1∕f Noise in the Simple Genetic Algorithm Applied to a Traveling Salesman Problem

Abstract

Complex dynamical systems are observed in physics, biology, and even economics. Such systems in balance are considered to be in a critical state, and [Formula: see text] noise is considered to be a footprint. Complex dynamical systems have also been investigated in the field of evolutionary algorithms inspired by biological evolution. The genetic algorithm (GA) is a well-known evolutionary algorithm in which many individuals interact, and the simplest GA is referred to as the simple GA (SGA). However, the GA has not been examined from the viewpoint of the emergence of [Formula: see text] noise. In the present paper, the SGA is applied to a traveling salesman problem in order to investigate the SGA from such a viewpoint. The timecourses of the fitness of the candidate solution were examined. As a result, when the mutation and crossover probabilities were optimal, the system evolved toward a critical state in which the average maximum fitness over all trial runs was maximum. In this situation, the fluctuation of the fitness of the candidate solution resulted in the [Formula: see text] power spectrum, and the dynamics of the system had no intrinsic time or length scale.