Topical Maps in Nonlinear Spectral Theory

• Topical maps are nonlinear, order-preserving, homogeneous functions on nonnegative cones that generalize classical nonnegative matrix theory.
• They extend Perron–Frobenius theory to nonlinear operators through refined irreducibility notions like facial, graphical, and indecomposability.
• The algorithmic reformulation via SAT solvers efficiently verifies these properties, ensuring the existence and uniqueness of strictly positive eigenvectors.

A topical map is a mathematical construct generalizing the action of nonnegative matrices, extending the classical Perron–Frobenius theory to nonlinear operators on the nonnegative cone. In the multiplicative formulation (“m‐topical”), a topical map $$f: \mathbb{R}^n_{\geq 0} \rightarrow \mathbb{R}^n_{\geq 0}$$ is continuous, order-preserving, and homogeneous, with the crucial property that $f(\mathbb{R}^n_{> 0}) \subseteq \mathbb{R}^n_{> 0}$. The study of irreducibility for topical maps involves conditions under which such a map admits an entrywise positive eigenvector, mirroring the irreducibility and spectral properties of classical nonnegative matrices.

1. Fundamental Properties and Definitions

In the m‐topical setting, the central objects are nonlinear maps $f$ satisfying, for all $x, y \in \mathbb{R}^n_{\geq 0}$,

• Order-preserving: $x \leq y \implies f(x) \leq f(y)$,
• Homogeneous: $f(\lambda x) = \lambda f(x)$ for all $\lambda > 0$,
• Interior mapping: $f(\mathbb{R}^n_{> 0}) \subseteq \mathbb{R}^n_{> 0}$.

This encompasses—via logarithmic coordinates—additively topical (max-plus) maps, Shapley operators, and general cone-preserving nonlinearities (Lins, 14 Oct 2025). The classical Perron–Frobenius theory states that an irreducible nonnegative matrix admits a unique (up to scaling) strictly positive eigenvector. Topical maps broaden this result to nonlinear settings, provided an analogue of irreducibility holds.

Given a topical map $f$, the set of faces of the cone $\mathbb{R}^n_{\geq 0}$, $$F_J = \{ x \in \mathbb{R}^n_{\geq 0}: x_i = 0\ \forall i \not\in J \}$$ for $J \subsetneq [n]$, organizes the possible “degenerate” states. Irreducibility in this context means the map cannot be restricted to a nontrivial face.

2. Generalized Irreducibility Notions

The study presents several nonlinear variants of irreducibility (Lins, 14 Oct 2025):

• Facially Irreducible: $f$ is facially irreducible if no nontrivial proper closed face is invariant under $f$.
• Graphically Irreducible: Associate a directed graph $G(f)$ with edges $i \rightarrow j$ if $\lim_{t \to \infty} f(1 + t e_{\{j\}})_i = \infty$. Then $f$ is graphically irreducible if $G(f)$ is strongly connected.
• Indecomposable (Gaubert–Gunawardena): No nonempty proper subset $J \subsetneq [n]$ admits $\lim_{t \to \infty} f(1 + t e_J)_i < \infty$ for all $i \not\in J$.
• Partial Irreducibility (weaker): There are no invariant nontrivial “parts” (equivalence classes under comparability).
• Imperturbability: No $I \subseteq J \subsetneq [n]$ with $f(e_I)_i > 0$ for all $i \in I$ and $\lim_{t \to \infty} f(1 + t e_J)_i < \infty$ for all $i \not\in J$.

These conditions organize a hierarchy: facial and graphical irreducibility each imply indecomposability; under convexity or subadditivity, they become equivalent.

Table: Notions of Irreducibility

Name	Definition	Relationship
Facial Irreducibility	No invariant proper face	Strongest
Graphical Irreducibility	$G(f)$ strongly connected	Strong (convex equiv.)
Indecomposable	No nontrivial block decomposition	Middle
Partial Irreducibility	No nontrivial invariant parts	Weakest
Imperturbability	No block/sub-eigenvector obstruction	Refines Partial

Facial and graphical irreducibility coincide with each other for subadditive (convex) topical maps (Lins, 14 Oct 2025).

3. Algorithmic Reformulation via SAT

A significant technical advance is the translation of these irreducibility conditions to Boolean satisfiability (SAT) statements. The key devices are the lower signature $\underline{f}$ and upper signature $\overline{f}$, monotone Boolean maps on $\{0,1\}^n$:

• $\underline{f}(e_J)_i = 1$ if $f(e_J)_i > 0$,
• $\overline{f}(e_J)_i = 1$ if $\lim_{t \to \infty} f(1 + t e_J)_i = \infty$.

Conditions can then be rewritten:

• Facial irreducibility: no nontrivial $x$ with $\underline{f}(x) \leq x$,
• Indecomposability: no nontrivial $x$ with $\overline{f}(x) \leq x$,
• Imperturbability: no nontrivial $x, y$ with $x \leq y$, $\underline{f}(x) \geq x$, and $\overline{f}(y) \leq y$.

These Boolean formulae, expressed in conjunctive normal form, can be solved efficiently by modern SAT solvers, particularly in high dimensions. For instance, decomposability can be encoded as the existence of $x$ with $$(x_1 \vee \cdots \vee x_n) \wedge (\neg x_1 \vee \cdots \vee \neg x_n) \wedge (x_i \vee \neg \overline{f}(x)_i)\ \forall i$$.

4. Existence and Uniqueness of Positive Eigenvectors

A central theorem guarantees that for any m‐topical map, a cone eigenvector $x \in \mathbb{R}^n_{\geq 0}$ exists:

$r(f) := \lim_{k \to \infty} ||f^k(u)||^{1/k}$

with $f(x) = r(f) x$ for some $x \in \mathbb{R}^n_{\geq 0}$ and $u \in \mathbb{R}^n_{> 0}$ arbitrary. The crucial step is showing irreducibility, in any of its forms, upgrades this existence result to positivity: $x \in \mathbb{R}^n_{> 0}$.

Moreover, local strict monotonicity at an eigenvector, as formulated in conditions (M) or (N) (Lins, 14 Oct 2025), further ensures uniqueness up to scaling. These advanced criteria quantify the absence of nontrivial invariant faces or blocks at the eigenvector’s location.

5. Applications and Contexts

The theory of topical maps and their irreducibility properties has immediate applications:

• Nonlinear Perron–Frobenius Theory: Uniqueness and existence of strictly positive eigenvectors for nonlinear extensions.
• Network Theory: Steady-state and bias vectors in nonlinear decision processes and max-plus algebra.
• Population Biology & Tensors: Stable distributions in nonlinear population models; positive eigenvectors in nonnegative tensor settings.
• Numerical Algorithms: Convergence and uniqueness in Sinkhorn–Knopp and Menon operator computations; certification via SAT.
• Game Theory: Unique fixed points for Shapley operators.

The ability to translate and verify irreducibility conditions via SAT solvers is especially consequential for high-dimensional operators, where direct combinatorial checks are infeasible.

6. Mathematical Formulations

Key formulae include:

• Hilbert’s projective metric:

$$d_H(x, y) = \inf\{ \log (\beta/\alpha) : \alpha x \leq y \leq \beta x \}$$

for $x, y \in \mathbb{R}^n_{> 0}$.

• Local condition (M) for uniqueness:

$$\forall J \subsetneq [n],\ \exists i \notin J\ \text{such that}\ f(u + t e_J)_i > f(u)_i\ \forall t > 0$$

7. Implications for Nonlinear Spectral Theory

Topical maps provide a unifying framework for understanding nonlinear generalizations of spectral theory. The relationships among various irreducibility properties clarify under what regimes the classical spectral radius theory carries over. Inferring from the paper, efficient computational strategies (SAT) and refined hierarchy of irreducibility offer new tools for analysis and certification of nonlinear spectral phenomena in diverse mathematical and applied contexts.