The two-nest ants process on triangle-series-parallel graphs

TL;DR

This paper studies a modified ants process with two nests on complex triangle-series-parallel graphs, proving convergence and describing the limiting behavior using advanced stochastic and combinatorial methods.

Contribution

It introduces a two-nest ants process on complex graphs and provides a rigorous convergence analysis with explicit limit descriptions.

Findings

The process converges under the specified conditions.

The limit behavior is characterized explicitly.

The analysis employs stochastic approximation and combinatorial techniques.

Abstract

The ants process is a stochastic process introduced by Kious, Mailler and Schapira as a model for the phenomenon of ants finding shortest paths between their nest and a source of food (seen as two marked nodes in a finite graph), with no other means of communications besides the pheromones they lay behind them as they explore their environment. The ants process relies on a reinforcement learning mechanism. In this paper, we modify the ants process by having more than one ants nest (and still one source of food). For technical reasons, we restrict ourselves to the case when there are two nests, and when the graph is a triangle between the two nests and the source of food, whose edges have been replaced by series-parallel graphs. In this setting, using stochastic approximation techniques, comparison with P\'olya urns, and combinatorial arguments, we are able to prove that the ants process converges and to describe its limit.