## 1 Introduction  
With the advancement of information technology, many complex systems in real life can often be described and represented in the form of complex networks, such as social networks, citation networks, scientist collaboration networks, and protein interaction networks. The majority of these real-life networks typically exhibit distinct community structures, where a network consists of multiple communities, and the connections between nodes within a community are highly dense, while connections between nodes of different communities are relatively sparse [1]. Community detection is a fundamental task in the analysis of complex networks, aiming to partition the entire network into several communities. This process holds significant importance for studying and analyzing the organizational structure and functionality of networks, as well as uncovering latent patterns within them. There has been a great deal of research in detecting and evaluating community structures in complex networks. In pursuit of this fundamental task of community detection, researchers have proposed numerous community detection algorithms based on various methods such as graph partitioning, statistical inference, clustering, modularity optimization, dynamics, and deep learning. More detailed reviews of community detection can be available through several more extensive review articles [2; 3; 4; 5; 6]. Among many methods,dynamics-based methods constitute a significant branch of community detection algorithms, which reveal the community structure by modeling the interactions between nodes in a network. Currently, some of the mainstream dynamic-based community detection algorithms include label propagation, random walk, Markov clustering, dynamic distance, and particle competition [7]. To further improve the accuracy and stability of community detection algorithms, competitive learning is applied to the field of community detection. Competition is a natural process observed in the natural world and many social systems with limited resources. Competitive learning, as a crucial machine learning strategy, has been widely applied in artificial neural networks for achieving unsupervised learning. Early developments include Adaptive Resonance Theory networks [8], Self-Organizing Feature Map networks [9], Learning Vector Quantization neural networks [10], Dual Propagation neural networks [11], and Differential Competitive Learning [12]. The competition among particles in complex networks can generate intricate patterns formed by predefined interactions among individuals within a system. The simple interactions between particles in a network can construct the complex behaviors of the entire network, enabling functions like community detection and node classification. Particle competition-based community detection methods were first proposed by [PERSON] and [PERSON] [13]. In this approach, a set of particles is initially placed randomly in the network. These particles engage in random walks and compete with each other based on predefined rules to occupy nodes. When each community is dominated by only one particle, the process reaches dynamic equilibrium, thus accomplishing the community detection task. [PERSON] et al. [14] introduced the Stochastic Competitive Learning (SCL) algorithm for unsupervised learning, refining the walk rules of the particle competition model. Particles move in the network based on a convex combination of random walk and preferential walk rules. After entering a silent state due to energy depletion, particles execute a jump resurrection step. The final attribution of nodes to communities is determined based on the relative control abilities of particles, achieving the community detection task. The SCL algorithm represents a nonlinear stochastic dynamical system, characterized by adaptability, local motion reflecting the whole, and has contributed to the advancement of complex network dynamics.