1. Introduction

2. Building the Model:

3. Simulations Model

• Effective Potential Energy (V): The potential energy due to pheromone concentration $C(\mathbf{r},t)$ at position $\mathbf{r}$ and time t can be modeled as:

4. Mechanism

5. Simulation Methods

6. Results

6.1. Time Graphs

The raw data are presented as output measures vs time for four quantities, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11. All variables measure the degree of order in the system. The time data show the phase transition from a disorganized to an organized state as an increase in AAE, Figure 7, of the order parameter, Figure 8, as a decrease in internal entropy, Figure 9, the amount of information, Figure 10 and the number of events per unit time, Figure 11. Those data are exponential in the region before the inflection point of the curves, where growth is unconstrained.

The AAE, Figure 7, changes similarly to density which serves as an order parameter, Figure 8, inversely to entropy, Figure 9, and similar to information, Figure 10, and flow, Figure 11. The system starts with a close to zero AAE and order parameter which increase to some maximum value and then saturate as the system is fixed in size. In the case of entropy, the system starts at maximum internal entropy and it drops to a minimum value as the system reaches the saturation point.

Pheromone is the amount of information in the system which is proportional to the degree of order, Figure 10, and flow rate is also proportional to all of the other measures, indicating the number of events as defined in the system, Figure 11. Both of these start at an initial minimum value and undergo a phase transition to the organized state, after which they saturate, due to the fixed size of the system. Those are measures directly connected to action efficiency and are some of the most important performance metrics for self-organizing systems.

Figure 7. Increase in Average Action Efficiency with Ant Population. The graph shows the progression of Average Action Efficiency (AAE) over time for different ant populations, ranging from 70 to 200 ants in increments of 10 (bottom curve to top). Initially, AAE increases steeply during the phase transition as ants explore and reinforce shorter paths. As the simulation approaches its limits, the increase slows, but AAE continues to rise gradually up to 1000 ticks due to the strengthening and annealing of the shortest path. Below time 100, the data for the average path time are not reliable, and those points are missing due to the specifics of the simulation.

Figure 7. Increase in Average Action Efficiency with Ant Population. The graph shows the progression of Average Action Efficiency (AAE) over time for different ant populations, ranging from 70 to 200 ants in increments of 10 (bottom curve to top). Initially, AAE increases steeply during the phase transition as ants explore and reinforce shorter paths. As the simulation approaches its limits, the increase slows, but AAE continues to rise gradually up to 1000 ticks due to the strengthening and annealing of the shortest path. Below time 100, the data for the average path time are not reliable, and those points are missing due to the specifics of the simulation.

Figure 7 provides insight into how the Average Action Efficiency (AAE) evolves during the simulation as the number of ants is incrementally increased from 70 to 200. The steep initial rise in AAE reflects the phase transition where ants begin to organize by reinforcing shorter paths. As the system nears its operational limits, the rate of increase diminishes, signifying stabilization in the path optimization process. Despite this, AAE continues to improve gradually up to 1000 ticks, indicating the ongoing refinement and annealing of the shortest path as the simulation progresses. This figure emphasizes the influence of agent population on the dynamics of self-organization and efficiency optimization.

6.1.1. a Note on the Rate of Self-Organization as a Function of the Size of the System

Another observation in this figure is related to the rate of self-organization as a function of the size of the system. AAE for larger populations of ants undergoes a phase transition from a disorganized to an organized state systematically earlier in time, which can also be seen in the rest of the metrics presented in this section of the paper. This indicates that the rate of self-organization as a characteristic of complex systems, also depends on the size of the system, which we will explore in the next parts of this paper. We consider this an important aspect of this study because the message is that for a system to self-organize faster and to achieve higher levels of organization, it has to be larger. The second part of this statement is the essence of the size-complexity rule and the scaling relationships in complex systems, observed for many years by other authors [65,66,67].

Figure 8. The density of ants versus the time as the number of ants increases from the bottom curve to the top. As the simulation progresses, the ants become more dense.

Figure 8. The density of ants versus the time as the number of ants increases from the bottom curve to the top. As the simulation progresses, the ants become more dense.

The data in Figure 8 show the whole run as the density increases with time as self-organization occurs. The more ants, the larger the final density and the earlier the transition to it. The increase of density depends on two factors: 1. The shorter average path length at the end, and 2. The increased number of ants.

Figure 9. The internal entropy in the simulation versus the time as the number of ants increases from bottom to top. Entropy decreases from the initial random state as the path forms.

Figure 9. The internal entropy in the simulation versus the time as the number of ants increases from bottom to top. Entropy decreases from the initial random state as the path forms.

Figure 9 shows the initial entropy when the ants are randomly dispersed which scales with the number of ants because the number of microstates corresponding to the same macrostate grows. When the ants form the path, the amount of decrease of entropy is larger for the larger number of ants, even though the absolute value of entropy at the end of the simulation also scales with the number of ants, since the number of microstates when the ants form the final path also scales with the number of ants. Entropy at the initial and final state of the simulation is seen to grow with the increase in the number of agents. In this dataset we also observe an earlier transition to order with increased population of ants.

Figure 10. The total amount of pheromone versus the time passed as the number of ants increases from bottom to top. As the simulation progresses, there is more pheromone for the ants to follow.

Figure 10. The total amount of pheromone versus the time passed as the number of ants increases from bottom to top. As the simulation progresses, there is more pheromone for the ants to follow.

The pheromone is a measure of the information in the system. Its change during the simulation is shown on Figure 10. It scales with the number of ants. Each simulation starts with zero pheromones as the ants are dispersed randomly and do not carry any pheromones, but as they start forming the path they lay pheromones from the food and nest respectively and the more ants are in the simulation, the more pheromones they carry. Larger systems contain more information as each agent is a carrier of information. The transition happens faster when more ants are present in the system.

Figure 11. The flow rate versus the time passed as the number of ants increases from bottom to top. As the simulation progresses, the ants visit the endpoints more often.

Figure 11. The flow rate versus the time passed as the number of ants increases from bottom to top. As the simulation progresses, the ants visit the endpoints more often.

Figure 11 shows the flow rate vs. time during the simulation. The number of events, which is the number of visits to the food and nest, is inversely proportional to the average path length, and respectively average path time, scales with the number of ants as expected. Initially, the number of crossings is zero but it quickly increases and is greater for the larger systems. After the ants form the shortest path, it saturates and stays close to constant for all simulations. The transition to order happens earlier in time for simulations with a larger population.

Other metrics from the simulation can also be used as a measure of the rate and degree of self-organization. We will explore those in the follow-up papers.

6.2. Power Law Graphs

All figures representing the relationship between the characteristics of this system at the most organized state at the end of the simulation demonstrate power law relationships between all of the quantities as theoretically predicted by the model, as seen in Figure 12 to Figure 45. They are all on a log-log scale. This serves as a simulation confirmation of the model, and on the other hand as a theoretical explanation for the simulation. The power law data correspond to scaling relations measured in many systems of different natures [65,66,67].

For clarity, the "#" symbol represents the number of ants in the simulation. For these data, the number of ants does not change within the simulation; rather, each point represents a simulation with a different number of ants .

6.2.1. Size-Complexity rule

Figure 12 shows the size-complexity rule: as the size increases, the action efficiency as a measure of the degree of organization and complexity increases. This is supported by many experimental and observational data on scaling relations by Geoffrey West, Bonner, Carneiro, and many others [19,65,66,67]. The data coincide with the Kleiber’s law [81], and other similar laws, such as the area speciation rule in ecology and others.

Figure 12. The average action efficiency at the end of the simulation versus the number of ants, on a log-log scale. As more ants are added, they are able to form more action-efficient structures by finding shorter paths.

Figure 12. The average action efficiency at the end of the simulation versus the number of ants, on a log-log scale. As more ants are added, they are able to form more action-efficient structures by finding shorter paths.

6.2.2. Unit-Total Dualism

The following graphs serve as empirical support for the unit-total dualism described in this paper. Figure 13 shows the unit-total dualism between the average action efficiency and total action.

Figure 13. The average action efficiency at the end of the simulation versus the total action as the number of ants increases on a log-log scale. As there is more total action within the system, the ants become more action-efficient.

Figure 13. The average action efficiency at the end of the simulation versus the total action as the number of ants increases on a log-log scale. As there is more total action within the system, the ants become more action-efficient.

The total action is a measure of all energy and time spent in the simulation by the agents in the system, as it can be seen in Figure 13. As the number of agents increases, the total action increases. This suggests a duality of decreasing unit action and increasing total action as a system self-organizes progressively . It also points to a dynamical action principle, as the unit action per one event decreases with the growth of the system, as seen in the increase of the average action efficiency, while the total action increases. This is an expression of the dualism for the decreasing unit action principle and the increasing total action principle, for dynamical action as systems self-organize, grow, evolve, and develop.

Figure 14 is an expression of the unit-total duality of entropy when the unit entropy per one event in the system tends to decrease its total internal entropy of the system increases.

Figure 14. Unit entropy at the end of the simulation versus internal entropy on a log-log scale. As the total entropy for the simulation increases, the entropy per agent decreases.

Figure 14. Unit entropy at the end of the simulation versus internal entropy on a log-log scale. As the total entropy for the simulation increases, the entropy per agent decreases.

6.2.3. the Rest of the Characteristics

Next, we show the rest of the power law fits between all of the quantities in the model, in Figure 15 to Figure 45. All of them are on a log-log scale, where a straight line is a power law curve on a linear-linear scale. These graphs match the predictions of the model and confirm the power-law relationships between all of the characteristics of a complex system derived there.

Figure 15. The average action efficiency at the end of the simulation versus the average time required to traverse the path as the number of ants increases on a log-log scale. Average action efficiency increases as the average time to reach the destination shortens, i.e. the path length becomes shorter.

Figure 15. The average action efficiency at the end of the simulation versus the average time required to traverse the path as the number of ants increases on a log-log scale. Average action efficiency increases as the average time to reach the destination shortens, i.e. the path length becomes shorter.

Figure 15 shows the average action efficiency at the end of the simulation versus the time required to traverse the path as the size of the system, in terms of a number of agents, increases. In complex systems, as the agents find shorter paths, this state is more stable in dynamic equilibrium and is preserved. It has a higher probability of persisting. It is memorized by the system. If there is friction in the system, this trend will become even stronger, as the energy spent to traverse the shorter path will also decrease. To the macro-state of AAE at each point, there is a growing number of micro-states, corresponding to the variations of the paths of individual agents.

Figure 16 shows the average action efficiency at the end of the simulation versus the density increase of the agents as the size of the system increases in terms of the number of agents. Density increases the probability of shorter paths, i.e. less time to reach the destination, i.e. larger action efficiency. In natural systems as density increases, action efficiency increases, i.e. level of organization increases. Another term for density is concentration. When hydrogen gas clouds in the universe under the influence of gravity concentrate into stars, nucleosynthesis starts and the evolution of cosmic elements begins. In chemistry increased concentration of reactants speeds up chemical reactions, i.e. they become more action efficient. When single-cell organisms concentrate in colonies and later in multicellular organisms their level of organization increases. When human populations concentrate in cities, the organization increases, and civilization advances [67].

Figure 16. The average action efficiency at the end of the simulation versus the density increase is measured as the difference between the final density minus the initial density as the number of ants increases, on a log-log scale. As the ants get denser, they become more action-efficient.

Figure 16. The average action efficiency at the end of the simulation versus the density increase is measured as the difference between the final density minus the initial density as the number of ants increases, on a log-log scale. As the ants get denser, they become more action-efficient.

As internal statistical Boltzmann entropy decreases by a greater amount during self-organization, as seen in Figure 17, the system becomes more action-efficient. Decreased randomness is correlated with a well-formed path as a flow channel, which corresponds to the structure (organization) of the system. Here, the increase of entropy difference obeys the predictions of the model being in a strict power law dependence on the other characteristics of the self-organizing complex system.

Figure 17. The average action efficiency at the end of the simulation versus the absolute amount of entropy decrease, as the number of ants increases, on a log-log scale. As the ants get less random, they become more action-efficient.

Figure 17. The average action efficiency at the end of the simulation versus the absolute amount of entropy decrease, as the number of ants increases, on a log-log scale. As the ants get less random, they become more action-efficient.

Figure 18 shows the average action efficiency at the end of the simulation versus the flow rate as the size of the system in terms of the number of agents increases. The flow rate measures the number of events in a system. For real systems, those can be nuclear or chemical reactions, computations, or any other events. In this simulation, it is the number of visits at the endpoints, or the number of crossings. As the speed of the ants is a constant in this simulation, the number of visits or the flow of events is inversely proportional to the time for crossing, i.e. the path length, therefore action efficiency increases with the number of visits.

Figure 18. The average action efficiency at the end of the simulation versus the flow rate as the number of ants increases, on a log-log scale. As the ants visit the endpoints more often, they become more efficient.

Figure 18. The average action efficiency at the end of the simulation versus the flow rate as the number of ants increases, on a log-log scale. As the ants visit the endpoints more often, they become more efficient.

Figure 19 shows the average action efficiency at the end of the simulation versus the amount of pheromone, or information, as the size of the system in terms of the number of agents increases. The pheromone is what instructs the ants how to move. They follow its gradient towards the food or the nest. As the ants form the path, they concentrate more pheromone on the trail, and they lay it faster so it has less time to evaporate. Both depend on each other in a positive feedback loop. This leads to increased action efficiency, with a power-law dependence as predicted by the model. In other complex systems, the analog of the pheromone can be temperature and catalysts in chemical reactions. In an ecosystem, as animals traverse a path, the path itself carries information, and clearing the path reduces obstacles and, therefore the time and energy to reach the destination, i.e. action.

Figure 19. The average action efficiency at the end of the simulation versus the amount of pheromone, or information, as the number of ants increases on a log-log scale. As there is more information for the ants to follow, they become more efficient.

Figure 19. The average action efficiency at the end of the simulation versus the amount of pheromone, or information, as the number of ants increases on a log-log scale. As there is more information for the ants to follow, they become more efficient.

Figure 20 shows the total action at the end of the simulation versus the size of the system in terms of the number of agents. The total action is the sum of the actions of each agent. As the number of agents grows the total action grows. This graph demonstrates the principle of increasing total action in self-organization, growth, evolution, and development of systems.

Figure 20. The total action at the end of the simulation versus the number of ants on a log-log scale. As there are more agents in the system, the total amount of action increases proportionally.

Figure 20. The total action at the end of the simulation versus the number of ants on a log-log scale. As there are more agents in the system, the total amount of action increases proportionally.

Figure 21 shows the total action at the end of the simulation versus the time required to traverse the path as the size of the system, in terms of the number of agents increases. With more ants, the path forms better and gets shorter, which increases the number of visits. The shorter time is connected to more visits and increased size of the system, which is why the total action increases. This graph also demonstrates the principle of increasing total action in self-organization, growth, evolution, and development of systems.

Figure 21. The total action at the end of the simulation versus the time required to traverse the path as the number of ants increases on a log-log scale.

Figure 21. The total action at the end of the simulation versus the time required to traverse the path as the number of ants increases on a log-log scale.

Figure 22 shows the total action at the end of the simulation versus the increase in the density of agents as the size of the system in terms of the number of agents increases. The larger the system is, it contains more agents, which corresponds to greater density, more trajectories, and more total action. This graph demonstrates as well the principle of increasing total action in self-organization, growth, evolution, and development of systems.

Figure 22. The total action at the end of the simulation versus the increase of density as the number of ants increases on a log-log scale. As the ants become more dense, there is more action in the system.

Figure 22. The total action at the end of the simulation versus the increase of density as the number of ants increases on a log-log scale. As the ants become more dense, there is more action in the system.

Figure 23 shows the total action at the end of the simulation versus the absolute decrease of entropy as the size of the system in terms of the number of agents increases. As the total entropy difference increases, which means that the decrease of the internal entropy is greater for a larger number of ants, the total action increases, because there are more agents in the system and they visit the nodes more often. Greater organization of the system is correlated with more total action demonstrating again the principle of increasing total action in self-organization, growth, evolution, and development of systems.

Figure 23. The total action at the end of the simulation versus the absolute increase of entropy difference as the number of ants increases, on a log-log scale. As the entropy difference increases, there is more action within the system.

Figure 23. The total action at the end of the simulation versus the absolute increase of entropy difference as the number of ants increases, on a log-log scale. As the entropy difference increases, there is more action within the system.

Figure 24 shows the total action at the end of the simulation versus the flow rate, which is the number of events per unit time, as the size of the system in terms of the number of agents increases. As the flow of events increases, which is the number of crossings of ants between the food and nest, the total action increases, because there are more agents in the system and they visit the nodes more often by forming a shorter path. This also demonstrates the principle of increasing total action in self-organization, growth, evolution, and development of systems.

Figure 24. The total action at the end of the simulation versus the flow rate as the number of ants increases, on a log-log scale. As the ants visit the endpoints more often, there is more total action within the system.

Figure 24. The total action at the end of the simulation versus the flow rate as the number of ants increases, on a log-log scale. As the ants visit the endpoints more often, there is more total action within the system.

Figure 25 shows the total action at the end of the simulation versus the amount of pheromone as a measure for information, as the size of the system in terms of the number of agents increases. As the total number of agents in the system increases, they leave more pheromones, which causes forming a shorter path, increases the number of visits, and the total action increases. Again, this graph demonstrates the principle of increasing total action in self-organization, growth, evolution, and development of systems.

Figure 25. The total action at the end of the simulation versus the amount of pheromone as the number of ants increases on a log-log scale. As there is more information for the ants to follow, there is more action within the system.

Figure 25. The total action at the end of the simulation versus the amount of pheromone as the number of ants increases on a log-log scale. As there is more information for the ants to follow, there is more action within the system.

Figure 26 shows the total pheromone as a measure of the amount of information at the end of the simulation versus the size of the system in terms of number of agents. As the total number of ants in the system increases, they leave more pheromones and form a shorter path, which counters the evaporation of the pheromones. This increases the amount of information in the system, which helps with its rate and degree of self-organization.

Figure 26. The total pheromone at the end of the simulation versus the number of ants, on a log-log scale. As more ants are added to the simulation, there is more information for the ants to follow.

Figure 26. The total pheromone at the end of the simulation versus the number of ants, on a log-log scale. As more ants are added to the simulation, there is more information for the ants to follow.

Figure 27 shows the total pheromone at the end of the simulation versus the average path time required to traverse the path as the size of the system in terms of the number of agents increases. As the total number of ants in the system increases, they form a shorter path as the degree of self-organization is higher, and as they visit the food and nest more often and there are greater number of ants they leave more pheromones. The increased amount of information in turn helps form an even shorter path which reduces the pheromone evaporation increasing the pheromones event more. This is a visualization of the result of this positive feedback loop.

Figure 27. The total pheromone at the end of the simulation versus the time required to traverse the path as the number of ants increases on a log-log scale. As it takes less time for the ants to travel between the nodes, there is more information for the ants to follow and as there is more pheromone to follow, the trajectory becomes shorter - a positive feedback loop.

Figure 27. The total pheromone at the end of the simulation versus the time required to traverse the path as the number of ants increases on a log-log scale. As it takes less time for the ants to travel between the nodes, there is more information for the ants to follow and as there is more pheromone to follow, the trajectory becomes shorter - a positive feedback loop.

Figure 28 shows the total pheromone as a measure of information at the end of the simulation versus the density increase as the size of the system in terms of the number of agents increases. As the total number of ants in the system increases, they form a shorter path as the degree of self-organization is higher, and as there are more ants, their density increases, and as they visit the food and nest more often and there is a greater number of ants and lower evaporation, they leave more information.

Figure 28. The total pheromone at the end of the simulation versus the density increase as the number of ants increases on a log-log scale. As the ants become more dense, there is more information for them to follow.

Figure 28. The total pheromone at the end of the simulation versus the density increase as the number of ants increases on a log-log scale. As the ants become more dense, there is more information for them to follow.

Figure 29 shows the total pheromone as a measure of the amount of information at the end of the simulation versus the absolute decrease of entropy as the size of the system in terms of the number of agents increases. As the total number of ants in the system increases, they form a shorter path as the degree of self-organization is higher. As there are more ants, the entropy difference increases. The entropy during each simulation decreases, and as they visit the food and nest more often and there is a greater number of ants and less evaporation they accumulate more pheromones.

Figure 29. The total pheromone at the end of the simulation versus the absolute increase of entropy difference as the number of ants increases on a log-log scale. As the entropy difference increases, there is more information for the ants to follow and greater self-organization.

Figure 29. The total pheromone at the end of the simulation versus the absolute increase of entropy difference as the number of ants increases on a log-log scale. As the entropy difference increases, there is more information for the ants to follow and greater self-organization.

Figure 30 shows the total pheromone as a measure of the amount of information in the systems at the end of the simulation versus the flow rate, which is the number of events (crossings of the edge) per unit of time, as the size of the system in terms of the number of agents increases. As the total number of ants in the system increases, they form a shorter path as the degree of self-organization is higher. They visit the food and nest more often, and as there are more ants, the number of visits increases proportionally, the evaporation decreases, and they accumulate more pheromones.

Figure 30. The total pheromone at the end of the simulation versus the flow rate as the number of ants increases on a log-log scale.

Figure 30. The total pheromone at the end of the simulation versus the flow rate as the number of ants increases on a log-log scale.

Figure 31 shows the flow rate in terms of the number of events at the end of the simulation versus the size of the system in terms of the number of agents. As the total number of ants in the system increases, they form a shorter path as the degree of self-organization is higher, visit the food and nest more often, and the number of visits increases proportionally.

Figure 31. The flow rate at the end of the simulation versus the number of ants, on a log-log scale. As more ants are added to the simulation and they are forming shorter paths in self-organization, the ants are visiting the endpoints more often.

Figure 31. The flow rate at the end of the simulation versus the number of ants, on a log-log scale. As more ants are added to the simulation and they are forming shorter paths in self-organization, the ants are visiting the endpoints more often.

Figure 32 shows the flow rate in terms of the number of events per unit of time at the end of the simulation versus the time required to traverse between the nodes as the size of the system in terms of the number of agents increases. As the total number of ants in the system increases, they form a shorter path as the degree of self-organization is higher, visit the food and nest more often, and as there are more ants, the number of visits increases proportionally.

Figure 32. The flow rate at the end of the simulation versus the time required to traverse between the nodes as the number of ants increases, on a log-log scale. As the path becomes shorter, the ants are visiting the endpoints more often.

Figure 32. The flow rate at the end of the simulation versus the time required to traverse between the nodes as the number of ants increases, on a log-log scale. As the path becomes shorter, the ants are visiting the endpoints more often.

Figure 33 shows the flow rate in terms of the number of events (edge crossings) per unit of time at the end of the simulation versus the time required to traverse between the nodes as the size of the system in terms of the number of agents increases. As the total number of ants in the system increases, they form a shorter path as the degree of self-organization is higher, this leads to an increase in density, and as there are more ants, the number of visits increases proportionally.

Figure 33. The flow rate at the end of the simulation versus the increase of density as the number of ants increases on a log-log scale. As the ants get more dense, they are visiting the endpoints more often.

Figure 33. The flow rate at the end of the simulation versus the increase of density as the number of ants increases on a log-log scale. As the ants get more dense, they are visiting the endpoints more often.

Figure 34 shows the flow rate in terms of the number of events (edge crossings) per unit of time at the end of the simulation versus the absolute decrease of entropy as the size of the system in terms of the number of agents increases. As the total number of ants in the system increases, they form a shorter path as the degree of self-organization is higher, the absolute decrease of entropy is larger, and as there are more ants, the number of visits increases proportionally.

Figure 34. The flow rate at the end of the simulation versus the absolute decrease of entropy as the number of ants increases, on a log-log scale. As the entropy decreases more, the ants are visiting the endpoints more often.

Figure 34. The flow rate at the end of the simulation versus the absolute decrease of entropy as the number of ants increases, on a log-log scale. As the entropy decreases more, the ants are visiting the endpoints more often.

Figure 35 shows the absolute amount of entropy decrease versus the size of the system in terms of the number of agents as it increases. As the total number of ants in the system increases, they form a shorter path as the degree of self-organization is higher, they start with a larger initial entropy and the difference between the initial and final entropy grows. More ants correspond to greater internal entropy decrease, which is one measure of self-organization. It is one of the scaling laws in the size-complexity rule.

Figure 35. The absolute amount of entropy decrease versus the number of ants, on a log-log scale. As more ants are added to the simulation, there is a larger decrease in entropy reflecting a greater degree of self-organization.

Figure 35. The absolute amount of entropy decrease versus the number of ants, on a log-log scale. As more ants are added to the simulation, there is a larger decrease in entropy reflecting a greater degree of self-organization.

Figure 36 shows the absolute amount of entropy decrease versus the average time required to traverse the path at the end of the simulation as the size of the system in terms of number of agents as they increase. As the total number of ants in the system increases, they form a shorter path, and the entropy decrease is greater, as the degree of self-organization is higher. When the path is shorter, this corresponds to shorter times to cross between the two nodes, the internal entropy decreases more.

Figure 36. The absolute amount of entropy decrease versus the time required to traverse the path at the end of the simulation as the number of ants increases, on a log-log scale. As it takes more time to move between the nodes with fewer ants, there is more of a decrease in entropy, and vice versa.

Figure 36. The absolute amount of entropy decrease versus the time required to traverse the path at the end of the simulation as the number of ants increases, on a log-log scale. As it takes more time to move between the nodes with fewer ants, there is more of a decrease in entropy, and vice versa.

Figure 37 shows the absolute amount of entropy decrease versus the amount of density increase at the end of the simulation as the size of the system in terms of the number of agents increases. As the total number of ants in the system increases, they form a shorter path as the degree of self-organization is higher, and as there are more ants their density increases, and the internal entropy difference increases proportionally.

Figure 37. The absolute amount of entropy decrease versus the amount of density increase as the number of ants increases on a log-log scale. As the ants become more dense, there is a larger decrease in entropy.

Figure 37. The absolute amount of entropy decrease versus the amount of density increase as the number of ants increases on a log-log scale. As the ants become more dense, there is a larger decrease in entropy.

Figure 38 shows the amount of density increase versus the size as it increases in terms of the number of agents. As the total number of ants in the system increases, they form a shorter path as the degree of self-organization is higher, and as there are more ants, the density increases proportionally.

Figure 38. The amount of density increase versus the number of ants, on a log-log scale. As more ants are added to the simulation, and they form shorter paths, density increases proportionally.

Figure 38. The amount of density increase versus the number of ants, on a log-log scale. As more ants are added to the simulation, and they form shorter paths, density increases proportionally.

Figure 39 shows the amount of density increase versus the average time required to traverse the path as the size increases in terms of number of agents. As the total number of ants in the system increases, they form a shorter path as the degree of self-organization is higher, visit the food and nest more often, the time to cross between the nodes decreases, and the density increases proportionally.

Figure 39. The amount of density increase versus the time required to traverse the path as the number of ants increases on a log-log scale. When there are more ants it takes less time to traverse the path, and there is more of an increase in density.

Figure 39. The amount of density increase versus the time required to traverse the path as the number of ants increases on a log-log scale. When there are more ants it takes less time to traverse the path, and there is more of an increase in density.

Figure 40 shows the average time required to traverse the path versus the increasing size of the system in terms of the number of agents. As the total number of ants in the system increases, they form a shorter path as the degree of self-organization is higher, visit the food and nest more often, and the time for the visits decreases proportionally, increasing action efficiency.

Figure 40. The time required to traverse the path versus the number of ants, on a log-log scale. As more ants are added to the simulation, it takes less time to move between the nodes because they form a shorter path at the end of the simulation.

Figure 40. The time required to traverse the path versus the number of ants, on a log-log scale. As more ants are added to the simulation, it takes less time to move between the nodes because they form a shorter path at the end of the simulation.

Figure 41 shows the final entropy at the end of the simulation versus the size of the system in terms number of agents. The final entropy in the system increases when there are more agents, and therefore more possible microstates of the system.

Figure 41. The final entropy at the end of the simulation versus population on a log-log scale. As the population increases, there is more entropy in the final most organized state.

Figure 41. The final entropy at the end of the simulation versus population on a log-log scale. As the population increases, there is more entropy in the final most organized state.

Figure 42 shows the initial entropy at the beginning of the simulation versus the size of the system in terms number of agents. The initial entropy reflects the larger number of agents in a fixed initial size of the system and scales with the size of the system as expected. The initial entropy in the system increases when there are more agents in the space of the simulation, and therefore more possible microstates of the system.

Figure 42. Initial entropy on the first tick of the simulation versus the population on a log-log scale. As the population increases, there is more entropy.

Figure 42. Initial entropy on the first tick of the simulation versus the population on a log-log scale. As the population increases, there is more entropy.

Figure 43 shows the unit entropy at the end of the simulation versus the size of the system in terms number of agents.

Figure 43. Unit entropy at the end of the simulation versus population on a log-log scale. As there are more agents, there is less entropy per path at the end of the simulation.

Figure 43. Unit entropy at the end of the simulation versus population on a log-log scale. As there are more agents, there is less entropy per path at the end of the simulation.

Figure 44 shows the unit information per one path at the end of the simulation versus the size of the system in terms number of agents. It shows that as the system increases in size, it has more ability to self-organize and to form shorter paths, therefore needing less information for each path, which in this system is one event in the system. More organized systems find shorter paths for their agents and need less information per path.

Figure 44. Unit information at the end of the simulation versus population on a log-log scale. As there are more agents, there is less information per path at the end of the simulation as the path is shorter.

Figure 44. Unit information at the end of the simulation versus population on a log-log scale. As there are more agents, there is less information per path at the end of the simulation as the path is shorter.

Figure 45 shows the unit information per one path at the end of the simulation versus the total information in the system. As the system grows, the total information in the system increases and it has more ability to self-organize and to form shorter paths, therefore needing less information for each path, which in this system is one event in the system. More organized systems find shorter paths for their agents, and need less information per path but increases the total amount of information in the system.

Figure 45. Unit information at the end of the simulation versus the total information in the system on a log-log scale. As there are more agents, there is less information per path at the end of the simulation as the path is shorter, and more total information as the size of the system in terms of the number of agents is larger.

Figure 45. Unit information at the end of the simulation versus the total information in the system on a log-log scale. As there are more agents, there is less information per path at the end of the simulation as the path is shorter, and more total information as the size of the system in terms of the number of agents is larger.

6.3. Comparison with Literature Data for Real Systems

Here we present for comparison data for other systems showing scaling behavior analogous to the model in this paper.

6.3.1. Stellar Evolution

Figure 46 reproduces Figure 5 from the paper [19] where the effect of size on the evolution of stars is presented. This relationship further highlights the predictive power of the model for self-organizing systems. Specifically, the Figure 46 below demonstrates how the "Progress of Nucleosynthesis" scales with the "Initial Total Number of Solar Masses," revealing distinct power-law relationships that align with the theoretical framework developed in this study. The figure underscores the robustness of these scaling relationships across different system characteristics.

In Figure 46 we see an example of data analysis for stellar systems, which shows that the level of organization in a system, in terms of the amount of progress of nucleosynthesis, as the fraction of the nucleons of a star converted to heavier than Hydrogen elements, is in a scaling law relationship with its size in terms of total initial number of nucleons, or total initial mass of the star. The powers of the exponents of the power law fits from the bottom to the top line are respectively: 0.86, 1.47, 1.8, and 2.6. They are close to many of the values of the power law relations in Table 5. This means that many of the power law dependencies in the simulation are in the range of data for stars as complex self-organizing systems. The composition of stars at the end of their life is measured directly with observations of the content of heavier elements in the nebulas that they produce after their explosions.

The closest characteristics from this simulation to the data in Figure 46 and the values of their corresponding exponents are: $\mathsf{\Delta }s$ corresponds to a decrease in internal entropy of the simulation which relates to the progress of nucleosynthesis because it measures the grouping of separate nucleons into heavier elements, which means that as the nucleons are grouping together their degrees of freedom of motion are reduced, which corresponds to decrease of internal entropy of a star. $\mathsf{\Delta }\rho$ relates to the progress of nucleosynthesis because as the nucleons are grouped in heavier elements, the density of the star in its volume where they are located increases, as nucleons are packed closer together. i relates to the progress of nucleosynthesis because as nucleons are linked together this is analogous to the ants being linked closer together through the pheromones on the final path. $\varphi$ relates to the progress of nucleosynthesis because as stars get larger, the number of events of grouping new nucleons is increasing, as in a shorter lifetime of the heavier stars, there are more events of connecting nucleons, as the fraction of heavier elements in the stars, which is the definition of the progress of nucleosynthesis, increases. Q relates to the progress of nucleosynthesis because it measures the total amount of action in terms of energy and time for the processes occurring in stars, and as stars get heavier, more nuclear reactions occur and a larger fraction of nucleons convert to heavier elements. The X-axis in solar masses is analogous to N in our simulation, which is the size of the system in terms of the number of agents, as for the star, the number of agents is the number of nucleons, which is directly proportional to their mass.

Figure 46. The relationship between the progress of nucleosynthesis and the initial total number of solar masses on a log-log scale, illustrating the power-law scaling inherent in self-organizing systems. It relates to the predictions from the model in this paper and the simulation results. The initial metalicity of stars varies from bottom to top from 0, 0.001, 0.004 and 0.02. Reproduced from [Butler, T.H.; Georgiev, G.Y. Self-Organization in Stellar Evolution: Size-Complexity Rule. In Efficiency in Complex Systems: Self-Organization Towards Increased Efficiency; Springer, 2021; pp. 53–80.] Reproduced with permission from Springer Nature.

Figure 46. The relationship between the progress of nucleosynthesis and the initial total number of solar masses on a log-log scale, illustrating the power-law scaling inherent in self-organizing systems. It relates to the predictions from the model in this paper and the simulation results. The initial metalicity of stars varies from bottom to top from 0, 0.001, 0.004 and 0.02. Reproduced from [Butler, T.H.; Georgiev, G.Y. Self-Organization in Stellar Evolution: Size-Complexity Rule. In Efficiency in Complex Systems: Self-Organization Towards Increased Efficiency; Springer, 2021; pp. 53–80.] Reproduced with permission from Springer Nature.

The corresponding values of the power law fits in Table 5 for those characteristics vs N respectively: For $\mathsf{\Delta }s$ is 1, for $\mathsf{\Delta }\rho$ is 1.11, for i is 1.09, for $\varphi$ is 1.104, and for Q is 0.983. We consider this as a good alignment between the results of this simulation and the characteristics of stars in their evolution in terms of the progress of nucleosynthesis. Another factor that makes the comparison valid is that the data for stars are at the end of the stellar life, and in our simulation, the data are at the end of each simulation at different populations, which means that the self-organization process is complete in both cases.

6.3.2. Evolution of Cities

In Figure 47 of the paper [82], Figure 1A, we see an example of data analysis for cities, which shows that the level of organization in a system, in terms of the GDP, is in a scaling law relationship with its size in terms of population. It is with an exponent of 1.126. This means that many of the power law dependencies in the simulation are in the range of data for cities as complex self-organizing systems.

Figure 47. The relationship between GDP and population for cities illustrates the power-law scaling in self-organizing systems. Caption: "A typical superlinear scaling law (solid line): Gross Metropolitan Product of US MSAs in 2006 (red dots) vs. population; the slope of the solid line has exponent, 1.126 (95% CI [1.101,1.149]) ). " Reproduced from [Bettencourt, L. M., Lobo, J., Strumsky, D., West, G. B. (2010). Urban scaling and its deviations: Revealing the structure of wealth, innovation and crime across cities. PloS one, 5(11), e13541. https://doi.org/10.1371/journal.pone.0013541], This figure is reproduced under a Creative Commons Attribution (CC-BY) International License (http://creativecommons.org/licenses/by/4.0/).

Figure 47. The relationship between GDP and population for cities illustrates the power-law scaling in self-organizing systems. Caption: "A typical superlinear scaling law (solid line): Gross Metropolitan Product of US MSAs in 2006 (red dots) vs. population; the slope of the solid line has exponent, 1.126 (95% CI [1.101,1.149]) ). " Reproduced from [Bettencourt, L. M., Lobo, J., Strumsky, D., West, G. B. (2010). Urban scaling and its deviations: Revealing the structure of wealth, innovation and crime across cities. PloS one, 5(11), e13541. https://doi.org/10.1371/journal.pone.0013541], This figure is reproduced under a Creative Commons Attribution (CC-BY) International License (http://creativecommons.org/licenses/by/4.0/).

The closest characteristics from this simulation to the data in Figure 47 and the values of their corresponding exponents are: $\mathsf{\Delta }s$ corresponds to a decrease in internal entropy of the simulation which relates to the GDP of cities as a measure of their productivity and therefore degree of self-organization. $\mathsf{\Delta }\rho$ relates to the GDP of cities, as larger cities are more dense. $\varphi$ relates to the GDP of cities, as to increase the GDP more events, i.e. more transactions are necessary. Q relates to GDP, because it is proportional to the total energy spent to produce the Gross Product. The X-axis in Figure 47 is population, analogous to N in our simulation, which is the size of the system in terms of the number of agents.

The corresponding values of the power law fits in Table 5 for those characteristics vs N, as mentioned above is respectively: For $\mathsf{\Delta }s$ is 1, for $\mathsf{\Delta }\rho$ is 1.11, for i is 1.09, for $\varphi$ is 1.104, and for Q is 0.983. We consider this as a good alignment between the results of this simulation and the characteristics of cities in terms of GDP.

6.3.3. Further Confirmation with Literature Data

The relationships between the results of this simulation and published results, in this section, are in good initial agreement with real-world data. They warrant greater confirmation and investigation in further simulations, as more realistic details, such as dissipation and obstacles are added to bring the simulations closer to the specifics of real-world systems. Further comparisons with results for other systems from published data will serve as additional tests and verifications. It will illuminate the correspondences with real systems and limitations of our model and point to a direction for its future improvement and refinement with the goal of helping to understand the mechanisms of self-organization and structure formation in evolving complex systems.

6.4. A Table Presenting the Fit Values for the Power Law Relationships in the Simulation

In Table 5 we show the values of the fit parameters for the power law relationships:

Table 5. This table contains all the fits for the power-law graphs. The "a" and "b" values in each row follow the equation $y=a{x}^b$, and the $R^2$ is shown in the last column.

Table 5. This table contains all the fits for the power-law graphs. The "a" and "b" values in each row follow the equation $y=a{x}^b$, and the $R^2$ is shown in the last column.

variables	a	b	$R^2$
$\langle \alpha \rangle$ vs. Q	$7.713·{10}^{-36}$	$6.787·{10}^{-2}$	$0.977$
$\langle \alpha \rangle$ vs. i	$1.042·{10}^{-35}$	$6.131·{10}^{-2}$	$0.981$
$\langle \alpha \rangle$ vs. $\varphi$	$1.510·{10}^{-35}$	$6.055·{10}^{-2}$	$0.982$
$\langle \alpha \rangle$ vs. $\mathsf{\Delta }s$	$1.020·{10}^{-35}$	$6.675·{10}^{-2}$	$0.978$
$\langle \alpha \rangle$ vs. $\mathsf{\Delta }\rho$	$1.647·{10}^{-35}$	$5.947·{10}^{-2}$	$0.964$
$\langle \alpha \rangle$ vs. $\langle t\rangle$	$1.622·{10}^{-34}$	$-6.175·{10}^{-1}$	$0.995$
$\langle \alpha \rangle$ vs. N	$1.168·{10}^{-35}$	$6.673·{10}^{-2}$	$0.977$
Q vs. i	$8.502·{10}^1$	$9.012·{10}^{-1}$	$1.000$
Q vs. $\varphi$	$2.000·{10}^4$	$8.897·{10}^{-1}$	$1.000$
Q vs. $\mathsf{\Delta }s$	$6.202·{10}^1$	$9.829·{10}^{-1}$	$0.999$
Q vs. $\mathsf{\Delta }\rho$	$7.133·{10}^4$	$8.784·{10}^{-1}$	$0.990$
Q vs. $\langle t\rangle$	$1.410·{10}^{19}$	$-8.888$	$0.972$
Q vs. N	$4.550·{10}^2$	$9.830·{10}^{-1}$	$1.000$
i vs. $\varphi$	$4.281·{10}^2$	$9.873·{10}^{-1}$	$1.000$
i vs. $\mathsf{\Delta }s$	$7.064·{10}^{-1}$	$1.090$	$0.999$
i vs. $\mathsf{\Delta }\rho$	$1.755·{10}^3$	$9.740·{10}^{-1}$	$0.988$
i vs. $\langle t\rangle$	$1.407·{10}^{19}$	$-9.887$	$0.976$
i vs. N	$6.445$	$1.090$	$0.999$
$i_u$ vs. $i_f$	$4.626·{10}^2$	$-1.281·{10}^{-2}$	0.873
$i_u$ vs. N	$4.516·{10}^2$	$-1.391·{10}^{-2}$	0.864
$\varphi$ vs. $\mathsf{\Delta }s$	$1.521·{10}^{-3}$	$1.104$	$0.999$
$\varphi$ vs. $\mathsf{\Delta }\rho$	$4.175$	$9.864·{10}^{-1}$	$0.988$
$\varphi$ vs. $\langle t\rangle$	$5.438·{10}^{16}$	$-1.002·{10}^1$	$0.977$
$\varphi$ vs. N	$1.427·{10}^{-2}$	$1.104$	$0.999$
$\mathsf{\Delta }s$ vs. $\mathsf{\Delta }\rho$	$1.301·{10}^3$	$8.939·{10}^{-1}$	$0.991$
$\mathsf{\Delta }s$ vs. $\langle t\rangle$	$4.439·{10}^{17}$	$-9.035$	$0.969$
$\mathsf{\Delta }s$ vs. N	$7.598$	$1.000$	$1.000$
$s_i$ vs. N	$7.598$	$1.000$	$1.000$
$s_f$ vs. N	$5.793$	$9.745·{10}^{-1}$	$1.000$
$s_u$ vs. N	$4.059·{10}^2$	$-1.298·{10}^{-1}$	$0.938$
$s_u$ vs. $sf$	$5.121·{10}^2$	$-1.329·{10}^{-1}$	$0.935$
$\mathsf{\Delta }\rho$ vs. $\langle t\rangle$	$1.103·{10}^{16}$	$-9.975$	$0.949$
$\mathsf{\Delta }\rho$ vs. N	$3.308·{10}^{-3}$	$1.110$	$0.991$
$\langle t\rangle$ vs. N	$7.061·{10}^1$	$-1.075·{10}^{-1}$	$0.970$

7. Discussion

8. Conclusions

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