The neighborhood of canonical deterministic PSO

Particle swarm optimization (abbr. PSO) is one of the most effective optimization algorithms. The PSO contains many control parameters. These causes, the performance of the searching ability of the PSO is significantly alternated. In order to analyze the dynamics of such PSO system rigorously, we proposed a canonical deterministic PSO (abbr. CD-PSO) systems which does not contain any stochastic factors, and its coordinate of the phase space is normalized. The found global best information influences the dynamics. This situation can be regarded as the full-connection state. On the other hand, there is the case where the best information in a limited population. Such information is called as Ibest. How to get the Ibest information from any population is equivalent to a network structure. Such network structure influences the performance of searching ability. In order to clarify a relationship between network structures of CD-PSO and its performance, we pay attention to the degree and the average distance used in graph theory. First, we consider the case where the CD-PSO has an extended cycle structure. Our numerical simulation results indicates the searching performance is depended on the average distance of the node, and the optimal average distance is existed. Next, we consider the case where the CD-PSO has a Small World network structure. The extended cycle structure has uniform symmetric property. On the contrary, a small world network has nonuniform property. Even in the case where the CD-PSO has the small world network structure, the searching performance is depended on the average distance.