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

Cécile Mailler, Zoé Varin

TL;DR

This work extends the ants process to a two-nest setting on triangle-series-parallel graphs and proves almost-sure convergence of the normalized edge-weights $W_e(n)/n$ to a random limit $\chi_e$. The authors couple stochastic-approximation techniques with generalized Pólya urn results and the ODE method, leveraging a decomposition into three SP subgraphs $G_1,G_2,G_3$ and the notion of effective conductance to analyze reinforcement. They show that the limit is determined by the SP-heights $\ell_i=h(G_i)$ and the nest-selection parameter $\alpha$, yielding three regimes: (i) concentration on edges along the shortest paths when $\ell_2\ge\ell_1+\ell_3$, (ii) a mixed regime with parameters $\beta_1,\beta_3$ when $\ell_2<\ell_1+\ell_3$ and $\ell_3<\ell_1+\ell_2$, and (iii) concentration on the other pair of edges when $\ell_3\ge\ell_1+\ell_2$. The analysis relies on restricting the two-nest process to each SP component as a single-nest process and applying Bendixson–Dulac to eliminate nontrivial ODE orbits, ultimately yielding almost-sure convergence to a fixed point of an associated vector field. The results formalize how reinforcement leads to transport-network-like structures in this graph class, while highlighting cases where the limit cannot be interpreted as an optimal transport network. This provides rigorous convergence guarantees and precise limits for a biologically inspired, reinforcement-based path formation model on a structured graph family, with potential implications for modeling collective transport systems.

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there are two nests, and when the graph is a triangle between the two nests and the source of food, whose edges have been replaced by series-parallel graphs.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

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

TL;DR

This work extends the ants process to a two-nest setting on triangle-series-parallel graphs and proves almost-sure convergence of the normalized edge-weights to a random limit . The authors couple stochastic-approximation techniques with generalized Pólya urn results and the ODE method, leveraging a decomposition into three SP subgraphs and the notion of effective conductance to analyze reinforcement. They show that the limit is determined by the SP-heights and the nest-selection parameter , yielding three regimes: (i) concentration on edges along the shortest paths when , (ii) a mixed regime with parameters when and , and (iii) concentration on the other pair of edges when . The analysis relies on restricting the two-nest process to each SP component as a single-nest process and applying Bendixson–Dulac to eliminate nontrivial ODE orbits, ultimately yielding almost-sure convergence to a fixed point of an associated vector field. The results formalize how reinforcement leads to transport-network-like structures in this graph class, while highlighting cases where the limit cannot be interpreted as an optimal transport network. This provides rigorous convergence guarantees and precise limits for a biologically inspired, reinforcement-based path formation model on a structured graph family, with potential implications for modeling collective transport systems.

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 Pólya urns, and combinatorial arguments, we are able to prove that the ants process converges and to describe its limit.
Paper Structure (16 sections, 19 theorems, 120 equations, 8 figures)

This paper contains 16 sections, 19 theorems, 120 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The multi-nest ants process
  3. Main results
  4. Discussion
  5. Acknowledgements
  6. Strategy and preliminary results
  7. Random walks on weighted graphs: the method of conductances
  8. Generalised Pólya urn processes
  9. The ODE method on stochastic approximations
  10. The single-nest process on series-parallel graphs
  11. Proof of Lemma \ref{['lemma:conv_rn']}
  12. Proof of the main result (Theorem \ref{['thm:trianglesSP']})
  13. The zeros of the vector field $F$
  14. Flow of the ODE
  15. Unstable zeros
  16. ...and 1 more sections

Key Result

Theorem 1.5

We assume that $G$ is a triangle-series-parallel graph as in Definition def:triangleSP and use the notation from this definition. We also assume that $\alpha\in(0,1)$. We let $\ell_i = h(G_i)$ and assume that $\ell_i\geq 1$, for all $i=1, 2, 3$. Finally, we let $({\bf W}(n))_{n\geq 0}$ be the (loop- where $(\chi_e)_{e\in E}$ is a random vector. To describe $(\chi_e)_{e\in E}$, we assume, without l

Figures (8)

  • Figure 1: Illustration of the loop-erased process: $(X_i^{(n+1)})_{i\geq0}$ corresponds to the black trajectory on the leftmost picture; then the (backward) loop-erased trajectory $\gamma_{n+1}$ is represented by the orange non-dotted path on the rightmost picture.
  • Figure 2: Illustration of the definition of series-parallel graphs and triangle-SP graphs. Subfigure \ref{['subfig:SPparallel']} represents the merging in parallel of two graphs $G_1$ and $G_2$. Similarly, Subfigure \ref{['subfig:SPseries']} illustrates the merging of $G_1$ and $G_2$ in series. Subfigure \ref{['subfig:triangleSP']} represents a triangle-SP graph (see Definition \ref{['def:triangleSP']}).
  • Figure 3: The $(\ell_1, \ell_2, \ell_3)$-triangle.
  • Figure 4: The probability that a random walk starting from $S$ reaches $T_1$ before $T_2$ is $\frac{C_{G_1}}{C_{G_1} + C_{G_2}}$.
  • Figure 5: Visual support for the proof of Lemma \ref{['lem:Zoe']}. On the left-hand side: the path $\gamma$ is decomposed into two parts $u$ (until the last visit at $\mathcal{N}$) and $s$ (after the last visit at $\mathcal{N}$). For better readability, we have coloured the first part of $u$ in purple, the second part in orange, and $s$ in black. The non-dashed part of $\gamma$ is $\mathrm{LE}(\gamma)$. In the middle: the path $u' s_1$, to which we apply $\psi$ recursively in the proof of Lemma \ref{['lem:Zoe']}. On the right-hand side: the path $u_2 s's_2$, which goes from $\mathcal{A}$ to $\mathcal{F}$. Recall that we define $\psi(\gamma) = \psi(u's_1) u_2s's_2$.
  • ...and 3 more figures

Theorems & Definitions (45)

  • Remark 1.1
  • Definition 1.2: See Subfigure \ref{['subfig:SPparallel']} and \ref{['subfig:SPseries']}
  • Definition 1.3
  • Definition 1.4: See Subfigure \ref{['subfig:triangleSP']}
  • Theorem 1.5
  • Definition 1.6: See Figure \ref{['fig:triangle']}
  • Example 1.7
  • Example 1.8
  • Example 1.9
  • Definition 2.1
  • ...and 35 more