# Search on an NK Landscape with Swarm Intelligence: Limitations and Future Research Opportunities

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## Abstract

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## 1. Introduction

#### Future Research Directions

- A payoff landscape or function that allows for continuous movement so that the full value of swarm in modeling agent (firm or human) behavior can be realized.
- Endogenous landscapes that allow payoffs to change as a function of agents positions on the landscape.
- Agent-specific landscapes that allow for researchers to explore heterogeneous returns to agent actions.
- Allowing information between agents, their locations, and their performance to be uncertain or incomplete.
- Including the cost of movements into the framework rather than implicitly assuming free movements on the landscape.

## 2. NK Landscape

#### 2.1. Background

^{N}. Each combination of component states has a ‘fitness’ associated with it, which is a measure of how well the system is performing in that particular state. Depending on the interactions among components, the fitness or performance landscape can be visualized as a surface with peaks (high fitness) and valleys (low fitness). A smooth landscape has fewer local optima, while a rugged landscape has many.

#### 2.2. Mathematical Development of NK

## 3. Swarm Intelligence Algorithms

#### 3.1. Background

#### 3.2. Swarm Intelligence to Model Agent Behavior

#### 3.3. Mathematical Development of Swarm

## 4. Search Algorithms

- One-mutant change or hill climb, where an agent chooses a new location from one of its one-mutant neighbors, such as 101 to 001, 111, or 100, if the fitness value of the new location is higher.
- Fitter-dynamics: an agent chooses a new location from the best of its one-mutant neighbors.
- Greedy dynamics (i.e., large or long jump), where an agent chooses a new location from all of its mutant neighbors, whoever has the highest fitness value.

## 5. Results

#### 5.1. Global Search

#### 5.2. Local Search

#### 5.3. Large N

^{20}= 1,048,576. In our simulations, we use N = 15. We also increase the number of firms to 100, which is a much smaller amount compared to the large space of over 1 million dimensions. As for K, we perform simulations for two cases: K = 14, the most rugged landscape, and the other is a randomly chosen K. In the latter case, we randomly choose K by setting a 50% chance that any other component has an influence on or not on component $i$. Specifically, for a component $i$, we randomly decide if any component $j\ne i$ has an influence or not on component $i$. Hence, on average, K is roughly 7. While we can make the landscape smoother, the result already seems quite conclusive, hence indicating no such need for a smoother landscape.

## 6. Discussion of Simulation Results and Identification of Limitations of the NK Landscape for Swarm-Based Search

#### 6.1. Discussion of Simulation Results

#### 6.2. Further Discussion of NK Limitations for Swarm-Based Search Studies

## 7. Directions for Future Research

#### 7.1. The Need for a Flexible Landscape

#### 7.2. Landscape and Search Process Extensions

#### 7.2.1. Endogenous Landscapes

#### 7.2.2. Agent-Specific Landscapes

#### 7.2.3. Incomplete Information

#### 7.2.4. Costly Movements

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**An example of a binary 3-dimensional NK landscape. Note: The battery (${x}_{1}$) depends on the screen, the CPU (${x}_{2}$) depends on the battery, and the screen (${x}_{3}$) depends on CPU. The fitness values are presented in parentheses (the global peak is 0.6333 at 1, 1, 1).

**Figure 10.**N = 6, K = 2, one-mutant search (randomly choose between exploitation 75% and exploration 25%). Each simulated firm is represented by a different color.

**Figure 11.**N = 6, K = 5, one-mutant search. Each simulated firm is represented by a different color.

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**MDPI and ACS Style**

Chen, R.-R.; Miller, C.D.; Toh, P.K.
Search on an NK Landscape with Swarm Intelligence: Limitations and Future Research Opportunities. *Algorithms* **2023**, *16*, 527.
https://doi.org/10.3390/a16110527

**AMA Style**

Chen R-R, Miller CD, Toh PK.
Search on an NK Landscape with Swarm Intelligence: Limitations and Future Research Opportunities. *Algorithms*. 2023; 16(11):527.
https://doi.org/10.3390/a16110527

**Chicago/Turabian Style**

Chen, Ren-Raw, Cameron D. Miller, and Puay Khoon Toh.
2023. "Search on an NK Landscape with Swarm Intelligence: Limitations and Future Research Opportunities" *Algorithms* 16, no. 11: 527.
https://doi.org/10.3390/a16110527