Phase Transition in Ant Colony Optimization

Abstract

Ant colony optimization (ACO) is a stochastic optimization algorithm inspired by the foraging behavior of ants. We investigate a simplified computational model of ACO, wherein ants sequentially engage in binary decision-making tasks, leaving pheromone trails contingent upon their choices. The quantity of pheromone left is the number of correct answers. We scrutinize the impact of a salient parameter in the ACO algorithm, specifically, the exponent α, which governs the pheromone levels in the stochastic choice function. In the absence of pheromone evaporation, the system is accurately modeled as a multivariate nonlinear Pólya urn, undergoing phase transition as α varies. The probability of selecting the correct answer for each question asymptotically approaches the stable fixed point of the nonlinear Pólya urn. The system exhibits dual stable fixed points for α≥αc and a singular stable fixed point for α<αc where αc is the critical value. When pheromone evaporates over a time scale τ, the phase transition does not occur and leads to a bimodal stationary distribution of probabilities for α≥αc and a monomodal distribution for α<αc.