A mathematical analysis of improved EigenAnt algorithm.

As a variant of Ant Colony Optimization, the EigenAnt algorithm finds the shortest path between a source node and a destination node based on negative feedback in the form of selective pheromone removal that occurs only on the path which is actually chosen for each trip. EigenAnt algorithm also could change quickly to reflect to the dynamic variety of initial pheromone concentrations and path length etc. However, in general, the solution of EigenAnt algorithm is not always convergent. In this paper, we propose an improved EigenAnt (iEigenAnt) algorithm in terms of both negative and positive feedback; that is, selective pheromone updates are decided by smart ants or stupid ones, which depends whether the amount of the pheromone at the selected path increases or not. The system modelled by our algorithm has a unique equilibrium as the shortest path. Besides, using mathematical analysis, we demonstrate that the equilibrium is global asymptotically stable, i.e., stable and convergent. Finally, we also implement the iEigenAnt algorithm under four different cases and apply it on travelling salesman problem problem, the simulation result shows that our iEigenAnt algorithm is faster convergent and more effective compared to the original EigenAnt algorithm, and some combinatorial optimisation problems can be effectively solved based on our iEigenAnt algorithm.