Skeleton Sensitivity Theorems

• Skeleton Sensitivity Theorems are formal frameworks that quantify how minimal structural representations preserve key invariants when subjected to perturbations across various mathematical domains.
• They establish explicit bounds and equivalence conditions ensuring the stability of structures in settings ranging from Banach spaces and dynamical systems to proximity graphs and TDA.
• Applications include characterizing Asplund spaces, enhancing cosmological data analysis sensitivity, and providing robust risk controls in selective machine learning certification.

Skeleton Sensitivity Theorems formalize and quantify the robustness, stability, and limitations of mathematical, algorithmic, or statistical “skeletons”—minimal, foundational structures that encode critical properties in systems ranging from Banach spaces and hypergraphs to proximity graphs and machine learning models. These theorems address how such skeletons respond to perturbations, changes in underlying data or model specifications, and methodological choices, and they often establish explicit bounds or equivalences that characterize sensitivity and invariance. Theorems related to skeleton sensitivity play a central role in functional analysis, graph theory, topological data analysis, cosmological statistics, and contemporary machine learning.

1. Projectional and Retractional Skeleton Sensitivity in Banach Spaces

Projectional skeletons are systems of bounded linear projections $\{P_s : X \to X\}_{s \in I}$ on Banach spaces, indexed by up-directed, order-complete partially ordered sets, satisfying compatibility, separability, and coverage conditions. In "Simultaneous projectional skeletons" (Cuth, 2013), skeleton sensitivity theorems appear as characterizations of Asplund spaces: a Banach space $X$ is Asplund if and only if every equivalent norm on $X$ induces a 1-projectional skeleton in its dual $(X, \|\cdot\|)^*$ and a retractional skeleton in its bidual unit ball.

The paper establishes:

• For each separable subspace, the associated projections build invariants that survive renorming and passage to duals.
• Existence and uniqueness theorems: For any countable family of $T_p(D)$-closed subspaces $\{Y_n\}$ in $C(K)$, a simultaneous skeleton $\{P_s\}$ can be constructed such that $P_s(Y_n) \subset Y_n$ for all $s, n$.
• Robustness: If a skeleton exists for $X$, it persists under all equivalent norms if and only if $X$ is Asplund.

These results quantify how structural features (such as convexity and topology of unit balls) are “skeleton sensitive”—they are preserved by the appropriate skeletons under renorming and dualization, precisely when $X$ is Asplund.

Property	Skeleton Existence	Sensitivity Consequence
$X$ Asplund	Skeletons in all normed duals	Renorming invariance
Non-Asplund $X$	Skeletons may not exist	Skeletons can fail under renorming

2. Multi-Sensitivity in Topological Dynamics

In "Analogues of Auslander-Yorke theorems for multi-sensitivity" (Huang et al., 2015), skeleton sensitivity theorems are interpreted along temporal “skeletons” of iteration times in dynamical systems. The notions of multi-sensitivity, thick sensitivity, and syndetically equicontinuous points furnish skeletons of times at which sensitivity or regularity is observed.

Notable results include:

• For transitive systems, thick sensitivity and multi-sensitivity are equivalent; i.e., the skeleton of sensitive times is both large (in the sense of thickness) and repetitive across opene sets.
• The main dichotomy theorem: A minimal system is either multi-sensitive (possesses a skeleton of sensitivity times) or an almost one-to-one extension of its maximal equicontinuous factor (the skeleton is absent or regular).
• Syndetically equicontinuous points provide a “skeleton” for regularity: Orbits are equicontinuous on a syndetic skeleton of times, with failures distributed sparsely.

These concepts refine classical sensitivity, allowing skeleton sensitivity theorems to characterize systems by the presence or absence of robust skeletons of sensitivity.

3. Skeleton Sensitivity in Proximity and Geometric Graphs

For β-skeletons—a class of proximity graphs parameterized by a continuous value $\beta$—sensitivity is explicitly quantified in "How beta-skeletons lose their edges" (Adamatzky, 2013) and "Improving SDSS cosmological constraints through $\beta$-skeleton weighted correlation functions" (Yin et al., 21 Mar 2024). Here, skeleton sensitivity refers to changes in the skeleton graph’s edge structure and derived statistics as parameters or underlying distributions are perturbed.

Core results:

• Edge disappearance as $\beta$ increases follows a power law: $$e(n, \beta) \sim c(n, \beta) \cdot \beta^{\alpha(n, \beta)}$$ with $\alpha(n, \beta) < 0$.
• Random planar sets lose edges more rapidly with increasing $\beta$ than structured sets—$|\alpha|$ is larger for random sets.
• In cosmological applications, β-skeleton weighted correlation functions yield increased sensitivity to cosmological parameters such as the matter density $\Omega_m$. Joint analysis of two-point correlation and skeleton-weighted mark functions leads to significant improvements in discrimination (as measured by $\chi^2$ statistics).

Sensitivity theorems thus manifest in the quantification of topological or statistical features’ reaction to parameter changes and perturbations.

Skeleton Type	Perturbation	Sensitivity Manifestation
β-skeleton	Increasing β	Edge loss per $\beta$ (power law)
Cosmic web skeleton	Varying $\Omega_m$	Enhanced $\chi^2$ discrimination

4. Higher-Dimensional Skeletons and Persistent Topological Features

"The Higher-Dimensional Skeletonization Problem" (Verovsek et al., 2017) develops homologically persistent skeletons (HoPeS) in arbitrary dimension $d$, generalizing MST-based skeletons to complexes that preserve persistent homology features across all scales. Skeleton sensitivity is encoded in:

• Greedy algorithms that produce minimal spanning $d$-trees, optimal in total weight and resistant to arbitrary additions (any extra $d$-faces add weight without improving homology).
• Theorems proving that, for every scale parameter $\alpha$, the reduced skeleton is homologically fitting (maintaining isomorphism of $H_d$ with the original complex) and remains optimal in weight.
• Explicit sensitivity to weight filtrations: Only persistent features reflected in barcodes (birth, death intervals) survive in the skeleton, yielding robustness against non-persistent noise.

“Skeleton sensitivity” here is reflected in the minimality and feature-preserving nature of homologically persistent skeletons; changes in weights or topology are reflected precisely in the skeleton structure.

5. Combinatorial Skeleton Sensitivity: Hypergraphs and Hamilton Cycles

The Skeletal Lemma for quasigraphs in 3-hypergraphs, as proved in (Kaiser et al., 2019), formalizes skeleton sensitivity in combinatorial optimization. The lemma asserts that unless a quasigraph π is acyclic and its final skeletal partition is π–skeletal, one can always find a strictly “better” quasigraph (with improved signature or fewer edges), iteratively steering toward a robust skeleton.

Key points:

• The “skeleton” is the terminal partition where each part is connected and anticonnected, and the contracted complement is acyclic.
• Sensitivity is measured by how changes in π (e.g., edge additions/removals) affect the overall partition signature via a lexicographic ordering of plane sequences.
• This iterative improvement process is robust and sensitive: The final skeletal partition controls key properties needed for applications such as Hamiltonicity in line graphs.

The lemma generalizes classical spanning tree results to hypergraphs and exposes the sensitivity of connectivity properties to the underlying skeleton structure.

6. Skeleton Sensitivity in Statistical Certification and Machine Learning

In selective classification for LLMs, "Selective Risk Certification for LLM Outputs via Information-Lift Statistics" (Akter et al., 16 Sep 2025) introduces skeleton sensitivity theorems as explicit quantitative guarantees:

• The η-robustness theorem: If the total variation between skeleton $S$ and $S^*$ is $\eta$, certified risk increases by at most $C\eta$.
• The κ-informativeness theorem: If mutual information between evidence $E$ and output $Y$ is less than κ, abstention is inevitable for at least $\Omega(\sqrt{\kappa})$ fraction of inputs.
• PAC-Bayes sub-gamma analysis: Risk bounds for selective classification remain valid under heavy-tailed statistics when standard Bernstein bounds fail, ensuring statistical robustness when the skeleton model is imperfect.

These theorems rigorously bound how sensitive risk certificates are to misspecification of the “skeleton” baseline, and how limitations in evidence translate into obligatory abstention.

Skeleton Error	Certified Risk Impact	Practical consequence
TV(S, S^*) = η	Risk increases by $\leq C\eta$	Graceful degradation
Low $\kappa$	Abstention $\geq \Omega(\sqrt{\kappa})$	Minimum abstention necessary

7. Cross-Domain Synthesis and Implications

Across domains, skeleton sensitivity theorems serve as structural invariants, robustness certificates, and failure-mode characterizations. Whether in Banach space decomposition, dynamical systems’ sensitivity timelines, skeleton graphs for spatial or topological analysis, or statistical certification for LLMs, skeleton sensitivity quantifies and guarantees the persistence or change of critical structure under parameter variation, perturbation, or misspecification.

Major implications include:

• Structural invariance under group actions (renorming, dualization): Characterized for classes such as Asplund spaces.
• Robust risk control in statistical learning: Errors in skeleton design incur only controlled increases in risk.
• Feature-preserving skeletonization in TDA: Only persistent topological structures survive, reflected explicitly in the skeleton.
• Algorithmic stability and tractability: Iterative improvement via skeletons yields canonical solutions for connectivity, minimality, or complexity.

Current research continues to refine bounds, generalize skeleton constructions to new domains, and probe the limits of sensitivity—especially where the underlying structure is high-dimensional, non-Euclidean, or driven by heavy-tailed non-Gaussian statistics.

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In summary, Skeleton Sensitivity Theorems provide the theoretical foundation for understanding the stability, robustness, and limitations of minimal structural representations (“skeletons”) under perturbations and methodological choices. They find rigorous and far-reaching applications in analysis, combinatorics, topology, statistics, and machine learning, offering quantitative control and principled characterizations throughout mathematical and computational science.