## 4 LNSSCL Algorithm
### Incorporating Node Local Similarity into Particle Preferential Movement Rule  
The stochastic competitive learning algorithm stipulates that particles navigate through the network based on a convex combination of random walk and preferential walk rules. The preferential walk rule ensures that particles preferentially access nodes under their control, reflecting the deterministic nature of particle traversal, and numerically equivalent to the particle's control ability ratio. However, this rule solely focuses on the particle's control ability over nodes, without considering the influence of node local similarity indicators. This may lead to a relatively high degree of randomness and weak inclination in the initial direction of preferential walk, thereby affecting the stability and accuracy of community detection results. To address these issues, an enhancement to the particle preferential walk rule is introduced by incorporating node similarity, enabling nodes with greater similarity  
Figure 1: Schematic of particle initialization position.  
to be more likely visited by particles. This modification enhances the directionality and determinism of particle traversal during the walk.  
The improved node similarity for the enhanced particle preferential walk rule not only considers cases where nodes share common neighbors, but also accounts for situations where nodes lack common neighbors. When there are shared neighbors between nodes, a similarity index considering both common neighbors and degree difference is employed to measure the degree of similarity between two nodes, as defined below:  
\[Sim(i,j)=\frac{\left|N(v_{i})\cap N(v_{j})\right|}{\left|N(v_{i})\cap N(v_{j}) \right|+\left|N(v_{i})\cup N(v_{j})\right|}\times\frac{\left|N(v_{i})\cap N(v_ {j})\right|}{1+\left|N_{s}-N_{h}\right|} \tag{11}\]  
For nodes \(v_{i}\) and \(v_{j}\), if \(\left|N(v_{i})\right|\geq\left|N(v_{j})\right|\), then \(N_{s}\) is set to \(N(v_{j})\), and \(N_{h}\) is set to \(N(v_{i})\); if \(\left|N(v_{i})\right|<\left|N(v_{j})\right|\), then \(N_{s}\) is set to \(N(v_{i})\), and \(N_{h}\) is set to \(N(v_{j})\). When there are no shared neighbors between nodes, further consideration is needed for nodes with a degree value of 1. For nodes without shared neighbors and with a degree value of 1, since their behavior is solely related to their unique first-order neighbor, their similarity value is set to 1. For nodes without shared neighbors and with a degree value greater than 1, their similarity is associated with the node's degree value. Based on the negative correlation between node degree and the unfavorable Hub Depressed Index, the degree value is inversely related to its similarity. The equation for calculating the similarity between nodes without shared neighbors is defined as follows:  
\[Sim(i,j)=\left\{\frac{\begin{matrix}1,&(d(v_{i})=1)\vee\left(d(v_{j})=1\right) \\ \frac{a_{ij}}{\max\left\{d(v_{i})d(v_{j})\right\}},&d(v_{i})\
eq 1,d(v_{j}) \
eq 1\end{matrix}\right. \tag{12}\]  
Building upon this, the equation for the particle's preferential walk rule incorporating node similarity is provided:  
\[P_{pref}^{(k)}(i,j,t)\triangleq\frac{a_{ij}\overline{Nu}_{j}^{(k)}(t)(1+Sim( i,j))}{\sum\limits_{u\in N(v_{i})}a_{iu}\overline{Nu}_{u}^{(k)}(t)(1+Sim(i,u))} \tag{13}\]  
where \(Sim(i,j)\) represents the similarity index between nodes \(v_{i}\) and \(v_{j}\), and \(\overline{Nu}_{j}^{(k)}(t)\) represents the control capacity of particle \(k\) over node \(v_{j}\). The improved preferential walk rule takes into account both the particle's control capacity over nodes and the similarity index between nodes as equally significant factors. This approach avoids the issue of randomness in the preferential walk direction that arises after particle initialization, thereby enhancing the inclination and certainty of particle movement throughout the entire preferential walk process.