Ants find the shortest path: a mathematical proof

In the most basic application of Ant Colony Optimization (ACO), a set of artificial ants find the shortest path between a source and a destination. Ants deposit pheromone on paths they take, preferring paths that have more pheromone on them. Since shorter paths are traversed faster, more pheromone accumulates on them in a given time, attracting more ants and leading to reinforcement of the pheromone trail on shorter paths. This is a positive feedback process that can also cause trails to persist on longer paths, even when a shorter path becomes available. To counteract this persistence on a longer path, ACO algorithms employ remedial measures, such as using negative feedback in the form of uniform evaporation on all paths. Obtaining high performance in ACO algorithms typically requires fine tuning several parameters that govern pheromone deposition and removal. This paper proposes a new ACO algorithm, called EigenAnt, for finding the shortest path between a source and a destination, based on selective pheromone removal that occurs only on the path that is actually chosen for each trip. We prove that the shortest path is the only stable equilibrium for EigenAnt, which means that it is maintained for arbitrary initial pheromone concentrations on paths, and even when path lengths change with time. The EigenAnt algorithm uses only two parameters and does not require them to be finely tuned. Simulations that illustrate these properties are provided.