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A note on irreducibility for topical maps

Brian Lins

TL;DR

We address irreducibility for multiplicatively topical maps on the nonnegative cone and clarify the relations among facial irreducibility, graphical irreducibility, and indecomposability, in addition to related conditions like partially irreducible and imperturbable. It surveys how these notions interact with map classes (e.g., $m$-convex and subadditive maps) and provides theoretical guarantees for the existence of a positive eigenvector via slice spaces and bounded invariant sets; uniqueness results are tied to real analyticity and (M)/(N) criteria. A central methodological contribution is translating irreducibility tests into Boolean satisfiability problems using upper and lower signatures $\overline{f}$ and $\underline{f}$, enabling SAT-based certification even in large dimensions, with a special Tarjan-based SCC check for $m$-convex maps. The results offer practical, scalable tools for certifying the existence—and in some cases the uniqueness—of entrywise positive eigenvectors in nonlinear Perron–Frobenius theory, with applications to max-plus, tensor-based, and population models.

Abstract

Topical maps are a nonlinear generalization of nonnegative matrices acting on the interior of the standard cone $\mathbb{R}^n_{\ge 0}$. Several analogues of irreducibility have been defined for topical maps, and all are sufficient to guarantee the existence of entrywise positive eigenvectors. In this note, we organize several of these notions, showing which conditions are stronger and when different types of irreducibility are equivalent. We also consider how to computationally check the conditions. We show that certain irreducibility conditions can be expressed as Boolean satisfiability problems that can be checked using SAT solvers. This can be used to confirm the existence of entrywise positive eigenvectors when the dimension is large.

A note on irreducibility for topical maps

TL;DR

We address irreducibility for multiplicatively topical maps on the nonnegative cone and clarify the relations among facial irreducibility, graphical irreducibility, and indecomposability, in addition to related conditions like partially irreducible and imperturbable. It surveys how these notions interact with map classes (e.g., -convex and subadditive maps) and provides theoretical guarantees for the existence of a positive eigenvector via slice spaces and bounded invariant sets; uniqueness results are tied to real analyticity and (M)/(N) criteria. A central methodological contribution is translating irreducibility tests into Boolean satisfiability problems using upper and lower signatures and , enabling SAT-based certification even in large dimensions, with a special Tarjan-based SCC check for -convex maps. The results offer practical, scalable tools for certifying the existence—and in some cases the uniqueness—of entrywise positive eigenvectors in nonlinear Perron–Frobenius theory, with applications to max-plus, tensor-based, and population models.

Abstract

Topical maps are a nonlinear generalization of nonnegative matrices acting on the interior of the standard cone . Several analogues of irreducibility have been defined for topical maps, and all are sufficient to guarantee the existence of entrywise positive eigenvectors. In this note, we organize several of these notions, showing which conditions are stronger and when different types of irreducibility are equivalent. We also consider how to computationally check the conditions. We show that certain irreducibility conditions can be expressed as Boolean satisfiability problems that can be checked using SAT solvers. This can be used to confirm the existence of entrywise positive eigenvectors when the dimension is large.
Paper Structure (5 sections, 13 theorems, 56 equations, 2 figures)

This paper contains 5 sections, 13 theorems, 56 equations, 2 figures.

Table of Contents

  1. Preliminaries
  2. Types of Irreducibility
  3. Other conditions like irreducibility
  4. Checking Irreducibility Conditions
  5. Unique eigenvectors

Key Result

Proposition 1.1

An m-topical map $f$ on $\mathbb{R}^n_{\ge 0}$ has an eigenvector $x \in \mathbb{R}^n_{\ge 0}$ with eigenvalue equal to the cone spectral radius where $u$ is any element of $\mathbb{R}^n_{>0}$.

Figures (2)

  • Figure 1: Summary of Theorem \ref{['thm:connect1']}. Facial and graphical irreducibility both imply indecomposability for all m-topical maps. Graphical irreducibility is equivalent to indecomposability and weaker than facial irreducibility for m-convex m-topical maps (dashed edges). For subadditve m-topical maps, all three types of irreducibility are equivalent.
  • Figure 2: Relationships between irreducibility conditions from Theorems \ref{['thm:connect1']} and \ref{['thm:connect2']}. Dashed arrows apply to m-convex maps. Partial irreducibility is equivalent to imperturbability for subadditive maps.

Theorems & Definitions (35)

  • Proposition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Lemma 1.4
  • proof
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • proof
  • ...and 25 more