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arxiv: 2506.07911 · v2 · pith:F2GIIMCInew · submitted 2025-06-09 · 🧮 math.AT · cs.DM· math.CT

Stability and Extension of Steady and Ranging Persistence

Yann-Situ Gazull This is my paper

Pith reviewed 2026-05-19 10:32 UTC · model grok-4.3

classification 🧮 math.AT cs.DMmath.CT
keywords persistent homologysteady persistenceranging persistencecategory theoryhypergraphstopological data analysisstabilityfeature characterization
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The pith

Steady and ranging persistence extend to general objects through category theory, enabling stability analysis and feature characterization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops extensions of steady and ranging persistence from graphs to arbitrary objects by means of category theory. It analyzes the stability of these generalized persistence constructions. The central result is a characterization of the features on these objects that produce balanced steady and ranging persistence. A sympathetic reader would care because this generalization allows persistence techniques to apply to richer data structures such as hypergraphs while maintaining desirable stability properties.

Core claim

By employing category theory, steady and ranging persistence are lifted to general objects, and the features that induce balanced forms of these persistences are characterized. Stability is studied for the resulting constructions, with concrete illustration on hypergraphs.

What carries the argument

Category-theoretic extension of steady and ranging persistence, which carries the definitions, stability results, and the feature characterization.

Load-bearing premise

The extension of steady and ranging persistence via category theory preserves the necessary structure to support a well-defined stability analysis and feature characterization.

What would settle it

Observation of a specific feature on hypergraphs where the induced persistence fails to be balanced despite satisfying the characterization conditions, or where stability does not hold as predicted.

Figures

Figures reproduced from arXiv: 2506.07911 by Yann-Situ Gazull.

Figure 1
Figure 1. Figure 1: A persistence function (on the left) and its representation as a persistence [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: On the left: three hypergraphs (nodes in black and hyperedges in blue) and [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A hypergraph filtration H0 ↣ H1 ↣ H2 in Hgph= m. e0 is a hub in H0 and H2 but not in H1 so the feature F h is not convex in Hgph= m. Hence, it is not convex in Hgph≤ m and Hgphm 2. The exclusivity feature: a hyperedge e is said to have an exclusivity in hypergraph H if and only if it possesses a node that does not belong to any other hyperedge: e has an exclusivity in (V, E) ⇐⇒ e ̸⊆ ∪e′∈E\{e}e ′ ⇐⇒ ∃v ∈ e … view at source ↗
Figure 4
Figure 4. Figure 4: A hypergraph filtration H0 ↣ H1 ↣ H2 in Hgph≤ m. The max-originality values of e0 are OH0 (e0) = 1, OH1 (e0) = 0 and OH2 (e0) = 2/3. Hence, the feature F O is not convex in Hgph≤ m (nor is it in Hgphm). Moreover, e0 has an exclusivity in H0 and H2, but not in H1. This implies that F x is not convex in Hgph≤ m (nor is it in Hgphm). 5.3 Experiments In this section, we present some results of steady and rangi… view at source ↗
Figure 5
Figure 5. Figure 5: The scene-hypergraph filtration of King Lear at t = 3 and t = 5. The play has 26 scenes, so the filtration starts at t = 0, and is constant after t = 25. The hypergraphs rapidly start being dense and unreadable, so only some of the first ones are represented [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Steady persistence of the hub feature F h for the scene-hypergraph filtration induced by King Lear. The 0-th scene starts being a hub at t = 1, and stops being a hub between t = 6 and t = 16 before becoming a hub again at t = 25. This induces a difference between the steady and ranging diagrams, which illustrates that F h is not convex in Hgph= m (see [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Ranging persistence of the hub feature F h for the scene-hypergraph filtration induced by King Lear. The first and the last scene (0-th and 25-th scenes) are the only hubs that appear in the filtration [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Persistence of F x for the scene-hypergraph filtration induced by King Lear. This feature is convex in Hgph= m, so the steady and ranging persistence diagrams are equal (see Theorem 4.8). In the scene-hypergraph, the exclusivity feature represents the scenes that feature a unique character. This persistence diagram is rich because most of the scenes introduce a new character. 16 [PITH_FULL_IMAGE:figures/f… view at source ↗
Figure 9
Figure 9. Figure 9: Persistence of F O for the scene-hypergraph filtration induced by King Lear. This feature is convex in Hgph= m so the steady and ranging persistence diagrams are equal (see Theorem 4.8). This diagram is sparse because the only max-original scenes are the 0-th, the 16-th and the 25-th scene. Results for the scene-hypergraph of King Lear [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The character-hypergraph filtration of King Lear at t = 3 and t = 5. The second hypergraph is the dual of the first hypergraph of [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Steady persistence of the hub feature F h for the character-hypergraph fil￾tration induced by King Lear. The character hubs are Kent, Gloucester and Goneril. Kent stops being a hub between t = 8 and t = 9. This induces a difference between the steady and ranging diagrams (see [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Ranging persistence of the hub feature F h for the character-hypergraph fil￾tration induced by King Lear. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Steady persistence of F x for the character-hypergraph filtration induced by King Lear. In the character-hypergraph, the exclusivity feature represents the characters that are the only character for some scene, i.e. the characters who have a monologue scene. In this play, only Edgar has a monologue scene. Although F x is not convex in Hgph≤ m (see [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Steady persistence of F O for the character-hypergraph filtration induced by King Lear. The persistence diagram is empty, which means that no character is max￾original, i.e. for every character e, there is another character e ′ that appears in more than half of the scenes of e. Although F O is not convex in Hgph≤ m (see [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Steady and ranging persistence of the hub feature [PITH_FULL_IMAGE:figures/full_fig_p028_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Persistence of the two convex features (for [PITH_FULL_IMAGE:figures/full_fig_p029_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Steady and ranging persistence of the hub feature [PITH_FULL_IMAGE:figures/full_fig_p030_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Persistence of the two features F x and F O for the character-hypergraph filtration induced by Romeo and Juliet. Although these two features are not convex for Hgph≤ m, their steady and ranging persistence diagrams for this filtration are equal. Note that every character-hypergraph filtration is in the category dual(Hgph=)m and F x is convex for dual(Hgph=)m (claimed but not proved here). 31 [PITH_FULL_I… view at source ↗
read the original abstract

Persistent homology is a topological data analysis tool that has been widely generalized, extending its scope beyond the field of topology. Among its extensions, steady and ranging persistence were developed to study a wide variety of graph properties. Precisely, given a feature of interest on graphs, it is possible to build two types of persistence (steady and ranging persistence) that follow the evolution of the feature along graph filtrations. This study extends steady and ranging persistence to other objects using category theory and investigates the stability of such persistence. In particular, a characterization of the features that induce balanced steady and ranging persistence is provided. The main results of this study are illustrated using a practical implementation for hypergraphs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript extends steady and ranging persistence—originally defined for graphs via filtrations tracking a feature of interest—to general objects in a categorical setting. It investigates stability of the resulting persistence modules and supplies a characterization of those features that produce balanced steady and ranging persistence. The main results are illustrated by a concrete implementation on hypergraphs.

Significance. If the categorical extension preserves the requisite functoriality and the stability bounds hold without extra axioms, the work would meaningfully broaden the scope of persistence-based methods beyond graphs to hypergraphs and other structured objects. The provision of a practical hypergraph implementation and the explicit characterization of balanced features are concrete strengths that could support reproducible follow-up work.

major comments (2)
  1. [§3] §3 (Categorical Extension): the construction defines feature maps as functors but does not verify that these functors commute with the filtration functors for arbitrary objects in the target category, so that the interleaving distance (or bottleneck stability) remains controlled under small perturbations of the feature. This verification is load-bearing for the claim of a well-defined stability analysis outside the hypergraph case.
  2. [§5] §5 (Characterization of balanced features): the stated characterization is shown to hold for the hypergraph implementation, yet the general categorical statement does not include an explicit check that the balance condition is preserved when the underlying category lacks the limits or colimits needed to form the persistence modules. Without this, the characterization risks being restricted to the illustrated examples rather than applying to the general extension asserted in the abstract.
minor comments (2)
  1. [§2] The notation distinguishing steady from ranging persistence could be introduced with a small commutative diagram in §2 to aid readers unfamiliar with the graph case.
  2. [Figure 4] Figure 4 (hypergraph example) would benefit from an explicit legend indicating which bars correspond to steady versus ranging persistence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment point by point below and indicate the revisions made to strengthen the presentation.

read point-by-point responses

  1. Referee: [§3] §3 (Categorical Extension): the construction defines feature maps as functors but does not verify that these functors commute with the filtration functors for arbitrary objects in the target category, so that the interleaving distance (or bottleneck stability) remains controlled under small perturbations of the feature. This verification is load-bearing for the claim of a well-defined stability analysis outside the hypergraph case.

    Authors: We appreciate the referee drawing attention to this point. In the categorical extension of §3, feature maps are defined as functors, and the commutation with filtration functors follows from the naturality of the steady and ranging constructions with respect to the underlying category. To make this explicit and to confirm that the interleaving distance remains controlled for arbitrary objects (without extra axioms), we have added a new lemma in the revised §3 that verifies the required commutation diagrams hold generally. This supports the stability analysis beyond the hypergraph case while preserving the functoriality of the original definitions. revision: yes

  2. Referee: [§5] §5 (Characterization of balanced features): the stated characterization is shown to hold for the hypergraph implementation, yet the general categorical statement does not include an explicit check that the balance condition is preserved when the underlying category lacks the limits or colimits needed to form the persistence modules. Without this, the characterization risks being restricted to the illustrated examples rather than applying to the general extension asserted in the abstract.

    Authors: We agree that greater clarity on categorical assumptions is helpful. The characterization of balanced features in §5 is proved using the limits and colimits that exist in the categories under consideration (including those needed to form the persistence modules). The hypergraph case provides a concrete check, but the general statement already presupposes these structures. In the revision we have added an explicit remark at the beginning of §5 stating that the result applies in any category possessing the requisite limits and colimits, thereby clarifying the scope without restricting the generality claimed in the abstract. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on independent categorical constructions

full rationale

The paper defines steady and ranging persistence on graphs, then uses category theory to extend the constructions to general objects before characterizing features that yield balanced versions and proving stability. No quoted step reduces a claimed prediction or characterization to a fitted parameter, self-citation, or definitional tautology; the extension and stability results are presented as consequences of functoriality and interleaving-distance arguments rather than being presupposed by the inputs. The hypergraph illustrations serve only as verification, not as the source of the general claims. This is a standard self-contained mathematical development.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are specified in the abstract; the work relies on standard category theory and persistent homology concepts without detailing additional postulates.

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