at a set rate, while a third aspect should predictably variate between one extreme to the next (Figure 2.4).
Figure 2.4 A small ConesWorld topography update of position only.
As the topography of ConesWorld is subject to change if the user defines multiple landscapes for it to generate during a given run, it is entirely possible that new maximum values for the function would be created. At the same time past maximum values might vanish as the cones lower in height. Due to this process of vanishing and reappearing maximums, agents need to adjust their movements and mechanisms to explore the environment for new viable maximum points. This can result in a dramatic downturn in the scoring for a group of agents during an update cycle.
As seen in Figure 2.5, those points on the graph where an upward trend is suddenly interrupted by a dramatic drop in score, followed by a rapid ascension to a steady upward trend is a clear indication of a topographical update occurring in the system. Here what had been a maximum suddenly becomes a non‐maximum, and the agents need to branch out to find the location of the new maximum point. Depending on how dramatically the landscape changes, the reclamation of the previous score may be instantaneous or take several generations to rediscover. For example, a topography where the height and radius of all cones remains static but their positions gradually change places, as seen in Figure 2.4, will result in momentary drop‐offs and then rapid return to the maximum score.
It should also be noted that as there is no guarantee that the absolute allowable maximum height for a cone will be present in any simulation. In fact, it is entirely possible for the overall maximum score to lower over the lifespan of a simulation given multiple topographical updates, as depicted in Figure 2.6. In this case, the height of the cones varied with each update resulting in lower possible maximums due to the shifting nature of the overall possible maximum.
To control this shift, a logistics function is utilized, which produces a change variable that fluctuates between a range of 0 and 1, and this rate of fluctuation may be controlled by the earlier described A‐values that may be set by the user.
Figure 2.5 The Knowledge Source fitness of a ConesWorld simulation undergoing regular topographical update every 20 ticks.
Figure 2.6 ConesWorld KS Fitness with a regular topographical update at every 50 ticks, with variable cone heights.
The Logistics Function
The logistics function (Figure 2.7) utilized by the CAT System, notably in the update mechanism for the topographic of the ConesWorld fitness function, is a recursive function that generates a value between 0 and 1. This value is then used by the update mechanism to select a fractional amount of the allowable rate of change that the system specifies.
Given an A value between 0 and 4, successive iterations of the logistics function will generate a fluctuation between the values of 0 and 1. The frequency of this fluctuation is dependent on the size of the A value. Due to the recursive nature of the function, low values of A will result in subsequent values of Y(n) approaching a steady output that will cease to fluctuate after a sufficient number of iterations. For this reason, using lower values of A in the ConesWorld simulation will result in the appearance of smooth, predictable, near‐linear transitions from one update to the next.
However, as the A value approaches 3, the resulting fluctuations becomes more self‐sufficient and will maintain a steady, regular frequency between two absolutes, which its peaks and valleys will trend toward. For these cases, it is possible to have subsequent steps of the logistics function vary, but in predictable ways. As each cone in the ConesWorld simulation freshly calculates the logistics function with the next iteration of Y(n), alternating cones querying it for their rate of change will receive alternating high and low rates from it.
After exceeding an A value of 3.33, the logistics function enters into a self‐sufficient, non‐maintaining erratic frequency with relatively unpredictable shifts in frequency, without ever stabilizing to a constant, repetitive cycle. As seen in Figure 2.8, the lower values of A, ranging from 0 to 3 in the forefront, quickly reduce to a singular output after a brief initial period. After A reaches 3, the frequency does not reduce, but rather becomes a self‐perpetuating frequency that continues indefinitely without diminishing.
In those rows of the graph beyond 3.33, the pattern becomes erratic, with each subsequent iteration of the logistics function taking a dramatic, seemingly unpredictable movement that bears little relation to the patterns produced by lower values of A. It should also be noted that even with this erratic fluctuation the resulting Y(n) values are still within the range of 0 and 1. However, should the A value exceed 4, then the system will destabilize and quickly break out of the given range.
Figure 2.7 The CAT System Logistics Function.
Figure 2.8 A three‐dimensional visualization of successive iterations of the logistics function across increasing values of A.
CAT Sample Runs: ConesWorld
The following data are the result of two separate runs of the CAT system's ConesWorld simulation. Both runs use similar parameters for their initializations. The population consists of 50 agents in each of them; the social topology between them is represented by lBest (each agent having a connection to 2 agents, resulting in a circular chain); the influence is calculated by Majority; the number of cones is 150 (it must be noted that some smaller cones can be consumed by larger cones and not be visible during the simulation); and the A‐values for height, radius, and position are all 3.5
The difference between the two runs is the usage of
