On the selection of the optimal topology for particle swarm optimization: a study of the tree as the universal topology

In this paper, we deal with the problem of selecting the best topology in Particle Swarm Optimization. Unlike most state-of-the-art papers, where statistical analysis of a large number of topologies is carried out, in this work we formalize mathematically the problem. In this way, the problem is to find the best topology in the set of all simple connected graphs of n nodes. To determine which is the best topology, each graph in this set must be measured with a function that evaluates its quality. We introduce the concepts of equivalent neighborhood and equivalent topology to prove that for any simple connected graph there is an equivalent tree. The equivalence between two topologies means that each particle belonging to these has the same local best in both. Therefore, the problem can be simplified in complexity to find the best tree in the set of all trees with n nodes. Finally, we give some examples of equivalent topologies, as well as the applicability of the obtained result.