

where:

Then, based on the eigenvalue function, the maximum eigenvalue

where:

where:

In the feature extraction module, transfer learning is introduced to build a target prediction model with excellent generalization performance. The algorithm is trained by a parameter-based migration method. Specifically, the overall loss function is defined

where:

In the optimization module, through the application of the backpropagation algorithm based on the stochastic gradient descent method, the minimization problem in the overall loss function is solved. The update of the weight and the update of the offset term parameters are then realized by the following rules

where:

To test the effectiveness of the proposed DSEM-TL algorithm, the learning rate corresponding to the deep self-encoder is set to 0.01, and the number of training iterations is set to 100,000. Furthermore, by comparing and analyzing the widely used community discovery solutions in the Girvan–Newman (GN) and the Lancichinetti–Fortunato–Radicchi (LFR) benchmark network, we test the effectiveness of the DSEM-TL algorithm in structural reconstruction and optimization in the complex network community. The selected commonly used community discovery schemes the subspace pursuit (SP) algorithm [

Dataset | Karate | Friendship 6 | Polbooks | Citeseer | Cora | Football |
---|---|---|---|---|---|---|

Number of nodes | 34 | 69 | 105 | 3312 | 2708 | 115 |

Number of sides | 78 | 220 | 441 | 4732 | 5429 | 613 |

Number of communities | 2 | 6 | 3 | 6 | 7 | 12 |

The ant colony optimization (ACO) algorithm is an intelligent optimization heuristic algorithm. It has strong robustness and well-integrated with other algorithms. It has strong applicability in solving discrete optimization and complex network community detection problems, and is more applicable in the network structure for small-scale datasets [

For the ACO algorithm, when the ants move from the starting node to a viable neighbor node, the generation of the solution can be expressed as

where:

In

As for quantum computing, the state of qubits can be expressed as

where:

The update of pheromone can be expressed as

where

A specific number of ants are randomly assigned among the complex network nodes using the QACO algorithm. The ants move from one node to another according to the heuristic information and the strength of the pheromone. Then, small groups appear in the population. We will integrate the small groups into the community according to the maximization of modularity and community division to realize the determination of the community structure in the complex network. In this process, the rotation and mutation mechanism based on quantum computing enhance the global search capability and solution diversity of the algorithm. Once the optimal solution or the maximum number of iterations appears, the entire optimization process terminates accordingly. The specific implementation process of the QACO algorithm is shown in

During the implementation of the QACO algorithm, the pheromone in the initialization phase can be expressed as

The heuristic information calculation between nodes can be expressed as

where:

where:

Furthermore, it is possible to update the pheromone intensity by using quantum rotation gates. The new pheromone intensity obtained at this time can be expressed as

where:

The extended GN benchmark network is used to verify the effectiveness of the QACO algorithm. At the same time, the QACO algorithm is compared with the interest factor-based ACO algorithm (IACO), the multi-objective discrete particle swarm optimization (MODPSO) community detection algorithm, the quantum distribution algorithm (QDM), the quantum genetic algorithm (QGA), and the locally optimized community detection algorithm (MABA) based on minimum spanning tree. In this way, we analyze the performance of the QACO algorithm in the detection of complex network community with NMI and Q selected as evaluation indicators.

All the nodes are treated as having integrity in the complex network community structure. We introduce the concept of game theory to excavate the implicit constraint conditions. The results are achieved through the DSEM-TL and QACO algorithms. The advantages of the deep self-encoder based on the DSEM-TL and QACO algorithms are organically integrated so that the accuracy of community detection in the community structure can be improved. To verify the effectiveness of applying the two optimized community detection algorithms in the reconstruction of the community structure, the DSEM-TL and QACO algorithms are integrated to complete the reconstructed community structure. In two typical game models Prisoner’s Dilemma and the Sprague-Grundy theorem both proposed algorithms are compared with the traditional node reconstruction (NR) and partition reconstruction (PR) method. For comparative analysis of complex networks, the random network (RN), small-world network (SWN), and scale-free network (SFN) are selected.

Analysis of the NMI corresponding to different community discovery schemes in

The performance comparison of the six complex network community discovery solutions on the GN and LFR benchmark network is shown in

In the above figure, Z_{out} represents the number of edges between each vertex community.

As shown in _{out} is greater than 6, DSEM-ML shows better performance on the GN benchmark network than other community discovery methods. Similarly, when the mixed parameter μ is greater than 0.65, the performance of DSEM-TL is superior to several other algorithms on the LRF benchmark network. This further validates the effectiveness of the DSEM-ML algorithm in complex network community discovery and community detection.

The above analysis reveals that the proposed DSEM-TL algorithm has the best performance. The reason is that the new method used for matrix preprocessing in the model construction process of DSEM-TL actually highlights the local information of the vertex. n additional, the DSEM-TL algorithm employs transfer learning to obtain low-dimensional feature matrices. Therefore, DSEM-TL has useful community discovery and community detection performances in complex networks. Furthermore, the DSEM-TL algorithm model based on deep learning has applicability in the artificial benchmark network. Accordingly, it is predicted that this complex network community discovery and community detection method based on deep learning has vast application potential in the reconstruction and optimization of community structure.

The NMI comparison results of the QACO algorithm with IACO, MODPSO, QDM, QGA, and MABA algorithms are shown in

We compare the results of the NMI and modularity measures of several algorithms, as shown in

The comparison between the proposed optimization scheme of complex network community structure and several community reconstruction methods is shown in

As shown in

The QACO algorithm can improve the accuracy of the GN benchmark network community division. The integration of quantum mechanisms gives the network a faster convergence speed, which leads to superior performance in complex network community testing. The introduction of quantum strategy has expanded the application of the ACO algorithm in solving community detection problems. The QACO algorithm is more applicable to the detection of complex network community structure than other algorithms, and therefore can play an active role in the reconstruction and optimization of complex network community structure. Notably, the proposed community structure reconstruction scheme has a high accuracy rate.

This article proposes the DSEM-TL algorithm model, which introduces transfer learning into deep self-encoders, and the QACO algorithm, which integrates the quantum strategy into the ACO algorithm. The results show that the two algorithms have significant advantages in the application of community discovery and community detection in complex network community structures. The combination of the two optimization algorithms shows the best accuracy in the reconstruction of the complex network community structure. However, the optimization of the complex network community structure still stays in the exploration stage. At this time, the coverage of the selected real network dataset is limited, and there has not been sufficient consideration of the characteristics of the complex network structure. Resolving these issues, which are due to several factors, is a focus of future research.

We thank LetPub (