## 5 Numerical Analysis
### Convergence Performance  
This study uses the proposed algorithm to solve the GN-based SUE in the Sioux Falls network and Anaheim network. As shown in Figure 7, the Sioux Falls network includes 24 nodes, 76 links, and 552 OD pairs. After filtering invalid paths, 5172 reasonable paths \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Path**} & \multicolumn{2}{c}{**Condition 1**} & \multicolumn{2}{c}{**Condition 2**} \\ \cline{2-5}  & **Flow** & **Cost** & **Flow** & **Cost** \\ \hline
1 & 83.94 & 25.86 & 56.03 & 12.36 \\
2 & 16.05 & 29.60 & 14.03 & 15.54 \\
3 & 0 & 12.01 & 27.09 & 13.80 \\
4 & 0 & 27.67 & 2.83 & 18.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Example network solutions and comparative analysis. are found in network, assuming node 23 is a point of interest (POI), and three wayfinding instructions before node 15, 20, and 24 are set for users to avoid choosing incorrect links (i.e., link 45, link 61, and link 74). Detailed input information is shown in Appendix C.  
In order to test the effectiveness and robustness of the solution algorithm, a larger network of Anaheim (i.e., with more links and feasible paths) was used. As shown in Figure 8, the Anaheim network consists of 416 nodes, 914 links, and 1406 OD pairs. After filtering out invalid paths, 28,466 feasible paths were identified in the network. During the process of solving this network equilibrium, the auxiliary solution in each iteration becomes more difficult to obtain due to the increased number of links and feasible paths. To ensure generalization, we conducted tests on the network without static wayfinding instructions, which provides the largest feasible region (i.e., all paths are available for HDVs). To illustrate the convergence performance of the different step size calculating methods mentioned above, the following parameters are used: \(v_{1}=5\times 10^{-4}\), \(v_{2}=1000\), which means step size updates after 1000 iterations in a predetermined method; \(d=1\) and \(d=2\) in sequence; \(\zeta^{(1)}=2\), \(\varepsilon_{1}=1.5\) and \(\varepsilon_{2}=0.2\); and \(r=10^{-4}\). Table 4 shows the specific step size in iteration \(n\) and the number of iterations required for convergence in two networks. It can be seen that predetermined step size and MSA method converge slowly compared to other methods. Note that MSA is a special case when \(d=0\) in MSWA (i.e., each iteration's auxiliary solution shares equal weight). The idea of increasing weights on closer auxiliary solutions significantly refines the convergence performance of MSA. We also note that if \(\varepsilon_{1}\) and \(\varepsilon_{2}\) in SRA are equal to 1, the SRA collapses to MSA, and changeable \(\varepsilon_{1}\) and \(\varepsilon_{2}\) makes step size in SRA decrease with iteration in a more flexible way than MSA and the flexibility speeds up convergence efficiently. It is also important to note that the time required for convergence does not necessarily improve with larger problems (i.e., more links and feasible paths), though more complex auxiliary  
Figure 7: Sioux Falls network with static wayfinding instructions. subproblems need to be solved in each iteration. Convergence time is also highly dependent on the distance between the initial feasible solution and the optimal solution. In our study, we start with a solution where demand is evenly distributed across all feasible paths for each OD pair. For a given OD pair, the flow on all paths sums to the OD demand and are non-negative, ensuring that the origin solution is feasible. However, the optimal solution is unknown in advance, and if it is relatively close to the initial solution, fewer iterations may be required to achieve convergence. Detailed convergence performance between step size methods is shown in Figure 9. Note that predetermined step size converges slowest in early iterations because of small original step size, and as the number of iteration increase, step size of MSA decreases rapidly, which restricts convergence speed noticeably.