## 2 Methodology
### Multipath Transmission Algorithm
#### 2.3.3 The Third Type of Simplification Operation  
The core idea of the third type of simplification operation is to find a new independent path by refining the link processing based on the link bottleneck on the first path. As illustrated in Figure 6, The red and yellow clouds on the left represent different heterogeneous networks. The source and destination nodes are node S and node D, respectively. The first calculated path is S-A-F-O-R-D. It can only go through link FO. At this point, FO is the only link of the network, and nodes O and F belong to the point cut set of the network, while link FO belongs to the edge cut set. When the link FO is deleted, the network topology will split into two disconnected subgraphs. In this situation, the first and second types of simplification methods become ineffective, necessitating the use of the third type of simplification. Based on link FO, the path is divided into two distinct subgraphs. Each subgraph will then re-determine the source and destination nodes and apply the second type of simplification  
Figure 5: Schematic diagram of the second type of simplification operation.  
again to find a second path that satisfies the multi-objective optimization requirements. Therefore, the second calculated path is S-E-F-O-T-D. The specific steps are as follows:  
Identify the link bottleneck: Locate the nodes on the first path where only two links are connected, i.e., the links that may have \"link bottlenecks\". These links are the weakest points in the path and require special handling.  
Path splitting: Split the first path into two parts based on unique links. This step refines the path processing to ensure the new path avoids these bottleneck links.  
Path computation: Apply the second simplification method again to find a second path that satisfies the multi-constraint requirements. This step ensures that a second path is found even in the presence of a link bottleneck.  
The main purpose of the three types of simplification operations is to significantly improve the overall reliability of the two paths. The first type of simplification ensures that the nodes on the second path are in different subnets from the nodes on the first path. If the subnet of a node on the first path fails, the node on the second path can still transmit data normally, improving the system's fault tolerance. The second type of simplification ensures that the two paths do not share links. If a link on the first path fails, the data on the second path can still be transmitted normally, enhancing the system's redundancy and data transmission reliability. The third type of simplification provides an effective path scheduling scheme for cases where data can only be transmitted over specific segments of links on the first path. This operation ensures that even with a link bottleneck, a second path can be found that satisfies the multi-constraint requirements, thereby ensuring the high reliability and availability of the system.  
The multipath transmission algorithm framework based on hierarchical culling, as proposed in this paper, offers an efficient and reliable multipath transmission scheme through three different types of simplification operations combined with the NSGA-II multi-objective optimization algorithm. This framework not only improves the independence of paths and the overall reliability of the system but also ensures the high efficiency and security of information transmission. It is particularly suitable for the multipath transmission requirements in complex heterogeneous network environments. By fine-grained management of the network topology graph and path planning, the algorithm effectively addresses various transmission challenges, providing robust technical support for achieving highly reliable network communication.