## 3 Background
### Stochastic Competitive Learning  
Stochastic Competitive Learning, as a classical particle competition model, constitutes a competitive dynamical system composed of multiple particles, achieving community detection through unsupervised learning [14]. In Stochastic Competitive Learning, multiple particles are randomly placed within the nodes of the network. Each particle serves as a community indicator, while the nodes in the network are treated as territories to be contended. The primary objective of particles is to expand their territories by continually traversing the network and gaining control over new nodes, while simultaneously strengthening their control over already dominated nodes. Due to the finite number of nodes in the network, natural competition arises among particles. When a particle visits any node, it enhances its control over the current node, consequently weakening the control of other competing particles over that node. Ultimately, each particle's control range tends to stabilize, leading to convergence. Byanalyzing the control ranges of particles after convergence, the underlying community structure of complex networks is unveiled [13]. As a stochastic nonlinear dynamical system, Stochastic Competitive Learning describes the state of the entire dynamic system through vectors \(\mathbf{p}(\mathbf{t})\), \(\mathbf{E}(\mathbf{t})\), \(\mathbf{S}(\mathbf{t})\), and matrix \(\mathbf{Nu}(\mathbf{t})\). Among these, vector \(\mathbf{p}(\mathbf{t})\) represents the current positions of each particle within the network; vector \(\mathbf{E}(\mathbf{t})\) signifies the energy possessed by each particle. When a particle visits a node under its control, its energy increases by \(\Delta\), whereas visiting a node controlled by a competing particle reduces its energy by \(\Delta\). This mechanism limits each particle's roaming range, thus minimizing remote and redundant network access. Vector \(\mathbf{S}(\mathbf{t})\) denotes the dynamic state of each particle: particles with energy are in an active state, continuously traversing the network; when energy depletes, particles enter a dormant state and randomly jump to one of the nodes under their control for revival. Matrix \(\mathbf{Nu}(\mathbf{t})\) records the visitation counts of each particle for all nodes in the network. The more a particle visits a particular node, the greater its control over that node. The particle with the highest visitation count for a node attains control over it. In Stochastic Competitive Learning, particles in an active state navigate through the network following a convex combination of random and preferential walk rules [34]. The particle walking rule, denoted as \(P_{transition}^{(k)}(i,j,t)\), is defined as follows:  
\[P_{transition}^{(k)}(i,j,t)\triangleq\lambda P_{pref}^{(k)}(i,j,t)+(1-\lambda) P_{rand}^{(k)}(i,j) \tag{3}\]  
where \(i\) denotes node \(v_{i}\). \(j\) denotes node \(v_{j}\). \(t\) represents the moment. \(k\) indicates the particle. \(P_{pref}^{(k)}(i,j,t)\) denotes the particle preferential walk rule. \(P_{rand}^{(k)}(i,j)\) represents the particle random rule. \(\lambda\in[0,1]\) represents the probability of a particle performing preferential walk, regulating the balance between random and preferential walking. When \(\lambda=1\), the particle exclusively follows preferential walking; when \(\lambda\in(0,1)\), the particle performs a combination of both random and preferential walking; and when \(\lambda=0\), the particle solely engages in random walking. The random walking mode guides the particle's exploratory behavior, where the particle randomly visits neighboring nodes without considering their control capacity. This mode reflects the particle's randomness, and the equation for random walking is defined as follows:  
\[P_{rand}^{(k)}(i,j)\triangleq\frac{a_{ij}}{\sum\limits_{u\in N(v_{i})}a_{iu}} \tag{4}\]  
The preferential walking mode guides the particle's defensive behavior, where the particle prioritizes visiting nodes it already controls, rather than nodes that are not yet under its control.