Swarm Intelligence

, Volume 7, Issue 1, pp 43–62 | Cite as

Ants find the shortest path: a mathematical proof

  • Jayadeva
  • Sameena Shah
  • Amit Bhaya
  • Ravi Kothari
  • Suresh Chandra


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.


Ant Colony Optimization Stability analysis Pheromone Collective foraging Stagnation Self-organization Selective removal of pheromone Distributed optimization Optimization Swarm intelligence 



The authors would like to thank reviewers 2 and 3 for constructive criticism, as well as reviewer 1, the Associate Editors and the Editor for detailed comments and suggestions. The work of the first author (J) was partially supported by a grant from the DST (Indo–Brazil International Collaboration). The work of the last author (AB) was partially supported by grants from FAPERJ (CNE) and CNPq (BPP, Brazil–India International Collaboration).


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Jayadeva
    • 1
  • Sameena Shah
    • 2
  • Amit Bhaya
    • 3
  • Ravi Kothari
    • 4
  • Suresh Chandra
    • 1
  1. 1.Department of Electrical EngineeringIndian Institute of Technology, DelhiNew DelhiIndia
  2. 2.Thomson Reuters Corp. R & DNew YorkUSA
  3. 3.Federal University of Rio de Janeiro, Department of Electrical EngineeringPEE/COPPE/UFRJRio de JaneiroBrazil
  4. 4.IBM India Research Lab.New DelhiIndia

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