Population Structure and Particle Swarm Performance James Kennedy Bureau of Labor Statistics Washington, DC Kennedy_Jim@bls.gov Rui Mendes Universidade do Minho Braga, Portugal rui@omega.di.uminho.pt Abstract: The effects of various population topologies on the particle swarm algorithm were systematically inves- tigated. Random graphs were generated to specifications, and their performance on several criteria was compared. What makes a good population structure? We discovered that previous assumptions may not have been correct. I. INTRODUCTION The trajectories of individual members of a particle swarm population have been analyzed in depth [3][[4][8], and those analyses have resulted in improve- ments in the performance of the algorithm. It has long been clear though that the uniqueness of the algorithm lies in the dynamic interactions of the particles. Even when changes are made to the formulas e.g., [1][2][4][8], the performance depends on an effect in- volving the entire population. The particle swarm algorithm can be described gen- erally as a population of vectors whose trajectories os- cillate around a region which is defined by each individ- ual’s previous best success and the success of some other particle. Various methods have been used to iden- tify “some other particle” to influence the individual. The two most commonly used methods are known as gbest and lbest (Figure 1). In the gbest population, the trajectory of each particle’s search is influenced by the best point found by any member of the entire population. The lbest population allows each individual to be influ- enced by some smaller number of adjacent members of the population array. Typically lbest neighborhoods comprise exactly two neighbors, one on each side: a ring lattice. A kind of lore has evolved regarding these so- ciometric structures. It has been thought that the gbest type converges quickly on problem solutions but has a weakness for becoming trapped in local optima, while lbest populations are able to “flow around” local optima, as subpopulations explore different regions [7]. The lore is based on experience and some data, but population topologies have not been systematically explored. The present research manipulates some sociometric variables that are hypothesized to affect performance. There is not room in this forum to discuss the No Free Lunch implications of the present strategy. We are convinced that it is worthwhile to seek an optimization algorithm that performs well on a variety of standard test functions, even if it is average across the full range of possible functions. II. CAUSAL FACTORS The present study focused on population topologies where connections were undirected, unweighted, and did not vary over the course of a trial. The usual particle swarm rule was used, that an individual gravitated to- ward a stochastically weighted average of its own previ- ous best point and the best point found by any member of its neighborhood. FIGURE 1. GBEST (LEFT) AND LBEST SOCIOMETRIC PAT- TERNS i . Watts [10][11] has shown that the flow of informa- tion through social networks is affected by several as- pects of the networks. The first measure is the degree of connectivity among nodes in the net. Each individual in a particle swarm identifies the best point found by its k neighbors; k, then, is the variable that distinguishes lbest from gbest topologies, and is likely to affect perform- ance. A second factor identified by Watts was the amount of clustering, C. Clustering occurs when a node’s neighbors are also neighbors to one another. The num- ber of neighbors-in-common can be counted per node, and can be averaged over the graph. Finally, Watts noted that the average shortest dis- tance from one node to another was an important graph characteristic for determining the spread of information through the network. The present research did not ma- nipulate this variable, which correlates very highly with both k and C. Previous investigation within the particle swarm paradigm had found that the effect of population topol- ogy interacted with the function being optimized [6]. Some kinds of populations worked well on some func-