## 2 Methodology
### Multi-Objective Optimization Strategy
NSGA-II effectively avoids falling into local optimal solutions during the iteration process through the use of the crowding distance operator as a diversity preservation mechanism and the ability to perform global searches based on the concepts of genetic algorithms and Pareto optimality. The optimization objectives proposed in this study include multiple goals, specifically maximizing reliability and minimizing delay. The objective of maximizing reliability is to optimize the reliability of the transmission path, thereby ensuring stability and resistance to interference during data transmission. The objective of minimizing delay is to optimize the delay of the transmission path, ensuring that data can be delivered to the target node rapidly and in a timely manner. Therefore, the objective function can be defined as follows:  
\[\min F=\min\{\text{Reliability}_{\text{max}},\text{delay}_{\text{min}}\} \tag{5}\]  
\[\min F=w_{1}D+w_{2}(-R) \tag{6}\]  
\[ \begin{array}{ll}s.t.&D=\sum t_{i}\\ &R=\prod r_{i}v_{i}\\ &r_{i}=\frac{\frac{1}{\lambda}\sum\limits_{j=1}^{N}A_{ij}c_{e}(v_{j})}{\varepsilon \left(\sum\limits_{j=1}^{N}A_{ij}\varepsilon(v_{i}v_{j})\right)}\\ &t_{i}=\frac{c_{i}}{v}\\ &R\geq 0.99\\ &D\leq 500\end{array}  \tag{7}\]  
The reliability of the transmission path, \(R\) is computed as the cumulative product of the reliability of each node. The reliability of each node, \(r_{i}\), is determined by node eigenvector centrality and node similarity. Eigenvector centrality represents the global importance of a node in the network topology, while the similarity between a node and its neighboring nodes measures the local importance of the network node. The delay of the transmission path, D is calculated as the sum of the delays of each link. The delay on each link segment, \(d_{i}\) contributes to the total path delay. In the process of multi-objective optimization, we limit the range of reliability and delay by setting optimization objective constraints. At the same time, based on the scenario in the algorithm, we prioritize reliability as the first optimization objective. When the reliability is greater than 0.99, we select paths with a delay of less than 500 as the selected object. In addition, this algorithm continuously collaborates on reliability and latency during the cross mutation screening process of offspring by using optimized NSGA-II algorithm, ultimately achieving Pareto-optimal solution. The optimization process is described as follows. The algorithmic process effectively solves the multi-objective optimization problem in heterogeneous networks, ensuring the reliability and timeliness of information transmission (Algorithm 2). The elitist strategy of NSGA-II optimization retains the best individuals from the previous generation, allowing the algorithm to avoid local optimal solutions and improve global search capability. Additionally, the algorithm maintains population diversity by calculating the crowding distance of individuals in the objective space, enhancing the stability and convergence of the algorithm. By performing the non-dominated sorting of the population, the algorithm effectively identifies superior solutions. These characteristics enable the algorithm to excel in multi-objective optimization problems, improving computational efficiency while ensuring the diversity and comprehensiveness of the optimization results. Generating the initial population using heuristics accelerates the convergence process of the algorithm and enhances the quality of the solutions, thereby achieving efficient, secure, and reliable multipath information transmission. ```
Input : heterogeneous network \(G(V,E)\), source node \(v_{s}\), target node \(v_{t}\), optimization parameters Output : Optimal transmission paths
1 Initial: The network with the input parameters. Initialize population: After determining the objective function, use heuristic methods to obtain a simple path between the source and destination to generate the initial population.