*3.3. Independent Multiple Paths*

Next, we examined maze-solving on multiple independent paths of different lengths. This structure includes cycles, which makes maze-solving difficult even in classical schemes. Figure 4a–c,e,f shows the numerical results for the networks with *M* paths, where the length of the *M*th path is 2*M*. After convergence, in all the examples shown here, the densities remain on all the paths between the start and the goal; however, the maximum densities are only observed on the shortest path, while smaller densities are observed farther from the shortest path. By regarding the path of the maximum densities as the correct path, the maze-solving was successful for these examples.

Figure 4d shows the time profiles of the edges on each path. The speed of convergence was higher than in the case of other structures of a similar scale. The reason for this is unclear, but a lack of branching on the paths may be responsible for the high speed. Among the three paths, the speeds of convergence did not differ significantly, and they did not depend on the distance from the sink unlike in the tree-like structure. In general, the convergence steps increased by the addition of other paths. However, a counter-intuitive decrease of the convergence steps was observed in Figure 4e,f.

The absolute values of amplitudes, after convergence, decrease as the length of the path becomes longer; however, they are not constant for the length of paths because a slight decrease was observed by additional paths. Table 2 lists the relation between the distances of paths and amplitude remaining on an edge. For Figure 4a, the amplitude is the inverse of the number of edges included in the path, which is the same as given in Table 1. However, the rule looks broken in the case of multiple paths.

**Table 2.** The relation between the distance and remaining amplitude for Figure 4 (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. Only the relative ratios are shown for Figure 4f because an appropriate rational number was not found).


**Figure 4.** The results of maze-solving for the structures with multiple independent *M* paths from the start to the goal. The length of *M*th path is 2*M*. (**a**) Amplitude distribution and the number of steps after convergence for *M* = 1. (**b**) Amplitude distribution and the number of steps after convergence for *M* = 2. (**c**) Amplitude distribution and the number of steps after convergence for *M* = 3. (**d**) Time profiles of the densities on selected edges for *M* = 3. (**e**) Amplitude distribution and the number of steps after convergence for *M* = 4. (**f**) Amplitude distribution and the number of steps after convergence for *M* = 5.

Most of the amplitudes were assigned rational expressions; however, the rule determining the absolute value (or a positive integer of the denominator) is not clear. However, the relative amounts of amplitudes among paths in each case were found to be in inverse

proportion to the distance between the start and goal exactly for these cases. The relative amounts of amplitude were determined not by the number of edges but by the distances.
