Contraction and Hourglass Persistence for Learning on Graphs, Simplices, and Cells

Indradyumna Roy; Mattie Ji; Vikas Garg

arxiv: 2604.17548 · v2 · submitted 2026-04-19 · 💻 cs.LG · math.AT· stat.ML

Contraction and Hourglass Persistence for Learning on Graphs, Simplices, and Cells

Mattie Ji , Indradyumna Roy , Vikas Garg This is my paper

Pith reviewed 2026-05-15 06:29 UTC · model grok-4.3

classification 💻 cs.LG math.ATstat.ML

keywords persistent homologycontraction homologyhourglass persistencegraph neural networkstopological data analysissimplicial complexescellular networksrepresentation learning

0 comments

The pith

By interleaving sequences of inclusions and contractions, Hourglass Persistence produces topological descriptors more expressive and stable than those from forward persistent homology alone for graph representation learning.

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

Standard persistent homology in GNNs builds features only by adding edges or simplices in growing filtrations, which the paper shows has inherent limits in capturing certain structures. The work analyzes contractions as a reverse topological operation and defines Contraction Homology to extract complementary information. Hourglass Persistence then alternates forward inclusions with contractions to create descriptors that improve expressivity, learnability, and stability. Efficient algorithms make the method compatible with end-to-end differentiable pipelines, and experiments report gains over prior persistence techniques on real graph data. The same construction applies to simplicial and cellular complexes.

Core claim

Forward persistent homology and contraction homology differ in expressivity. Interleaving inclusions and contractions yields Hourglass Persistence descriptors that boost expressivity, learnability, and stability. The approach extends to simplicial and cellular networks and supplies efficient algorithms pluggable into GNN pipelines, producing consistent empirical gains over many existing PH methods on standard real-world graph datasets.

What carries the argument

Hourglass Persistence: topological descriptors formed by interleaving a sequence of inclusions and contractions.

If this is right

Contraction homology captures topological information distinct from forward persistent homology.
Interleaving inclusions and contractions increases expressivity, learnability, and stability of the resulting descriptors.
Efficient algorithms allow direct integration into end-to-end differentiable GNN pipelines.
The framework extends directly to simplicial and cellular networks while preserving the same benefits.
Empirical evaluations show consistent improvements over prior persistent homology methods across standard real-world graph datasets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Bidirectional filtrations may prove useful for other topological data analysis tasks outside graphs, such as point clouds or meshes.
The stability gains could translate to better robustness when graphs contain noise or missing edges.
Task-specific interleaving schedules might further optimize performance on particular graph domains like molecular or social networks.
Combining Hourglass Persistence with other structural encodings could compound improvements in representation quality.

Load-bearing premise

Persistence information obtained from contraction sequences is complementary to forward persistent homology and does not discard essential topological features needed for downstream learning tasks.

What would settle it

A dataset of graphs where the persistence diagrams produced by contraction sequences are identical to those from inclusion sequences, or where models augmented with Hourglass Persistence show no accuracy or stability improvement over standard PH baselines.

Figures

Figures reproduced from arXiv: 2604.17548 by Indradyumna Roy, Mattie Ji, Vikas Garg.

**Figure 2.** Figure 2: illustrates an example of contraction homology. For now, we would like to focus on a specific class of CH that contracts the following sub-graphs known as intermediate complexes. Definition 5. Let H ⊂ G be a subset. The closure of H is the union of H and all vertices with incident edges contained in H. Let ∅ = G−1 ⊂ G0 ⊂ ... ⊂ Gn = G be a filtration of G induced by f, we define the intermediate complexes I… view at source ↗

**Figure 3.** Figure 3: Example graph pairs where (a) color filtration with same [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: Example of a sequence of maps arising in the hourglass persistence for a filtration of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: Example of graphs G and H with height filtration such that FB-persistence can differ but not (f, −f)-FB persistence. Note that the expressivity analysis of extended persistence had been carried out in Yan et al. (2025). Proposition 4. Let f be a filtration function. There exist filtration functions f1, f2 such that the sequence of topological maps in (f1, f2)-FB persistence is exactly the sequence of to… view at source ↗

**Figure 6.** Figure 6: The contraction steps in Figure [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

read the original abstract

Persistent homology (PH) encodes global information, such as cycles, and is thus increasingly integrated into graph neural networks (GNNs). PH methods in GNNs typically traverse an increasing sequence of subgraphs. In this work, we first expose limitations of this inclusion procedure. To remedy these shortcomings, we analyze contractions as a principled topological operation, in particular, for graph representation learning. We study the persistence of contraction sequences, which we call Contraction Homology (CH). We establish that forward PH and CH differ in expressivity. We then introduce Hourglass Persistence, a class of topological descriptors that interleave a sequence of inclusions and contractions to boost expressivity, learnability, and stability. We also study related families parametrized by two paradigms. We also discuss how our framework extends to simplicial and cellular networks. We further design efficient algorithms that are pluggable into end-to-end differentiable GNN pipelines, enabling consistent empirical improvements over many PH methods across standard real-world graph datasets. Code is available at \href{https://github.com/Aalto-QuML/Hourglass}{this https URL}.

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.

Desk Editor's Note private letter to a colleague

The paper defines Contraction Homology from contraction sequences and Hourglass Persistence by interleaving them with inclusions, claiming this boosts expressivity over standard forward PH for GNNs.

read the letter

The main thing here is that the authors treat contractions as a first-class topological move on graphs, define Contraction Homology from those sequences, prove it captures different information than ordinary persistent homology, and then build Hourglass Persistence by alternating the two operations. That interleaving is the central new construction, and they supply efficient algorithms meant to drop straight into differentiable GNN pipelines. They also sketch how the same ideas carry over to simplicial and cellular complexes. The experiments report consistent gains over prior PH baselines on standard graph datasets, and the code is released, which helps reproducibility. Those pieces are the concrete advances. The expressivity separation is the part that matters most theoretically; if the proof holds, it directly answers the one-directional limitation the abstract flags. The practical side looks workable because the algorithms are designed to be pluggable rather than requiring a full rewrite of existing models. The soft spots are modest but real. The complementarity claim between contractions and inclusions is asserted after the expressivity result, but the letter does not show a detailed argument that the combined descriptor never drops essential features. Stability under the interleaving also needs checking in the full proofs, since mixing operations can sometimes introduce new instabilities even if each piece is stable on its own. The empirical gains are described as consistent, yet without seeing the exact data splits, ablation tables, or variance numbers it is hard to judge how robust they are across different graph families. This is aimed at people already working on topological descriptors inside GNNs or simplicial networks. A reader who wants new, computable invariants that are not just the usual filtration will find the constructions and the code useful. It is worth sending to peer review: the theoretical distinction and the algorithmic framing are developed enough that referees can give targeted feedback, and the empirical claims are testable once the full details are examined.

Referee Report

2 major / 2 minor

Summary. The paper introduces Contraction Homology (CH) as the persistence of contraction sequences on graphs, simplices, and cells, proves that CH differs in expressivity from standard forward persistent homology (PH), defines Hourglass Persistence by interleaving inclusion and contraction operations to improve expressivity, learnability, and stability, studies related parametrized families, supplies efficient algorithms integrable into differentiable GNN pipelines, and reports empirical gains over existing PH baselines on standard real-world graph datasets.

Significance. If the expressivity separation and complementarity claims hold, the work supplies a new family of topological descriptors that capture features orthogonal to those obtained from inclusion-based PH, with direct applicability to graph, simplicial, and cellular representation learning. The provision of pluggable algorithms and publicly available code constitutes a concrete strength for reproducibility and downstream use.

major comments (2)

[Theoretical results on expressivity] The central claim that forward PH and CH differ in expressivity is load-bearing for the motivation of Hourglass Persistence; the manuscript should supply an explicit pair of graphs (or simplicial complexes) together with the filtration values at which the two descriptors first disagree, so that the separation can be verified independently of the general proof.
[Experimental evaluation] Table reporting empirical results: the improvements over baseline PH methods are stated as 'consistent' but the table does not list standard deviations across random seeds or the precise train/validation/test splits used; without these the magnitude of the reported gains cannot be assessed for statistical reliability.

minor comments (2)

[Definitions and related families] Notation for the two paradigms parametrizing the related families is introduced without a compact summary table; a single table listing the parameter ranges and the corresponding Hourglass variants would improve readability.
[Algorithmic contributions] The extension to cellular networks is asserted but the algorithmic complexity statement for the cellular case is given only in prose; an explicit big-O expression in terms of the number of cells would clarify the scaling claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive recommendation of minor revision and the constructive feedback. We address each major comment point by point below.

read point-by-point responses

Referee: [Theoretical results on expressivity] The central claim that forward PH and CH differ in expressivity is load-bearing for the motivation of Hourglass Persistence; the manuscript should supply an explicit pair of graphs (or simplicial complexes) together with the filtration values at which the two descriptors first disagree, so that the separation can be verified independently of the general proof.

Authors: We agree that an explicit example would make the expressivity separation more concrete and easier to verify. In the revised manuscript, we will add a simple counterexample consisting of two small graphs (a 4-cycle and a path of length 3) with explicit filtration values where the persistence diagrams of forward PH and CH first differ at a specific threshold, accompanied by the direct computation of both descriptors. This will complement the existing general proof. revision: yes
Referee: [Experimental evaluation] Table reporting empirical results: the improvements over baseline PH methods are stated as 'consistent' but the table does not list standard deviations across random seeds or the precise train/validation/test splits used; without these the magnitude of the reported gains cannot be assessed for statistical reliability.

Authors: We appreciate the emphasis on statistical reliability. In the revised version, we will update all experimental tables to report mean performance with standard deviations over 10 random seeds and will explicitly list the train/validation/test splits (including any fixed splits from prior work) for each dataset. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation introduces Contraction Homology via contraction sequences on graphs, proves an expressivity separation from forward persistent homology, and constructs Hourglass Persistence by interleaving the two operations. These steps rely on explicit topological definitions and algorithms rather than fitting parameters to the same data or reducing via self-citation chains. The claimed complementarity and empirical gains are presented as consequences of the new descriptors, not as inputs renamed as outputs. The paper remains self-contained against external benchmarks with no load-bearing step that collapses by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claims rest on the new definitions of contraction sequences and their persistence; no free parameters are mentioned. The work relies on standard algebraic topology axioms for simplicial complexes and persistent homology.

axioms (1)

standard math Standard properties of persistent homology on simplicial complexes hold under edge contractions
Invoked when defining Contraction Homology and claiming it differs from forward PH.

invented entities (2)

Contraction Homology no independent evidence
purpose: Persistence descriptor based on contraction sequences
New concept introduced to capture topological features during graph shrinking.
Hourglass Persistence no independent evidence
purpose: Interleaved inclusion-contraction topological descriptor
New class of descriptors claimed to boost expressivity and stability.

pith-pipeline@v0.9.0 · 5502 in / 1326 out tokens · 44348 ms · 2026-05-15T06:29:58.393379+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

We study the persistence of contraction sequences, which we call Contraction Homology (CH). We establish that forward PH and CH differ in expressivity. We then introduce Hourglass Persistence, a class of topological descriptors that interleave a sequence of inclusions and contractions
IndisputableMonolith/Foundation/AlexanderDualityProof.lean linking_forces_d3_cert echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

FB-persistence is strictly more expressive than forward and backward persistence combined... Hourglass persistence is more expressive than FB-persistence
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

the sequence of contractions... G → G/IC_n(G) → ... → * ... intermediate maps are the natural quotient maps

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

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Cristian Bodnar, Fabrizio Frasca, Nina Otter, Yu Guang Wang, Pietro Li `o, Guido Montufar, and Michael M

URLhttps://arxiv.org/abs/2302.09826. Cristian Bodnar, Fabrizio Frasca, Nina Otter, Yu Guang Wang, Pietro Li `o, Guido Montufar, and Michael M. Bronstein. Weisfeiler and lehman go cellular: CW networks. In A. Beygelzimer, Y . Dauphin, P. Liang, and J. Wortman Vaughan (eds.),Advances in Neural Information Processing Systems, 2021a. URLhttps://openreview.net...

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Advances in Mathe matics 134(1), 90–145 (1998), https://doi.org/10.1006/aima.1997.1650

ISSN 0001-8708. doi: https://doi.org/10.1006/aima.1997.1650. URLhttps://www. sciencedirect.com/science/article/pii/S0001870897916509. Robin Forman. A user’s guide to discrete morse theory.S ´eminaire Lotharingien de Combinatoire [electronic only], 48:B48c, 35 p., electronic only–B48c, 35 p., electronic only, 2002. URLhttp: //eudml.org/doc/123837. Rudolf F...

work page doi:10.1006/aima.1997.1650 1997
[3]

The GUDHI Project.GUDHI User and Reference Manual

URLhttps://proceedings.mlr.press/v235/papamarkou24a.html. The GUDHI Project.GUDHI User and Reference Manual. GUDHI Editorial Board, 3.11.0 edition,

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Introduction to Topological Data Analysis (TDA)

URLhttps://gudhi.inria.fr/doc/3.11.0/. Bastian Rieck, Christian Bock, and Karsten Borgwardt. A Persistent Weisfeiler-Lehman Procedure for Graph Classification. In Kamalika Chaudhuri and Ruslan Salakhutdinov (eds.),Proceedings of the 36th International Conference on Machine Learning, volume 97 ofProceedings of Machine Learning Research, pp. 5448–5458. PMLR...

work page arXiv 2019
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Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Russ R Salakhutdinov, and Alexander J Smola

URLhttps://arxiv.org/abs/2402.16346. Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Russ R Salakhutdinov, and Alexander J Smola. Deep sets.Advances in neural information processing systems, 30, 2017. Bingxu Zhang, Changjun Fan, Shixuan Liu, Kuihua Huang, Xiang Zhao, Jincai Huang, and Zhong Liu. The expressive power of graph neural netw...

work page doi:10.1109/tkde.2024 2017
[6]

For completeness, we will also use a special case of this lemma to show how persistence modules can derive the usual interpretation of persistence pairs for a graph filtration

In Appendix B.1, we establish a lemma that will be helpful in streamlining the proof of Proposition 6 in Appendix B.2. For completeness, we will also use a special case of this lemma to show how persistence modules can derive the usual interpretation of persistence pairs for a graph filtration. We will also introduce some concepts helpful in understanding...

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In Appendix B.2, we prove Proposition 3, Proposition 5, and Proposition 6 (i.e., the ex- pressivity parts of the section)

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keeping track of one vertex representative per component

In Appendix B.3, we will explain the two algorithms for Proposition 4 and prove it. B.1 KEYLEMMA AND THEUSUALPERSISTENTPAIRSINTERPRETATION FORGRAPH FILTRATIONS Before we give a proof of Proposition 6, we first note that the description of death times for con- nected components holds on an elevated generality that both the proofs of Proposition 6 and (late...

work page 2021
[9]

We fix one of these vertices to be called the supernode(denoted∗), in the sense that any time we compare a vertexvwith∗to decide which one to kill off, we always kill offv

In the filtration stepX −1 →X 0, we pick 1 vertex per connected component inX 0 and mark a tuple(0,−)corresponding to that. We fix one of these vertices to be called the supernode(denoted∗), in the sense that any time we compare a vertexvwith∗to decide which one to kill off, we always kill offv

work page
[10]

Fori≥0, if the stepX i →X i+1 is a filtration, then one treats this in algorithmic time as a procedure to spawn every new vertex that appears first and mark them as tuples(i,−), and then spawn edges between them. For each edge between spawned, if an edge is joined between 2 different components represented by verticesvandw, we mark the vertex born later t...

work page
[11]

Fori≥0, if the stepX i →X i+1 is a contraction, say contracting a subgraphG. We loop through each connected componentX j ofG, we find the vertexv j representing the componentX j belongs in and killv j unless (1) it got killed already in some index j′ < j when looping through the Xj’sor (2) it is the supernode ∗

work page
[12]

Afteri=N, for every tuple not dead yet, we mark it to die at time∞. Remark 2.Note that the statement of Lemma 1 does not requireX N to terminate with only the point∗left, so the case of hourglass persistence in Definition 9 is a special case in this lemma. A more specific case of Lemma 1 is when the maps are all inclusions. This will recover the usual pro...

work page 2026
[13]

We pick a non-zero vectorvofH 0(X0), and consider the sequence of linear subspaces generated by the iterated image ofvinH 0(X•)until it becomes0

work page
[14]

If there is still a non-zero vectorwinH k(X0), we pickwand repeat the process (if the linear map sendswoutside

Then we remove this sequence of linear subspaces off ofH0(X•). If there is still a non-zero vectorwinH k(X0), we pickwand repeat the process (if the linear map sendswoutside. Otherwise, we choose a non-zero vector from what is left inH k(X1)(if it exists) and look at its iterated images ahead, and so on. Let us also recall the following standard fact in a...

work page
[15]

We will go from the remark and see why it is equivalent to the first itme

Remark 1 says thatCis killed if and only if the iterated image of1 C gets landed inW i. We will go from the remark and see why it is equivalent to the first itme. Based on the description of how the induced map onH 0(−)behaves, we see that1 C can land intoW i at timej ′ > jif and only if there is some0≤k≤isuch thatf j′−1 ◦...◦f j(C)andf j′−1 ◦...◦f j(Ck)r...

work page 2026
[16]

Att= 0, we instantiatedimH D(X0)many tuples of the form(0,∞)

work page
[17]

Fort >0, we instantiedimH D(Xt)−dimH D(Xt−1)many tuples of the form(t,∞)

work page
[18]

independent cycles

We end att=n. We first look at how Lemma 2 specializes for graphs, before proving it. WhenD= 1, a 1- dimensional cell complex is the same as a graph, and the lemma recovers the usual way to calculate 1-dimensional PH’s. Indeed,H 1(−)of a graph corresponds exactly to its list of independent cycles. For completeness, by “independent cycles”, we mean the fol...

work page 2026
[19]

We spectulate whether or approach can be applied with edge contractions in graph pooling

independently from persistent homology. We spectulate whether or approach can be applied with edge contractions in graph pooling. F THEUSE OFLARGELANGUAGEMODELS We used large language models to aid in the writing process of the introduction, to help check gram- mar / suggest edits for the main paper, and to double-check some algorithmic discussions arisin...

work page

[1] [1]

Cristian Bodnar, Fabrizio Frasca, Nina Otter, Yu Guang Wang, Pietro Li `o, Guido Montufar, and Michael M

URLhttps://arxiv.org/abs/2302.09826. Cristian Bodnar, Fabrizio Frasca, Nina Otter, Yu Guang Wang, Pietro Li `o, Guido Montufar, and Michael M. Bronstein. Weisfeiler and lehman go cellular: CW networks. In A. Beygelzimer, Y . Dauphin, P. Liang, and J. Wortman Vaughan (eds.),Advances in Neural Information Processing Systems, 2021a. URLhttps://openreview.net...

work page doi:10.1007/s00200-014-0247-y 2016

[2] [2]

Advances in Mathe matics 134(1), 90–145 (1998), https://doi.org/10.1006/aima.1997.1650

ISSN 0001-8708. doi: https://doi.org/10.1006/aima.1997.1650. URLhttps://www. sciencedirect.com/science/article/pii/S0001870897916509. Robin Forman. A user’s guide to discrete morse theory.S ´eminaire Lotharingien de Combinatoire [electronic only], 48:B48c, 35 p., electronic only–B48c, 35 p., electronic only, 2002. URLhttp: //eudml.org/doc/123837. Rudolf F...

work page doi:10.1006/aima.1997.1650 1997

[3] [3]

The GUDHI Project.GUDHI User and Reference Manual

URLhttps://proceedings.mlr.press/v235/papamarkou24a.html. The GUDHI Project.GUDHI User and Reference Manual. GUDHI Editorial Board, 3.11.0 edition,

work page

[4] [4]

Introduction to Topological Data Analysis (TDA)

URLhttps://gudhi.inria.fr/doc/3.11.0/. Bastian Rieck, Christian Bock, and Karsten Borgwardt. A Persistent Weisfeiler-Lehman Procedure for Graph Classification. In Kamalika Chaudhuri and Ruslan Salakhutdinov (eds.),Proceedings of the 36th International Conference on Machine Learning, volume 97 ofProceedings of Machine Learning Research, pp. 5448–5458. PMLR...

work page arXiv 2019

[5] [5]

Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Russ R Salakhutdinov, and Alexander J Smola

URLhttps://arxiv.org/abs/2402.16346. Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Russ R Salakhutdinov, and Alexander J Smola. Deep sets.Advances in neural information processing systems, 30, 2017. Bingxu Zhang, Changjun Fan, Shixuan Liu, Kuihua Huang, Xiang Zhao, Jincai Huang, and Zhong Liu. The expressive power of graph neural netw...

work page doi:10.1109/tkde.2024 2017

[6] [6]

For completeness, we will also use a special case of this lemma to show how persistence modules can derive the usual interpretation of persistence pairs for a graph filtration

In Appendix B.1, we establish a lemma that will be helpful in streamlining the proof of Proposition 6 in Appendix B.2. For completeness, we will also use a special case of this lemma to show how persistence modules can derive the usual interpretation of persistence pairs for a graph filtration. We will also introduce some concepts helpful in understanding...

work page

[7] [7]

In Appendix B.2, we prove Proposition 3, Proposition 5, and Proposition 6 (i.e., the ex- pressivity parts of the section)

work page

[8] [8]

keeping track of one vertex representative per component

In Appendix B.3, we will explain the two algorithms for Proposition 4 and prove it. B.1 KEYLEMMA AND THEUSUALPERSISTENTPAIRSINTERPRETATION FORGRAPH FILTRATIONS Before we give a proof of Proposition 6, we first note that the description of death times for con- nected components holds on an elevated generality that both the proofs of Proposition 6 and (late...

work page 2021

[9] [9]

We fix one of these vertices to be called the supernode(denoted∗), in the sense that any time we compare a vertexvwith∗to decide which one to kill off, we always kill offv

In the filtration stepX −1 →X 0, we pick 1 vertex per connected component inX 0 and mark a tuple(0,−)corresponding to that. We fix one of these vertices to be called the supernode(denoted∗), in the sense that any time we compare a vertexvwith∗to decide which one to kill off, we always kill offv

work page

[10] [10]

Fori≥0, if the stepX i →X i+1 is a filtration, then one treats this in algorithmic time as a procedure to spawn every new vertex that appears first and mark them as tuples(i,−), and then spawn edges between them. For each edge between spawned, if an edge is joined between 2 different components represented by verticesvandw, we mark the vertex born later t...

work page

[11] [11]

Fori≥0, if the stepX i →X i+1 is a contraction, say contracting a subgraphG. We loop through each connected componentX j ofG, we find the vertexv j representing the componentX j belongs in and killv j unless (1) it got killed already in some index j′ < j when looping through the Xj’sor (2) it is the supernode ∗

work page

[12] [12]

Afteri=N, for every tuple not dead yet, we mark it to die at time∞. Remark 2.Note that the statement of Lemma 1 does not requireX N to terminate with only the point∗left, so the case of hourglass persistence in Definition 9 is a special case in this lemma. A more specific case of Lemma 1 is when the maps are all inclusions. This will recover the usual pro...

work page 2026

[13] [13]

We pick a non-zero vectorvofH 0(X0), and consider the sequence of linear subspaces generated by the iterated image ofvinH 0(X•)until it becomes0

work page

[14] [14]

If there is still a non-zero vectorwinH k(X0), we pickwand repeat the process (if the linear map sendswoutside

Then we remove this sequence of linear subspaces off ofH0(X•). If there is still a non-zero vectorwinH k(X0), we pickwand repeat the process (if the linear map sendswoutside. Otherwise, we choose a non-zero vector from what is left inH k(X1)(if it exists) and look at its iterated images ahead, and so on. Let us also recall the following standard fact in a...

work page

[15] [15]

We will go from the remark and see why it is equivalent to the first itme

Remark 1 says thatCis killed if and only if the iterated image of1 C gets landed inW i. We will go from the remark and see why it is equivalent to the first itme. Based on the description of how the induced map onH 0(−)behaves, we see that1 C can land intoW i at timej ′ > jif and only if there is some0≤k≤isuch thatf j′−1 ◦...◦f j(C)andf j′−1 ◦...◦f j(Ck)r...

work page 2026

[16] [16]

Att= 0, we instantiatedimH D(X0)many tuples of the form(0,∞)

work page

[17] [17]

Fort >0, we instantiedimH D(Xt)−dimH D(Xt−1)many tuples of the form(t,∞)

work page

[18] [18]

independent cycles

We end att=n. We first look at how Lemma 2 specializes for graphs, before proving it. WhenD= 1, a 1- dimensional cell complex is the same as a graph, and the lemma recovers the usual way to calculate 1-dimensional PH’s. Indeed,H 1(−)of a graph corresponds exactly to its list of independent cycles. For completeness, by “independent cycles”, we mean the fol...

work page 2026

[19] [19]

We spectulate whether or approach can be applied with edge contractions in graph pooling

independently from persistent homology. We spectulate whether or approach can be applied with edge contractions in graph pooling. F THEUSE OFLARGELANGUAGEMODELS We used large language models to aid in the writing process of the introduction, to help check gram- mar / suggest edits for the main paper, and to double-check some algorithmic discussions arisin...

work page