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

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there are two nests, and when the graph is a triangle between the two nests and the source of food, whose edges have been replaced by series-parallel graphs.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 Pólya urns, and combinatorial arguments, we are able to prove that the ants process converges and to describe its limit.

💡 Research Summary

The paper extends the ant foraging model—originally introduced by Kious, Mailler, and Schapira as a stochastic reinforcement process on a single nest and a food source—to a setting with two nests and one food source. The underlying graph is a triangle formed by the two nests (N₁, N₂) and the food node (F), but each side of the triangle is replaced by an arbitrary series‑parallel subgraph (G₁, G₂, G₃). The length (or “height”) of each subgraph, denoted ℓ₁, ℓ₂, ℓ₃, is the shortest distance between its source and sink. A parameter α∈(0,1) specifies the proportion of ants that start from N₁; the remaining 1−α start from N₂.

At each discrete time step an ant is launched from one of the nests according to α, performs a random walk whose transition probabilities are proportional to the current edge weights (pheromone levels), stops when it reaches F, then retraces its path backwards while erasing loops (the “loop‑erased” version). Every edge traversed on this backward path receives a unit increase in weight. Thus the weight vector W(n) evolves as a Markov chain with linear reinforcement.

The main result (Theorem 1.5) proves almost‑sure convergence of the normalized weight vector W(n)/n to a random limit χ. The structure of χ depends on the relative magnitudes of ℓ₁, ℓ₂, ℓ₃ and falls into three regimes:

1. ℓ₂ ≥ ℓ₁ + ℓ₃ – The middle subgraph G₂ is essentially never reinforced; ants from N₁ use G₁, while ants from N₂ use the concatenation G₃ ∪ G₁. Consequently all edges of G₂ have asymptotic weight zero.

2. ℓ₁ ≤ ℓ₂ < ℓ₁ + ℓ₃ and ℓ₃ < ℓ₁ + ℓ₂ – This is the generic case. Each subgraph Gᵢ receives reinforcement at a deterministic proportion βᵢ (i = 1,2,3) that is an explicit rational function of α and the ℓ’s. In particular β₁ and β₃ lie strictly between 0 and 1, showing that even though G₃ does not belong to any shortest N→F path, a positive fraction of ants still travel through it because of the interaction between the two nests.

3. ℓ₃ ≥ ℓ₁ + ℓ₂ – The side G₃ is never used; ants from N₁ travel via G₁ and ants from N₂ via G₂, both following shortest routes.

The proof hinges on two observations. First, restricted to any single subgraph Gᵢ, the process behaves exactly like the original single‑nest ant process, for which KMS22a already established convergence to shortest paths. Second, the vector of counts (N₁(n), N₂(n), N₃(n))—the number of times each subgraph has been reinforced—forms a stochastic approximation scheme. By applying the ODE method of Benaïm and results of Pemantle on reinforced random walks, the authors show that the normalized count vector converges almost surely to the deterministic fixed point described above.

To illustrate the theorem, the authors analyze several concrete “(ℓ₁,ℓ₂,ℓ₃)-triangles” where each side is a simple chain of ℓᵢ edges. In the (1,1,2) case the limiting weights match the intuitive picture that ants from N₂ must pass through N₁, yielding a road of width 1−α between the nests and a road of width 1 from N₁ to F. In the symmetric (1,1,1) case the limiting weights are α, α(1−α), and 1−α on the three sides, a distribution that does not correspond to any obvious notion of an “optimal transport network”. Similar observations hold for larger triangles, reinforcing the conclusion that the multi‑nest ant process does not directly model the optimal network formation observed in slime‑mould experiments.

The paper concludes with open problems: extending convergence results to arbitrary graphs (e.g., k‑cycles or complete graphs with multiple nests), investigating whether a modified multi‑city version could better capture Physarum’s optimal transport behavior, and relating the model to other reinforced random walk frameworks. Overall, the work provides a rigorous analysis of a natural generalisation of ant colony optimisation, reveals subtle interaction effects between multiple nests, and clarifies the limits of its applicability to optimal network modelling.