Methods for revealing patterns of non-local order in complex collective phenomena

 This topic arose from research done in Computing Special Topic 85333 and Math- ematics Special Topic 84333 [25]. An overview of quantum chaology [15, 16, 17] and structural complexity [20, 21, 22] was made with respect to how they describe global behaviour. Computer experimentation was used in an attempt to determine if there were any scaling properties in the distribution of prime numbers so as to find some patterns [26] that may be connected with non -local order as predicted by struc- tural complexity. The connections between the zeta function from number theory to quantum chaology [15] were also under investigation to in order to develop more tools for computational experiments. Following this research, there was a proposal by my supervisor Victor Korotkich to test his conjecture that the eigenvalue spectrum of matrices [13] associated with the dynamics of a robot soccer team would be a solid invariant that may serve as a global characteristic. The proposal was put forward for me to realise within a Masters of Applied Science topic. This required setting up a virtual simulation and computation laboratory using a combination of mathematics, statistics and programming. The laboratory had to be developed from scratch since this was the first time robot soccer team dynamics had been investigated in this way. It is analogous to the approach physicists take when searching for a theoretical particle. Because the methodology was new, the development of the new instrument (sim- ulators downloaded from the Internet, mathematical software and statistical analysis software) needed to be mastered and synthesised into something effective to detect the conjectured invariant. Once detected, adjustments to parameters were needed so that a clearer pattern could be discerned. The adjustments are analogous to the changing of lens and focussing required on a microscope to get better resolution. The data produced by robot soccer simulators was processed to test the conjecture. Nonparametric statistical analysis was applied to gain an understanding of the patterns that were found in the distribution of eigenvalues and eigenvalue spacings and create an experimental methodology to analyse this type of data. The computer simulation laboratory was set up to implement and test the technique. According to the statistical evidence it seems that each robot soccer team has an invariant in its collective dynamics. Several robot soccer simulators were found on the Internet. These simulators were evaluated as to whether they were able to be modified so as to extract the robot soccer teams' coordinates during a game, whether the teams strategies could be modified and whether they were suitable for use in the experiments. Features of the UNIX operating system, C++ (an implementation of the C++ programming language developed for the UNIX operating system) and JAVA compilers and languages also had to be studied in order to modify and run these simulators. The data produced by these simulators needed to be processed to test the conjec- ture. Because of the size of matrices (429 by 429) that arose by applying this approach, the MATLAB programming environment was used to evaluate and organise the data for statistical analysis. This data was then processed using the SPSS statistical pack- age and Excel spreadsheet software. Much reading and comprehension was gained in the advanced statistics topic of nonparametric analysis to gain understanding of the patterns that were found in the distribution of eigenvalues and eigenvalue spacings and create an experimental methodology to analyse this type of data. To present the results that were found, many features of MATLAB, postscript format, Excel, SPSS and aTEX needed to be understood. All the understanding, skills, software analysis programs, and statistical techniques synthesise together as a virtual computer laboratory s from which many other experiments can be performed. Here a powerful synergism has arisen that may be adapted and applied to understanding the dynamics of other systems. The laboratory has been successful in confirming the existence of the conjectured invariant using solid scientific methods. This study belongs to a new area of fundamental research and has already shown promising results which have been presented at the 12th Australian Joint Conference on Artificial Intelligence AI'99 and the International Conference "With Fuzzy Logic in the New Millennium". That the research is significant is demonstrated by the publication of the results by Springer-Verlag in their Lecture Notes on Artificial Intelligence [49] series and acceptance for publication in their Studies in Fuzziness and Soft Computing [51] series.