## 4 Problem Modeling and Solution
### Logical Topologies Construction
Combining the time complexity of Algorithm 1, the overall computational complexity of the problem is \(O\big{(}\big{(}L\times P^{2}\big{)}\times N+R\times N^{2}\big{)}\), which can also be written as \(O\big{(}\big{(}L\times P^{2}+R\times N\big{)}\times N\big{)}\). Since the multi-topology construction process is performed offline as a pre-processing step, it does not affect the real-time performance of mobile users. This approach balances the number of paths in each logical topology, thereby balancing the transmission capacity from the source node to each destination node. It is important to note that if the shortest path between a node pair has a high ranking number, it may not be selected. A higher ranking number for the shortest path indicates that its bottleneck bandwidth is smaller, making it more prone to congestion under heavy traffic. Our goal is to maximize the average transmission capacity, so we may need to discard the shortest path if its transmission capacity is poor. Even so, each selected path is the shortest among the multiple paths that share the same bottleneck link. Overall, the paths we select tend to be shorter. ## 5 Performance Evaluation

### Evaluation Metrics

In the previous section, we introduced a method for multi-topology construction based on path ranking, referred to as the RANK method. This section presents a series of experiments aimed at evaluating the impact of the RANK method on mobile user performance. In our research on routing strategies, we found that choosing an algorithm depends on the objectives intended to be achieved for a specific network [24]. Therefore, we used three well-established performance metrics to quantify mobile user performance: average packet delay, average packet loss rate, and throughput. ### Simulation Setup

The experimental platform was set up on Ubuntu 20.04 and included Mininet 2.3.0, Open vSwitch 2.17.9, and the Ryu 4.34 controller [25]. Mininet [13] is a network simulator widely used for developing, testing, and evaluating SDN applications. It creates a realistic virtual network, running real kernel, switch, and application code on a single machine, emulating the behavior of a real network. For a performance comparison, we employed two other multi-topology construction methods: AMPLE [6] and the k-shortest paths approach, along with a single-path switching method as the baseline. The single-path switching method routes all mobile data flows through the shortest path from the source node to the new destination node. For multi-topology routing, each intermediate node (excluding the source and destination nodes) was assigned a flow entry capacity constraint of four, corresponding to the constraint in Algorithm 2. Using the AMPLE method, we constructed four logical topologies and removed some paths to meet the flow entry capacity constraints of each node. Similarly, with the k-shortest paths method, we selected four shortest paths for each node pair to construct four logical topologies, and then deleted some paths to satisfy the flow entry capacity constraints of each node. In a typical network architecture, switches are organized into three layers: the core layer, the aggregation layer, and the access layer. They form an efficient and reliable network together. The access layer may consist of multiple switches connecting to devices accessed by mobile users (such as base stations or wireless access points). The aggregation switches are responsible for aggregating the traffic from the access layer and forwarding it to the core layer. Access switches are typically connected to multiple aggregation switches to achieve load balancing. To simulate a complex network topology with many paths between node pairs, this experiment employed a mesh topology. We generated a network topology containing 14 nodes based on the National Science Foundation Network (NSFNET) backbone [26], as shown in Figure 2 below. In this experiment, we assume that the transmission delay of a link is proportional to its length.