*3.1. Tree-like Structure*

We first examine maze-solving for the tree-like structures. This structure has no cycles, and there is only one path from the start to the goal. Figure 2a–c show the results of the amplitude distribution and the number of steps after convergence for the tree-like structures of 2*<sup>N</sup>* leaves for *N* = 1, 2, and 3. From the results, we observed that only the shortest path emerges as a chain of the eternally remaining densities, whereas the densities on the dead ends vanish during the evolution. The number of convergence steps seems to increase by digits according to the increase of *N* in these cases. The case of *N* = 4 was also examined. However, the distribution did not converge even after 10<sup>6</sup> steps that took three days.

Figure 2d shows the time profiles of the densities on selected edges, where the label 0–3, for example, denotes the edge between nodes 0 and 3. The densities fluctuate strongly at first and then converge to zero or to positive values. The speed of convergence varies according to the position of the edge; the greater the distance to the sink node is, the slower the speed of convergence.

To consider the influence of the extra branching at dead ends on the convergence steps, the cases of decreased and increased extra branching based on Figure 2c were examined. For the case with decreased branching as shown in Figure 2e, the convergence steps decreased, which was an intuitive result. However, the decrease in convergence steps was more for the case with increased extra branching as shown in Figure 2f. This counterintuitive dependence is difficult to explain for the present. However, it can be suggested that the extent of asymmetry in the network accelerated the convergence.

For the cases of Figure 2c,e,f, the absolute values of the converged amplitudes on the correct paths, including self-loops, were all 0.08. That value seems to have been determined by the distance between the start and the goal nodes for the case of the network without cycles. Table 1 lists the relation between the distance between the start and the goal and the absolute value of the amplitude remaining on an edge for each case. Edges indicates the number of edges on a path including self-loops. (Edges = 2 × Distance + 2).

A rational expression approximating the amount of amplitude was attached for each case. For these cases, the amplitudes can be expressed by the inverse of the number of edges included in the path. Namely, the sum of amplitudes along the path is "1.0" for all the cases. However, note that the "1.0" does not indicate all the amplitudes injected into the system because that is not the square sum of the amplitudes.

**Figure 2.** The results of maze-solving for the tree-like structure with 2*<sup>N</sup>* leaves. (**a**) Amplitude distribution and the number of steps after convergence for *N* = 1. (**b**) Amplitude distribution and the number of steps after convergence for *N* = 2. (**c**) Amplitude distribution and the number of steps after convergence for *N* = 3. (**d**) Time profiles of the densities on selected edges for *N* = 3. The inset focuses on the vibrational behavior of each profile. (**e**) Amplitude distribution and the number of steps after convergence for the case where the branches were eliminated from the dead ends in (**c**). (**f**) Amplitude distribution and the number of steps after convergence for the case where the branches were added to the dead ends in (**c**).

**Table 1.** The relation between the distance and remaining amplitude for Figure 2 (The distance between the start and goal on the correct 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).


## *3.2. A Line with Branches*

To investigate the dependence of the convergence steps on the placement of the branches, the maze-solving for various patterns of a line with shallow dead ends was examined. Figure 3a shows the result for a simple line constructed based on the correct path of Figure 2c. The number of convergence steps decreased by two orders of magnitude from the case shown in Figure 2c. Figure 3b shows the result for a line with four shallow dead-ends. Nearly the same result as Figure 2e was obtained, as the difference between them was only the length of the dead ends.

**Figure 3.** The results of maze-solving for a line with various placements of shallow dead ends. (**a**) Amplitude distribution and the number of steps after convergence for a single line of five edges. (**b**) Amplitude distribution and the number of steps after convergence for a line with four shallow dead ends. (**c**) The structures and the numbers of steps of convergence for a line with a single shallow dead-end at four positions. (**d**) The structures and the numbers of steps of convergence for a line with two shallow dead-ends at three patterns. (**e**) The structures and the numbers of steps of convergence for a line with three shallow dead-ends at four patterns.

Figure 3c–e shows the results for patterns of placement of one to three dead ends, respectively. The distribution of the amplitudes is omitted, but ±0.08, which is the same as Figure 2c,e,f, is on the correct path, and 0.00 is on the dead-end edges in all cases. The

convergence steps varied not only by the number of dead ends but also by the positions. As the trend shows, the convergence steps became larger with increasing dead ends; however, exceptions were observed depending on the positions of the dead ends. The convergence steps became larger for the case where dead ends were attached closed to the goal node. This seems counterintuitive considering the quick convergence near the sink, which was observed in Figure 2d.

The number of convergence steps for the case of Figure 3e was much larger than for Figure 3b or Figure 2c. For these cases, the asymmetry significantly decelerated the convergence speed, which is in contrast to the acceleration due to asymmetry observed in Figure 2e,f. A maze without cycles can be solved by this method; however, the dependence of the convergence steps on the network structure is difficult to predict intuitively.
