• Minsol Kim
• Bhargav Solanki

This repository hosts our group's submission for Task Sheet 2 of Collective robotics and scalability. The submission includes all necessary code, data visualizations (plots).

• Task2.1a: Calculate the average number of neighbors per firefly for vicinity distances r ∈ {0.05,0.1,0.5,1.4}
• Task2.1b: Determine the minimum and maximum number of concurrently flashing fireflies during the very last cycle (last L = 50 time steps starting from t = 4950)

• count_neighbors.py and simulate_fireflies.py are used for the task2.1a in the file Task2_1_a.py
• plot1.png is the result of Task2_1_a.py
• The plot1.png shows the number of flashing fireflies over time for different radii. Each subplot represents a different radius at which fireflies interact with each other, and the graph tracks how the number of fireflies flashing synchronously changes over the course of 5000 time steps.

Analysis of Each Radius

1. Radius 0.05 (Blue Line)

   • The blue line shows moderate fluctuations in the number of flashing fireflies, but the overall trend is fairly consistent without extreme peaks or troughs. This suggests that with a small radius, fireflies have fewer neighbors, resulting in less synchronization across the entire swarm.

2. Radius 0.1 (Green Line)

   • The green line demonstrates increased fluctuations compared to the smaller radius, which may indicate a better but still limited interaction among fireflies. The synchronization appears to improve slightly, possibly due to fireflies influencing more of their neighbors, but still does not achieve full synchronization.

3. Radius 0.5 (Red Line)

   • With the red line, there's a noticeable increase in the amplitude of fluctuations, and the pattern appears more chaotic. This suggests that a larger radius allows a firefly to influence and be influenced by more neighbors. The increased interactions can cause more dynamic changes in the state of flashing, leading to greater synchronization or desynchronization at different intervals.

4. Radius 1.4 (Purple Line)

   • The purple line shows very regular and high amplitude fluctuations, indicating strong synchronization across the swarm. At this radius, essentially every firefly is a neighbor of every other firefly due to the size of the radius relative to the square, allowing for very effective synchronization. The high consistency and amplitude in flashing suggest that most fireflies flash in unison or nearly so.

General Observations

• Increasing Radius and Synchronization: As the radius increases, the synchronization among fireflies tends to improve, with more fireflies flashing together. This is evident from the increase in the amplitude of fluctuations—larger radii tend to show more pronounced peaks and valleys, indicative of more fireflies flashing in unison.
• Stability and Chaos: Larger radii can also introduce more chaotic dynamics as fireflies are influenced by a larger number of neighbors. This can lead to rapid changes in the state of synchronization, which might explain the more chaotic appearance of the red graph.

Conclusion

From these observations, a larger radius generally leads to better synchronization among fireflies, as seen with the radius of 1.4, where the fluctuations are pronounced and regular, indicating strong collective behavior. However, the optimal radius for achieving synchronization without introducing too much chaos would need to balance these aspects, possibly favoring a slightly smaller radius than 1.4 to reduce the chaos while still maintaining good synchronization. Further analysis could involve looking at the specific dynamics at each radius to determine the best balance between radius size and synchronization efficiency.

• analyze_synchronization.py is used for the task2.1b in the file Task2_1_b_m.py

• plot2.png is the result of Task2_1_b_m.py

• The plot2.png shows the average synchronization amplitude as a function of the vicinity radius for a group of simulated fireflies. The synchronization amplitude refers to half the difference between the maximum and minimum number of fireflies flashing concurrently during the last cycle of each simulation. This metric is useful to understand how tightly coupled the flashing behavior is across the swarm.

Analysis of the Graph

The graph shows a non-linear relationship between the vicinity radius and the synchronization amplitude:

• Low Radius (0.0 to ~0.6): Initially, the amplitude is relatively lower and fluctuates as the radius increases. This region indicates partial synchronization, where only nearby fireflies influence each other significantly due to the smaller interaction radius. The fluctuations in this region could be due to local clusters of fireflies achieving synchronization while others remain out of phase.

• Intermediate Radius (~0.6 to ~1.0): As the radius increases, the amplitude generally rises, peaking around a radius of 1.0. This peak represents a high degree of synchronization across the entire swarm, suggesting that the interaction radius is optimal for most fireflies to influence and be influenced by their neighbors effectively, leading to a more unified flashing pattern.

• High Radius (~1.0 to 1.4): Beyond the peak, the amplitude sharply declines as the radius continues to increase. This decline can be interpreted as too large a radius causing too much interaction, which paradoxically might reduce the synchronization. When every firefly influences nearly every other, individual groups might start flashing in response to different subsets of the swarm, leading to a breakdown in the overall synchronization.

Interpretation and Implications

• Optimal Radius: The vicinity radius of around 1.0 appears to be optimal for achieving the highest synchronization amplitude. This radius likely balances the breadth of influence each firefly has over others, maximizing the number of fireflies that can synchronize their flashing without introducing discrepancies caused by overly widespread influences.

• Swarm Density Considerations: Higher synchronization at optimal radi suggests that the density and distribution of fireflies are appropriate for such a radius to work effectively. If the density were significantly different, the optimal radius might shift. This observation is crucial for understanding how firefly swarms might adapt their behavior in different ecological conditions or how similar principles could be applied in technological applications like drone swarming algorithms.

Conclusion

The results make sense within the context of collective behavior and complex systems where interactions are known to lead to emergent properties such as synchronization. The findings suggest focusing on a vicinity radius of around 1.0 for further simulations and potential real-world applications where synchronous behavior is desired.