*3.4. Ladder-like Structure*

As in other small examples of mazes with cycles, the ladder-like structures with *L* paths were examined. The difference from the previous subsection is that the edges are shared among the different paths. Figure 5a,b,d shows the results of *L* = 1, 3, and 4, respectively. For the cases shown here, the shortest paths are indicated by the chain of maximum densities, while smaller densities are observed farther from the shortest path, meaning that the maze-solving was successful. The absolute values of amplitude after convergence seem to correspond to the distance of each path by considering Figure 5b,d. However, this is not the case in Figure 5a.

For the cases of *L* = 2 and 5, undesirable eternal vibrations were observed and mazesolving could not work, which will be discussed in a later subsection. For the case of *L* = 6, the convergence was difficult to realize owing to the limitation of the computational times. However, it was not an eternal vibration judging from the actual calculations.

Figure 5c shows the time profiles of the edges in each path of Figure 5b. For the three paths, the speeds of convergence did not differ significantly, and they did not depend on the distance from the sink unlike what was observed in the tree-like structure. The convergence is faster than for the tree-like structure but slower than for the independent multiple paths. The number of convergence steps in Figure 5d is smaller than that in Figure 5b, exhibiting the difficulty faced in predicting the convergence speed from the structure of the maze.

Table 3 lists the relation between the distances of paths and amplitude remaining on an edge in Figure 5. The amplitudes can be expressed by rational numbers; however, the meanings of the denominator numbers are not clear as in Figure 4. The absolute values of the amplitudes in Figure 5 are generally smaller than those of the same scale case in Figure 4.

The ratio among the paths seems to have meaning; however, the reason has not been determined except for the relation between the longest and second-longest paths. The relative ratio of the longest path and the second-longest path is thought to be in inverse proportion to the ratio of the length of the non-shared part of each path. This hypothesis is complemented in the next subsection.


**Table 3.** The relation between the distance and remaining amplitude for Figure 5 (The waypoint of a path, the distance between the start and goal on the path, the number of edges in the path, the amplitude remaining on an edge on the path, and an approximate rational expression of the amplitude).

**Figure 5.** The results of maze-solving for the ladder-like structures with *L* paths from the start to the goal. (**a**) Amplitude distribution and the number of steps after convergence for *L* = 1. (**b**) Amplitude distribution and the number of steps after convergence for *L* = 3. (**c**) Time profiles of the densities on selected edges for *L* = 3. (**d**) Amplitude distribution and the number of steps after convergence for *L* = 4.
