# Search on an NK Landscape with Swarm Intelligence: Limitations and Future Research Opportunities
## 3 Swarm Intelligence Algorithms
### Mathematical Development of Swarm
The results are reported in Figure 7.

Figure 7: \(N=6\) and \(K=2\). Each simulated firm is represented by a different color. Panel A of Figure 7 presents the result of pure exploitation. In this simulation, every firm seeks to move toward the best fitness among its peers. As we can see, in two iterations, all fish stop moving as they all reach a local peak of 1.242677 (the third highest fitness level, see Figure 6) as opposed to a global peak of 1.279608. This is because each firm can jump to the best position quickly. Since this happens so quickly, without exploration, everyone stops. Panel B of Figure 7 presents the result of pure exploration. However, a firm will only move if a new position is better than its current one (i.e., firms have memory). We can see that all firms move toward the global peak, which is 1.279608. Given that there are only 64 nodes but there are 20 firms, it is understandable that one firm will very soon find the peak, and eventually every firm will find the peak. Panel C presents the results of the random combination of exploration and exploitation. We set 75% for exploitation and 25% for exploration and each firm randomly choosing between the two. We reach a local peak at 1.273777 (the second-best fitness value, see Figure 6). This is in between pure exploitation and pure exploration, as expected. It is clear by looking at Panel C that exploitation brings very quickly every firm to 1.242677, but instead of getting stuck there, after six iterations, one firm finds a higher position in the landscape and brings everyone else there. Hence, all firms will end up at the global peak. In terms of visualizing the climbing along the landscape as commonly depicted in the literature, such as Figure 3B, we plot firm #4 in Figure 8.

Figure 8 is the same landscape as Figure 5A, with the path of firm #4 marked. The positions through which firm #4 travels are given below the graph. It is clear that the collapsed landscapes given in Figure 7 do not provide an accurate description of the exact path on which each firm travels. Next, we present the results of \(N=6\) and \(K=5\), which we report in Figure 9. Similar to Figure 7, there are three panels for pure exploitation, pure exploration, and combination. In this case, the landscape is extremely rugged. Again, only pure exploration can reach the peak, which is shown in Figure 9B. Like the previous result, pure exploitation (Figure 9A) is stuck at a local peak for similar reasons. Finally, the combination will eventually reach the global peak (Figure 9C). One interesting observation is the contrast between Figures 7B and 9B, where the case of pure exploration is present in both landscapes. In a smooth landscape, such as Figure 7B, each firm reaches the global peak gradually and slowly. This is expected, as neighborhoods are just slightly better than the current position. On the contrary, in the case of a rugged landscape, large improvements in fitness can be very nearby, and hence jumps in fitness are more likely. Furthermore, the global peak is likely to be just nearby (for some firms). This is observed in Figure 9B. Figure 8: An example search path. Figure 9: \(N=6\) and \(K=5\). Each simulated firm is represented by a different color. As a conclusion, we find that exploration outperforms exploitation. This is consistent with the finding by [PERSON] (2022) [47]. This indicates that, in a high-dimensional NK landscape, following peers too closely is very likely to land on a local peak. ### Local Search

In this sub-section, we provide the results of a local (one-mutant) search. Each firm can still see the global best but cannot move there in one step. Instead, it must move to its immediate neighbor position. For example, 101 can only move to 001, 111, or 100. The main two reasons for such a constraint are that jumps are costly, and firms are not easily (politically or economically) able to perform large jumps.