Convergence results for continuous-time dynamics arising in ant colony optimization

TL;DR

This paper analyzes the long-term behavior of continuous-time models inspired by ant colony optimization algorithms for shortest path problems, proving stability and convergence properties.

Contribution

It provides a comprehensive stability analysis of these models, including a new result on the EigenAnt model's global stability and convergence rate.

Findings

Local stability of shortest path equilibrium established

EigenAnt model shown to be globally stable

Convergence speed proportional to path length differences

Abstract

This paper studies the asymptotic behavior of several continuous-time dynamical systems which are analogs of ant colony optimization algorithms that solve shortest path problems. Local asymptotic stability of the equilibrium corresponding to the shortest path is shown under mild assumptions. A complete study is given for a recently proposed model called EigenAnt: global asymptotic stability is shown, and the speed of convergence is calculated explicitly and shown to be proportional to the difference between the reciprocals of the second shortest and the shortest paths.