Contraction and Hourglass Persistence for Learning on Graphs, Simplices, and Cells
Authors: Mattie Ji
Indradyumna Roy
Vikas K Garg
Venue: ICLR 2026 Poster (2026)
Keywords: Graph neural networks
topological neural networks
persistent homology
Primary area: learning on graphs and other geometries & topologies
TL;DR: We provide an interleaving of inclusions and contractions to discuss persistence for learning on graphs, simplices, and cells.

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 https://github.com/Aalto-QuML/Hourglass.

Published as a conference paper at ICLR 2026
CONTRACTION AND HOURGLASS PERSISTENCE FOR LEARNING ON GRAPHS, SIMPLICES, AND CELLS
Mattie Ji Indradyumna Roy2..:; Vikas Garg2.4
University of Pennsylvania. AAalto University. IIdian Institute of Technology Bombay. Yai Ltd. mj mji130sas upenn. edu, indradyumna roy@aalto. fi, vgargécsa edu
ABSTRACT
Persistent homology (PH) encodes global information, such as cycles, and thus increasingly integrated into graph neural networks (GNNs), PH methods in GNNs typically traverse an increasing sequence of subgraphs. In this work, we first ex- pose limitations of this inclusion procedure. To remedy these shortcomings, we analyze contractions as 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, class of topological descriptors that interleave a sequence of inclusions and contractions to boost ex- pressivity, 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 improve- ments over many PH methods across standard real-world graph datasets. Code is available at https: / gighub..... com/ Aalto- MatoooMMMMHHorglaaaae.e
INTRODUCTION
Graph Neural Networks (GNNs) (Hamilton, 2020) is powerful paradigm for learning on structured data, yet their expressive power is fundamentally limited by the Ceisfeiler-Lehmee (WL) hierar- chy (Xu et al., 2019; Morris et al., 2019). In particular, peessage-passing GNNs struggle to capture higher-order topological signals (ex. the presence, interaction, and disappearance of cycles) that often drive downstream performance in molecular learning, physical systems, and network science (Garg et al., 2020; Chen et al., 2020; Tahmasebi et al., 2021). Persistent Homology (PH) (Edels- brunner et al., 2002) from Topological Data Analysis (TDA), offers way extract such signals by tracking the life of topological features in a filtration. Accordingly, PH-based descriptors are in- creasingly used augment GNNs and oogeer-order neural architectures (Papamarkou al., 2024), and to boost expressivity (Ballester & Rieck, 2024; Zhang et al., 2025; Li & Leskovec, 2022).
However, many PH pipelines in graph learning rely on inclasion-based filtrations ("forward persis- tence") (Rieck et al., 2019; Hofer et al., 2020; Immonen et al., 2023; Ballester & Rieck, 2024; Ying al., 2024; Ji al., 2025a;b). This one-sided view can leave information on the table. For instance, a forward pass on graphs, cycles born can never die; and while new connected components can ap- pear vertices arrive. This is also undesirable from the perspective of metrizability, as bottleneck distances can be ill-defined due to the presence of different number of permanent features.
To resolve this, we draw our insights from topological contractions, which offer the complementary picture to inclusions. Contractions have wide variety of uses in the study of persistent homology. Discrete Morse theory (Forman, 1998; 2002) uses the idea of simple contractions/collapess to sim- plify complexes. Contractions have since been used in many contexts to speed up PH calculations (Diotko & Wagner, 2013; Botnan & Spreemann, 2015; Boissonnat & S., 2016; Dey et al., 2016; Dey & Slechta, 2020; Boissonnat et al., 2023). The PH literature also has ways to resolve the afore- mentioned metrizability problem, one of which being extended persistence (Agarwal et al., 2004; Cohen-Steiner et al., 2009) by considering both forward and reverse pass on the input data.
"Equal Contributions.

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Persistence Goes Forward and Backward (Section 3):
Construction Backward PH and Barkwaddd PH Incomparability for Forward vs. Backward PH
Definition 6, Definition 7
Proposition 1
Construction of (a, --B PH and Hourglass Persistence
Definition Definition 9
Hourglass PH FB-Persistence > Forward + Backward PH
Theorem 1, Proposition
Relation to Extended Persistence and Extensions (Section 4) Introduce (f.g)-FD Persistence, with (a, T-B as Example FB-Persistence # Extended Persistence
Definition 10, Proposition 4
Proposition 5
[-F persistence Forward PH in f
Proposition 6
Extensions Higher Complexes and Stability (Section 5, 6):
Proposition 7; Theorem
Algorithmic Design and Experiments (Section 7):
Alg Alg 2; Table 2
Figure 1: Overview of the paper. Each row summarizes a core item and the section where it appears.
In this work, we study the PH of sequences of contractions (which we refer to as contraction ho- mology (CH)) for graph representation learning and their higher-order counterparts. While PH can be thought of as a fine-forwarding process, CH can be thought of as fime-reversing process that operates by contracting substructures. We then study a particular form of CH which we call back- ward persistence (BH), defined by contracting a sequence of intermediate complexes. We compare the expressivity of forward PH and backward PH and show that neither subsumes the other.
Motivated by our discovery, we then propose Barward-Backward (FB) persistence, which concate- nates an inclusion phase with a contraction phase. Intuitively. FB links how features appear (for- ward) with how they subsequently vanish (backward), assigning finite lifetimes structures that would otherwise persist to infinity in forward PH. We show FB FB-persistence is strictly more expres- sive than forward and backward considered in isolation.
Because of the modular nature of how we set up the intermediate complexes, we can interchange the sequence of intermediate complexes that included and contracted at each step. Guided by this combinatorial insight, we introduce Hourglass persistence, which interleaves inclusions and contractions in arbitrary order, subject only to the constraint that piece must be included before it can be contracted. Hourglass also gives a tradeoff between runtime and PH information of the whole complex (see Section E.4), which may be beneficial for evaluating large graphs.
Finally, we propose a general framework of (f.g)-FF persistence with respect to two filtration functions ff and g. which extends both extended and FB-persistence. We then extend our framework to ordeer-order structures, design algorithms to compute (f: ff.g)-FB persistence over graphs, and integrate them into GNNs for empirical validations.
Our work differs from many prior works in several ways. (1) The typical usage of collapses to speed up PH calculations is for a sequence inclusions and aims to alter the output as minimally as pos- sible. Instead, we study the PH of contractions that can significantly alter the topology and their interactions with inclusions. (2) Rather than contracting superlevel sets BH, we make the sub- structures more granular by quotienting intermediate complexes. In fact, we show in Proposition that -8-persistence and extended persistence are different. (3) To the best of our knowledge, ex- tended persistence used in GNNs (Zhao & Wang, 2019; Zhao al., 2020; Carriere al., 2020; Yan et al., 2022) does not compute their descriptors using the contraction perspective for experiments, but will demonstrate the utility of contractions in the experiments.
We summarize our paper in Figure and leave proofs and additional details to the Appendix.
BACKGROUND AND PRELIMINARIES
Let G E) be finite undirected graph, possibly with self-loops and multi-edges A common theme of enhancing graph neural networks (GNNs) is to incorporate persistent topological descrip- tors derived from given filtration of G. For our purpose, filtration is as follows.
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Definition 1. Let f: +:VUE-BE be function. We say f is a filtration function G ffr every edge e = (v, w) f(v) I f(e) and f(w) I f(e). The ordered set (-- has only finitely many elements, which gives list of sequential subgraphs:
G-1 Go C. G1 .... Gn =.
We say I is vertex-based filtration for every edge = w), f((e) = max(v, w). We say that an edge-based filtation if f(t) = 0 for all v E V and f(e) > for all e E.
For consistency, the function ʃ should not change after relabeling the vertices and edges, rather should only depend on the intrinsic features the graph G comes with. This property is called per- mutation equivariance (Ballester & Rieck, 2024). Common examples includes degree filtration.
Definition 2. Suppose G is colored graph (i.e., vertices have labels) equipped with an additional structure of coloring function V - C, where C is a collection of colors. A vertex-color filtration is a vertex-based filtration f V U E + I such that f(u) = f(w) for all v, w with c(v) = c(w). Intuitively, f preserves the vertex coloring.
A special case of Example includes degree filtration where c is the degree function and f = c. A common scenario where colored graphs arise is from molecular graphs (Hoogeboom et al., 2022; Xu ct al., 2022; 2023; Song et al., 2024), where the colors are the type of atoms (c.g,, oxygen, carbon).
A common descriptor extract data from given filtration is persistent homology.
Definition 3. Let X. (Xo 4 X1 f
In-3, Xn-1 Iin Xn) be sequence of topological spaces (ex. graphs) X, with maps fo X, - X,+1- The k-th persistent homology of X. is the result of applying H& (in 2//2coofficients) to X., that is, it is the sequence of linear maps:
Hx(X.) = (Xo) (oOb, H& (x1) (fa),,
(x((n-)) (fa-t),, Hx())).
An element v. E Hg(X.) is said to born at i if & im((fe-1)-). We say v dies at 2 i if d is the first element such that (fa-1)= o... (f)) O. The persistence pair associated v is the pair (b, d). If no such d exists, we mark the pair as (b, (b)) If the incoming complex is of the form
1-1, Xo +.... +....XX,, Xx) (e.g. from filtration), by H₂ (X.) we mean H₂ of the sequence with the X-1-term truncated.
X. (X-1 =0
Suppose the vector spaces are all finite dimensional, the k-th persistence diagram (i.e., barcodes in Ghrist (2007)) of X. the collection of persistent pairs for a specific choice of basis of the Hx(X.)'s obtained from interval decomposition of Hx(X.) (see Theorem 2.7-8 of Chazal et al. (2016) or Theorem 4.7 of Lesnick (2025) for more details).
Appendix B.1 shows its equivalence to the usual definition of graph PH. Here H-(-) are connected components, (-) are independent cycles, H2(-) are voids, and H& are k-dimensional voids. Remark 1. Explicitly, the k-th persistence diagram of X. can be computed by first picking non- zero vector v of Ha (Xo), and consider the sequence of linear subspace generated by the iterated image of in H& until it becomes O. Then we remove this sequence of linear subspaces from Hx(X.). If there is still a non-zero vector u in Hx)))), we pick w and repeat the process (if the linear map sends the complement, we consider it becomes 0). Otherwise, we choose non-zero vector from what is left in (x1))) ii exists) and look at its iterated images ahead, and so on.
PERSISTENCE GOES FORWARD AND BACKWARD
For now, we will focus on the case of graphs, but many constructions here can be readily generalized to the setting of simplicial and cell complexes (see Section 5). Classically, there are numerous applications in TDA and GNNs that uses an inclusion-based persistent homology (Edelsbrunner & Harer, 2008; Edelsbrunner & Morozov, 2012; Immonen et al., 2023; Ballester & Rieck, 2024). Concretely, an inclusion- baaddd PH may be viewed as an example of Definition 3 as follows.
PH). Let f G - I be a filtration function, and consider applying Ho(-) and H-(-) to G. G- = C Go C C Gn = G, where each map is the inclusion map. Then the persistent diagram in Definition 3 recovers all birth/death pairs (b, d) such that b < d (i.e., it omits trivial births). See Appendix B. for a formal proof. We can recover the trivial births by counting the number of unmarked simplices at each time.
Example 1
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The interest that initiated this work was: what if we reverse the inclusion steps with contractions? Definition 4. Let G be graph, the contraction homology (CH) of G is the persistence diagrams associated to any sequence of subgraph contractions of G down to a single point.
Figure 2 illustrates an example of contraction homology. For now, would like to focus specific class of CH that contracts the following sub-graphs known as intermediate complexes.
Definition 5. Let H C G be a subset. The closure of H the union of H and all vertices with incident edges contained in H. Let 0 = G-1 f. we define the intermediate complexes (G, f') as the closure of G; G;-1 O,...,R n. We omit the symbol when the context is clear.
Go ... Gn G be a filtration of G induced by
Definition 6 (Backward PH). Let G. G-1 0 c Go C. ... Gn G be filtration of G, consider a sequence of contractions with respect to G. as (G.)² below:
G Ga+1 == GG) - ==+2:= LU
+ C/(+,UICn-2(GG) + -
where *n+1 is the point representing the total contracted subcomplex the previous step, and so on. The intermediate maps are the natural quotient maps. We define the i-th backward persistent diagram as the persistent diagram associated with H²((G.))) ).
t=
t=2
t=3
t=4
Graph Sequence
1st Homology (Cycles)
2./2(ahea)
2/2((a)) 2/2
7./2(ab)
1 (1,
(1.0) -
Persistent Pairs
212
psinorphism
1/)(((,1)
isansophism
-2/2(1)
0
(1,4)
2/2((0,11)
(2, 3)
Figure 2: Contraction Homology of a graph G. The blue simplices indicate what is contracted.
We observe that the backward PH introduces new information that forward PH may ignore, A reason for this distinction is that a cycle that is born in forward PH never die, but backward PH can kill cycles. Conversely, backward PH can never make a new connected component apart from the initial step, but forward PH can spawn new components later.
There exists graphs G, H. with permutation equivariant filtrations such that the forward PH of their filtrations cannot tell apart G, H but backward PH can. Similarly, there are graphs that backward PH cannot tell apart but forward PH can. Examples can be found in Figure 3.
Proposition 1.
(8)
(b)
(d)
a
G
H G
H
G
Figure 3: Example graph pairs where (a) color filtration with same forward-based PH but different backward-based PH, (b) degree filtration with same backward-based PH but different forward- based PH, (c) degree filtration with same forward and backward PH but different FB-persistence, (d) degree filtration with same FB- Pe-eersiseence but different hourlgass persistence.
Another motivation for us to introduce backward based PH is from the viewpoint metrizability and stability. Classically, the bottleneck distance Cohen-Steiner et al. (2006) between 2 persistence
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diagrams PD1, of the same size is the infimum of the value maXpePD; lpp T(p)ll>o where и ranges over all bijections æ : PD1 + PD2. The bottleneck distance is finite for persistence diagrams coming from the same graph, but for diagrams, coming from two different graphs, this distance may very well be infinite. The reason why is because the number of tuples of the form (-,00) for both graphs may be different, which adds an co to the distance function.
One might argue that there can be workarounds by setting the time to die at a finite time N after both filtrations ended, or make die -1. These modifications however can be quite sensitively altered by, for example, having many features die around N or -1. One solution would be to extend the (inclusion-basee PDs by killing off the permanent features topologically via a sequence of contractions. Formally, this motivates our definition of
(FB) persistence.
Definition 7 (Forward- Borward-Backkaat Backward Persistence). Let G. = (0 G-1 C Go ... C Gn G) be a filtration of G. The i-th FB-persistence is the diagram (G. + (G.)"), where G. denotes the sequence of contractions in Definition 6 and denotes the concatenation of the two sequences.
We now establish the expressivity benefits due to FB-persistence.
Theorem 1. FB- TB-persistence is strictly more expressive than forward and backward persis- tence combined. More precisely, for any graph G with filtratio f. the FB-persistence with respect to f can recover the correspondent forward and backward persistence. However, there exists graphs G, H with a permutation equivariant filtration f such that their forward and backward persistence the same, but their FB-persistence differ (see Figure 3(c)).
A high level reason why Theorem holds is that deciding how we concatenate the values from the forward persistent diagrams with the values from the backward persistent diagrams is rather subtle.
Motivated by the perspective of color-based filtrations (Ballester & Rieck, 2024; Immonen et al., 2023; Ji et al., 2025a), we observe that there is canonical reason contract in the order of IC, (G), IC,-- (G), in Definition 6 and Definition 7. Since the intermediate complexes satisfy G and (ii) (G) and (G) can only intersect at vertices, any sequence of contracting the subgraphs in the list single point set *. Likewise, we can also see the filtration step as a special case of spawning the pieces TCn( X)), (X), in ascending order. There is no reason why we should have included them in this order. This motivates the following construction.
() U (G)
IC. would have terminated at the
Definition 8 () persistence). Let T, be two permutations of the list [m] {o,..., n.. The i-th (a, -)FFBBBpp persistence is the persistence diagram associated with the persistence module obtained by applying H-(-) to the sequence
In where for S i S n, Y, = Ujs, (Co ()) (G), and for 1 i n. 2, = Z LIC+u)(G)
0 Y-1 ... CYn G --
where カ5-1 is previously defined j 1 2 1, and *o is any point in (G)
In other words, we apply Definition 3 to the sequence where we filtrate G by spawning the interme- diate complexes the order o, and then contracting G in the order т. In the case where id the identity and re is the reverse list bijection. FB-peessstence is (id,
core motivation for Definition comes from cosor- filtrations (Example 2). Indeed, recall that degree filtration is special case of Example 2. The perspective of coloring filtration is that we can spawn the vertices in any permutation of the set of possible degree values here, which may lead to more information and reduce bias for the model. Furthermore, to reduce bias, one could also argue why we should wait until the entire graph has been filtrated to start the contraction process?
Indeed, the whole process would still terminate to point long complex being contracted has appeared earlier in time, and this yields further flexibility in the pro- cess of switching back and forth between the forward steps and the backward steps (analogous to the imagery of an hourglass). This motivates us to the definition of hourglass persistence, which may be regarded as "color-based version" of FB- Fe--essisteece that also has well-defined metric for persistence diagrams on different graphs.
ensure the intermediate
Definition 9. Let f X R be filtration function with associated intermediate complexes IC(X)). An hourglass persistence diagram is the persistence diagram of any sequence of inclusions and contractions, provided that TC(X,) is included in the sequence before is contracted.
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Figure gives illustration of an example of sequence of maps that occurs in hourglass persis- tence. To demonstrate its expressivity, we have:
Proposition 2. Hourglass persistence more expressive than FB-persistence (Figure 2(c)).
m
[C(())
IC,(X):
Inchade (1)
Tnclude (2)
Contrtt (1)
Include (0)
Contract (2)
Contrnct (0)
Figure 4: Example of a sequence of maps arising in the hourglass persistence for a filtration of G.
Hourglass persistence has promising potential to tackle problem that PH-based methods sometimes struggle to scale on large graphs/simplicial complexes/cell complexes, see Appendix E.4.
4 RELATIONSHIP TO EXTENDED PERSISTENCE AND EXTENSIONS
Extended and FB-persistence are different as follows: For filtration function f. extended per- sistence is obtained by considering a concatenation back. By Proposition 2.22 of Hatcher (2002), this is essentially akin to shrinking the superlevel sets GO = {a E V U E ((x) 2 a} as a goes down, but the superlevel sets and the intermediate complexes coming from are in general quite different. We make this precise in Proposition 5,
PDs with filtrating by f first and by -f
We will now introduce a unifying perspective of both methods:
Definition 10 ((F+ Persistence). Let f,g be filtration functions G with Gl, G being their induced filtrations, respectively. The i-th (f.9)-BB Persistence of G is the persistence diagram associated with the sequence:
1 = GL C Gb/ -. =G
G -..
where [G (EE--.........)) is the filtration of G induced by f. and the sequence Go == Go G -. G
sequence of contractions following the subgraphs that appear in the filtration induced by g. More precisely. Gi = G/ G/ (G, G G/(=1 U (G. ()), G3 == G/(#2 U (C, ())) etc., where #2 is the point representing the total contracted subcomplex from before.
The idea of combining different filtration directions has also appeared in zigzag (Carlsson & de Silva, 2010) and bipath (Aoki et al., 2025) filtration. We explain in Appendix E how Definition 10 differs from the two. Definition 10 also generalizes extended persistence due to the following result. Note that the following is not difficult to prove; we only include it here for completeness.
Proposition 3. The extended persistence of filtration function f (as defined in Section 2 Yan et al. (2022)) has the same expressive power as (f.-f)-FB persistence.
Note that the expressivity analysis of extended persis- tence had been carried out in Yan et al. (2025).
3
3
Proposition 4. Let J be filtration function. There exist filtration functions f1, f2 such that the sequence of topo- logical maps in (f1, F2) persistence exactly the se- quence of topological maps in (α, T)B persistence. If is vertex-based, we provide explicit O(N log N) al- gorithm to compute (f1, that becomes linear time for FB- "B-persistence (see Appendix B.3).
9
3
3
6
11
Figure Example of graphs G and H with height filtration such that FB FB-FF----re persistence can differ but not (f, -) persistence.
Now establish clear separation between FB persis- tence and extended persistence.
Proposition There exist embedded graphs G, H with height filraatioss (Figure 5) where FB- persistence can tell them apart, but extended persistence cannot.

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Besides the choice of -f or fo, another candidate be f itself. Here show that (f. hourglass persistence does not bestow further information.
Proposition 6. f)FB persistence has the same expressive power as forward-PH for For grapht, (fF persistence can be computed by:
Suppose there are n-steps in the filtration, cycle born in filtration at m-tti dies at n i. A cycle born in the contraction al t = n + i corresponds exactly to the occurrence of a connected component ( i that will be merged into a pre-existing connected component in the filtration steps. The death time is n + j. and j > i is the 1st time in filtration when the merge above occurs. The component death times are recorded as usual by keeping track of vertex representatives.
EXTENSION TO SIMPLICIAL AND CELLULAR COMPLEXES
We now extend the framework to higher dimensions, viewing graphs as I-dimensional simplicial complexes (for simple graphs) and 1-dimensional cell complexes. This extension can be readily integrated into simplicial and cellular networks (Bodnar et al., 2021a;b; Kim et al., 2020).
5.1 EXTENSION TO SIMPLICIAL COMPLEXES
Definition 3 works for any sequence of topological spaces, so our constructions may be defined directly. Unfortunately, the quotient of a simplicial complex may not be simplicial complex without further tri- angulations.
We can however adapt a trick to conduct "simpli- cial quotients" (see Cohen-Steiner et al. (2006); Dey & Wang (2022)) as follows. Add a disjoint vertex 04 to the simplicial complex at the beginning, and whenever simplex a is asked to be contracted, one instead adds a simplex [u+, a to keep this as filtration. On graphs, this means that one adds an edge from u to v4 for every vertex being contracted and a triangle (Vi, U2 v+) for every edge (VL, v2) being contracted (see Figure 6). One then computes the persistent diagram of this filtration instead.
t=
I 3
t= 4
Figure 6: The contraction steps in Figure inter- preted as adding simplices instead.
Proposition 7. For all methods defined in Section 3, 4, the simplicial quotient method will recover the k-th dimensional persistence diagrams for l 0 but may differ the level of = For k = O, the pairs can be directly computed by keeping track of one vertex representative per component.
5.2 EXTENSION TO CELLULAR COMPLEXES
The simplicial version adds more simplices to the set-up, which may make the computations more costly. We should try exploit the contraction steps geometrically, they would reduce the data. To do this, we would like to relax the collection of objects we are working with.
regular cell complex (Hansen & Ghrist, 2019) generalization of simplicial complexes that adds more flexibility by allowing X be built off of k-dimensional disks (called k-cells) as opposed to k-dimensional triangles. Intuitively, a cell complex built out inductively by starting with the O-cells (i.e., discrete points), attaching 1-cells to O-cells, then attaching 2-cells to the complex, and so on. This process yields poset structure on the cells where 7 S a indicates part of a is attached onto We refer to Section 2 of Bodnar et al. (2021a) for thorough explanation.
Now we may extend filtration function on a regular cell complex X as follows.
Let Cell(X) denote the collection of cells on X. A function f Cell(X) - IR is filtration function if f(T) & ( all t & a in Cell(X), which induces a filtration on X.
Definition 11.
An important feature of regular cell complexes is that they closed under quotients of subcom- plexes. Thus, every construction we discussed in Section 3 and 4 extends mutatis mutandis that is, one can just replace the word "graph" with the word "cell complex".

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6 STABILITY
Recall one motivation for "concatenating" inclusions and contractions was to compare metrics for diagrams different spaces. We would still like to ensure the diagrams stable if they are on the same space. This is a desired property as we would like perturbations in the input filtration to not affect the oulput drastically. To properly discuss (bottleneck) stability, however, we need to make distinction between combinatorial time and function time.
So far, the persistence diagrams in our construction have all been combinatorial time we use some function(s) to make sequence of maps, and the pairs (i,j) record the birth and death steps in the sequence. However, we want notion of bottleneck stability, we should work with function time that is, we change the pair (i,j) to the az).
When extending function time to include contractions, we observe that there are some subtleties in defining function time for (f.g) g)FB persistence, because the functions f and g can be independent. It can be the case that the values of g are mixed with the values of f on the real line, but define function time we would like the values of to appear after the values of f. To get well-defined notion of function time, we require f and to both positive, and that the corresponding function time of (a) an inclusion step i in g) persistence to be the i-th value of and (b) contraction step at n + i in gfB persistence to max(f) f+)+ + the i-th value of g.
Theorem 2 (Stability of (f.g)-FB persistence), Let X be a graph, simplicial or cellular complex. For 2 pairs of filtrations (f.9). (f', gg) on X, we have the following for all i 2 0:
dø dB (X, PH. (X, g')) s 2 f fo 19 I oo + 1aa (000(mma)))))))
We remark that establishing an appropriate function time for hourglass persistence is challenging, because the function values for each step is not as canonical since we are interleaving the steps of contractions and inclusions. Hourglass persistence does satisfy the condition of what Chazal et al. (2009) called "tame". which has stability in combinatorial time (see Theorem 4.4 therein).
7 ALGORITHM DESIGN AND EXPERIMENTS
Algorithm FORWARDINCLUSSIN
We now present a practical algorith- mic framework that supports any filtration scheme composed of a sequence of inclu- sions and contractions, assuming that all contraction intermediates have already ap- peared previously in the forward filtration.
1: Input: Filtration f: Graph G
2: Output: PD.. PD,, cycle basis B, union-find UF
3: Initialize UF V; PDu, PD,,
4: Sort edges c1,..., e. by f(e,)
5:
for j 1..m do
6:
(u, v) + Ej
7:
if UF. find UF. find(w) then
7.1 FORWARD INCLUSION WITH AUXILIARY BOOKKEEPING.
8:
// Cycll-creetiin edge
9: Build y E (0, 1)mh from e, and path u ~
10:
B [7) PD + (f(e,),
11:
else
In addition to the standard union-find struc-
12:
To UF. find(u); r. +- UF fin((r)
ture, which incrementally tracks connected-
y arg argmax,e(rr.ra) f(r)
component memberships during the forward
14:
PD. + PD, U ((f()), f(e))))
filtration, we maintain two further struc-
15:
UF. merge(u, v)
tures: (i) neighborhood information for the spanning forest being built, and (ii) funda-
16:
I| record mutual neighbors in spanning forest UF. ubrs U(u <> v}
17:
mental cycle basis over F2. Each new edge
18: for each root of UF do add (f(r), oo) to PD.
(u, v) is then handled in two cases:
19: return PD., PD,, B, UF
(1) If and are in different components, e a spanning-tree edge. We update the union-find and record u and v as neighbors in the spanning forest, thereby incrementally extending the spanning structure maintained during the filtration.
(2) If u and are already connected, e a cycle-creating edge. Together with the unique forest path u we v. this defines new cycle Ck+1. given existing basis (CI...., Cx). Since e does not appear in any earlier cycle, Ck+1 is linearly independent of (Ct,..., Ck} in the cycle space over Fz. The basis is thus extended (Ci,..., Ck2 Ck+1), 1. and new interval (f(e), oo) is added.
8

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We denote by PDy and PD1 the persistence diagrams of Hu and H1, respectively. Algorithm describes the forward stage of our framework. Its imput is a filtration function f V E - R a graph G (V, E), and its output consists of the persistence diagrams PD, and PD fundamental cycle basis B represented by indicator vectors in {0, 1)", and a union-find structure UF that maintains connected-componee
information and parent pointers for the spanning forest.
7.2 BACKWARD CONTRACTION WITH SUPERNODE BOOKKEEPING.
The backward stage uses a contraction func-
Algorithm BACKWARDCONTRACTION
tion g order contractions. We maintain supernode that accumulates contracted ver-
1: Input: contraction function gi persistence data
(PDg, PD,, B) and union-find UF from Alg.
tices; a vertex merges when scheduled by g.
2: Output: updated PDp, PD2 with finite deaths
and an edge contracts once both endpoints
3: Initialize supernode S 4- stack B, list L
reside in the supernode. Algorithm 2 sum-
4: for elements V U E in order of g(x) do
marizes the contraction process.
5: if E then
Node contraction
Vertex contractions. The contraction of vertex into the supernode has two effects: (1) If the vertex belongs to different con- nected component, the younger component 9:
6:
S <- S (x)
7:
if UF. find(x) o UF. (S) then // kill younger component y PD. + PD. U ((f(v)
8:
is killed, and its Ho interval is closed. (2) If the vertex lies in the same component, a new supernode cycle is created. Unlike forward
else
B. push(g(2))
// Birth supernode cycle
12:
mergeee,,
cycles, these are not tied to any edge but
13:
else
b Edge contraction
arise from merging disconnected subgraphs of the same component.
14:
=e== E with v E S
15:
remove from all B; reduce basis
some y becomes dependent then // Close forward cycle
Edge contractions. An edge is contracted
17:
once becomes a self-loop on the supern-
18:
PD + (birth(7),
ode. Each such contraction kills one cycle:
19:
B - B (7)
(1) If the edge participates in some forward
20:
else
cycle, we remove it from all cycle indicators
Close supernode cycle
in B and reduce the basis. a younger cy-
22:
T <- B. pop(): L LU ((.g(z))
cle becomes dependent on older ones, it is killed, and its H1 interval is closed. (2) If no forward cycle is removed, the contraction kills the most recent supernode cycle. In either case, the contraction assigns a finite death time to an H1 interval, completing the bookkeeping of backward updates.
23:
PD, + PD, UL
24: return PDo, PD1
7.3 EXPERIMENTAL SETUP AND RESULTS
Datasets. We evaluate on four standard graph classification datasets (Morris et al., 2020): NCT109, PROTEINS, IMDB-BINARY and NCI1 (Accuracy); graph-regression dataset ZINC (Dwivedi et al., 2023) (MAE) and the OGBG-MOLHIV (Hu et al., 2020) molecular property dataset (AUC). All methods use the same GIN (Xu et al., 2019) GCN (Kipf, 2016) backbone and the same DeepSets (Zaheer et al., 2017) pooling for encoding persistence diagrams.
Baselines and ablations. We compare five variants of persistence-based representations. (i) PH (Horn et al., 2021) uses a fixed vertex filtration and assigns each edge the maximum filtra- tion of its endpoints, producing standard inclusion-based persistence. (ii) RePHINE (Immonen et al., 2023) learns both vertex and edge filtrations and, importantly, augments every OD persistence point with the filtration value of the earliest incident edge, providing additional carly-connectivity information. (iii) Fwd-only learns general vertex-edge filtration but does not include RePHINE's augmentation, and disables all contractions; this isolates the effect of learning the forward filtra- tion alone. (iv) Bwd-only learns only the contraction schedule, isolating the backward component of our design. (v) Ours implements the full learnable inntusionncontractton (forward-backward) framework, jointly learning both vertex/edge filtrations and contraction order.
Observations. Table 1 reports performance across all six benchmarks. We observe: (1) Our method achieves the best or second- best performance in 9 of 12 settings, underscoring the value of jointly learning both inclusion and contraction schedules. (2) RePHINE remains a strong base- line and consistently outperforms standard PH in nearly all cases, reflecting the benefit of learning

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Published as a conference paper at ICLR 2026
Table 1: Comparison of PH variants across six datasets using GIN and GCN backbones. Classifica- tion accuracy and AUC scores are reported in percentage (%, 1) and ZINC regression evaluation in MAE (4). Best and second-bes best results per row are shown in bold and underline, respectively.
Dataset
PH
RePHINE
Fwd-only
Bwd-only
Ours
NCT109 (Acc.90 96, 1)
76. 4004
17
77.00=1.03
26.35=0.0
17.89
PROTEINS (Ace. %. f)
69.35
69.94
70.24
70.54
+
GIN
IMDB-B (Ace.90 96, 1)
68.67
70.67 +0.94
74.67 +0.47
-4.3
72.00+2.16
NCUI (Acc. 96, 1)
79.24
78.75
76.72+1113
75.75=0.44
81.27 +0000
ZINC (MAE, L)
0.43
0.41 +:001
0.62:10.01
0.61
0.40::001
MOLHIV (AUC%, 1)
74.34:::57
12.88
70.00
1..5
72.341074
NC1109 (Acc.%, 1)
76.59
79.50
71.91 +0:52
74.58=071
75.87
PROTEINS (Ace. 96, t)
70.54
68.75
69.35
1..5
72.32:1146
GCN
IMDB-B (Ace. 96, 1)
65.00
62.67:3.30 10.00
64.33
68.0012.16
NCU (Ace,9t 1)
78.43
78.91 1:0.0
75. 1.00
76.24
78.67
ZINC (MAF, .)
0.49=0002
0.46+
0.86
0.87=
0.44:
MOLHIV (AUC%; 1)
75.12
71.02:2.18 75.40
71.55=1221
76.37
both vertex and edge filtrations. (3) The ablations further clarify the role of each component: Fwd- only, which learns vertex-edge filtrations but disables contractions, typically improves over standard PH; Bwd-only, which learns only contraction order, also improves over PH on several datasets and performs roughly on par with Fwd-only overall, with no clear winner between them. (4) RePHINE significantly outperforms both Fwd-only and Bwd-only ZINC and MOLHIV, highlighting the effectiveness of its additional OD augmentation capturing early connectivity structure. (5) Our full model almost always surpasses both ablations, demonstrating that inclusion and contraction encode complementary structural information, and that jointly learning them is essential for achieving the strongest performance across classification, regression, and molecular prediction tasks.
7.4 COMPARISON WITH EXTENDED PERSISTENCE
We further compare our framework to methods based on extended persistence. which incorporate both sublevel and su- perlevel information. PersLay (Carriere et al. 2020) implements this by first com- puting extended-perriiseence diagrams and then applying learnale layer top. In contrast, motivated by Proposition 3. which shows that extended persistence has the same expressive power as (J-BB persistence, we integrate extended persis-
Table 2: Accuracy (%) with GIN and GCN backbones. Best score per row is in bold.
GIN
GCN
Dataset
ExtP
PersLay
Ours
ExtP
PersLay
Ours
NC1109
78.21
68.28
78.21
77.48
68.28
77.48
PROTEINS
74.11
66.07
73.21
72.32
66.07
72.32
IMDB-B
63.00
70.00
73.00
66.00
70.00
70.00
NCT1
78.59
68.86
81.51
80:05
68.86
80.78
tence directly into our aawaard-backward framework with -f as a backward schedule. In Table 2, observe: (1) Our (f- adaptation of extended persistence consistently outperforms PersLay under both GIN and GCN backbones, showing that embedding extended- extendedl-prrsistence structure di- rectly into the forward barwwrd-backward framework is more effective than processing extended- perddddteesstteecee model further exceeds both LersLay and the extended variant, indicating that jointly learning inclusion and contrac- tion provides expressive topological features beyond what extended persistence alone can capture.
diagrams through a post-hoe learnable layer. (2) Our full (J, (orwad-owrdddbkkkee
8 CONCLUSION
We formulate backward, fee---oowadbbbcwwwae and hourglass persistence and analyze their pressive powers, certified by minimal witness graphs and constructive proofs, accompanied by al- gorithms that realize the general (f.g) framework. These constructions extend simplicial and cellular complexes and admit a functional stability guarantee, offering principled tools for inclu- sion-come con-contraction schedules beyond the classical PH. While hourglass persistence satisfies combi- natorial stability, establishing a functional stability result is compelling direction for future work. Another interesting direction would be investigate the tradeoff between runtime and information of hourglass persistence.
10

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Published as a conference paper at ICLR 2026
ETHICS STATEMENT
This paper presents work whose goal to advance the interaction of topological descriptors with neural networks. Topological neural networks have many usage in the applied science, including medicine, finance, recommendation algorithms. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
REPRODUCIBILITY STATEMENT
We provide the code at https::///// /tgphtgttth.t////////////tt.hhht thing needed to reproduce our results: complete source code for methods in PyTorch and C++; scripts to download datasets and generate deterministic splits with fixed seeds; exact configuration files and pinned envi ronmertment...m yml with a README giving step-by-step commands run all experiments; config files specifying hyperparameters and training protocols (optimizer, learning rate, batch size, epochs, scheduler, early stopping, and seeds) and a Jupyter notebook to generate results table. The appendix contains the full statements and proofs of all theorems, and the paper includes extensive diagrams that illustrate the separation examples and clarify our theoretical claims.
Holll/Horrlasss containing every-
ACKNOWLEDGMENTS
This research was conducted while MJ and IR were participating during the 2025 Aalto Science In- stitute international summer research programme at Aalto University. We are grateful to the anony- mous program chair, area chairs and the reviewers for their constructive feedback and service VG acknowledges the Academy of Finland (grant 342077), Habb (grant 41 1025), and the Jane and Aatos Erkko Foundation (grant 7001703) for their support. We thank Amauri H. Souza for help- ful conversations on the code repository for Immonen et al. (2023). MJ would like to thank Mark Behrens and Erin Chambers for extensive helpful conversations and support. MJ would also like to thank Yuqin Kewang for helpful conversations.
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Bingxu Zhang, Changjun Fan, Shixuan Liu, Kuihua Huang, Xiang Zhao, Jincai Huang, and Zhong Liu. The expressive power of graph neural networks: A survey, IEEE Transactions Knowledge and Data Engineering, 37(3) 1474, March 2025. ISSN 2326- 3865. doi: 10. 2024 3523700. URL http: dow.d...i.ooooooooooooooo.o..... erg/ 10 1121 /TDEE 2 022 3523700.
Qi Zhao and Yusu Wang. Learning metrics for persistence-basel summaries and applications for graph classification. Curran Associates Inc., Red Hook, NY, USA, 2019.
Qi Zhao, Ze Ye, Chao Chen, and Yusu Wang. Persistence enhanced graph neural network. In Silvia Chiappa and Roberto Calandra (eds.), Proceedings of the Twenty Third International Con- ference Artificial Intelligence and Statistics, volume 108 of Proceedings of Machine Learning Research, pp. 1896-2906 PMLR, 26-28 Aug 2020. press/v10s zha html.
URL hctps: /ttp:://proceeee ngs mlr..
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Contraction and Hourglass Persistence for Learning on Graphs, Simplices, and Cells (Appendix)
A PROOFS FOR SECTION 3
In this section, we will provide the proofs for the results in Section 3.
Proof of Proposition 1. Let G be a path graph on 4 vertices colored in the order B, B, R. Let H be a star graph of 4 vertices such that the unique vertex with degree 3 has color B, and the other vertices have color B. R, R (see Figure 3()).. Consider filtration of G and by first spawning the induced subgraph on blue vertices, and then the induced subgraph on blue and red vertices. In other words, we have filtrations of the form
P and 0 c P2 H,
where P3 is the path graph filtrations are the same (see Figure 4 in Immonen et al. (2023)). However for backward-base PH, observe that G/ TC₁ (G) creates cycle but 1 does not create cycle. Thus, they can be told apart in the contraction stage.
vertices, both colored blue. The inclusion-based PH of both
For the second part, consider the following pair of graphs in Figure 3(b), equipped with a vertex- based filtration using degree. In this case, observe that the filtration of G (left) and H (right), respectively, is a sequence of discrete vertices until all the edges are spawned at the last time. This is because every edge connected
vertex of maximal degree equal for both G and H. Thus, the backward persistence for both G and H would be to quotient out the entire graph, which cannot distinguish G and H since they have the same number components and independent cycles. On the other hand, FB- 18-persistence can tell G' and H apart since in the filtration step spawning degree vertices, 3 vertices are spawned for G while vertices are spawned for H.
We first see how the TB-persistence diagram can recover the forward persis- tence diagrams and the backward persistence diagrams. Indeed, at n be the time the filtration com- pletes and appears at the last step of the filtration) are of the form (b, d) where d is possibly greater than n. From here, we can recover the persistence diagram in the forward filtration as by sending each pair
Proof of Theorem 1.
about to start contraction. The tuples that born before or on time (which
{ (b. d) d (b. 0c) ifdd>
(b, d), with S n +
where we note that if the pair (b, d) has death after time then it must have died in the contraction step, which means that the feature lived to co in the filtration step.
Similarly, we may obtain the backward persistence diagram by focusing on the persistent pairs (b, d) such that d n. We can recover them by the function
(0, d), i
(b, d) with > n
(b,d),
The reason why is because all the features that have not died yet the start of contraction in FB-PH are the same as the features that are born at the initial step of the contraction scheme in backward PH. Thus, the elder rule applies to the same features when we decide what pairs to kill off. The only difference that in backward PH, the features from the filtration steps all appeared at the same time, it does not matter which one to kill if we do have to kill them, but for FB persistence we would need to be more careful. Evidently, though, this gives a straightforward reduction of backward PH from FB PH.
To that FB PH is strictly more expressive than forward and backward PH. we consider the graphs G and H Figure 3(c) using degree as a vertex vertex-based filtration. Using the gudhi library (Project, 2025)'s persistence pairs computation,
directly apply it to the following code in Python:
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import guchi
VG [0, 1, 2, 3, 4, 5, 6, 71
EG [[0, 1), (1, 2), (1, 3) (2, 3), 13, 4), (4, 5), (4, 6), (4, 7) (6,
7)]
v_valuss [1, 3, 2, 4, 1, 2,
VH 10, 1, 2, 3, 4, 5, 6, 71
EE 100, 1), (1, 21, (2, 3), 12, 3), (3, 4), (3, 5), (5, 6) 15, 70, (6,
1)1
H_values [1, 3, 2, 4, 3, 2, 21
10
11 Compute PDs for G
12 stG guchi. SimplexTres() for in range (0, len (VG)):
14 current_v VGlil
15 v_val G_values[i]
16
stG.insert ([current_w), filtraticnev_val)
18
Adding edges
for range len ( 19
20
current_e [G[i]
e_val max valvel cuwrent_ee0111 C v_values |tw.auusssuurttteece.. |current_ell]]
stG. ins.rn (current_e. filtration=e_vet vall
23
stG. orC.m G_dgms stG.perrisstent (nin_persistenneeeeeeeeee persistence_dim_maxree print (G_dgms)
24
25
28
Compute PDs for 29 stH .uchi.SimplexTie. SimplexTTee()
30 for i in range (0, len (VH)
current_v VH[i]
32 v_val H_valuesli]
33
stH. insert ([current_v], filtrationev_val)
34
Adding edges
16
for
range (0, (kH)
37
current_e
EHlil
38
e v_val= = max (8 (H_ paalues/current Alalus [current_e e[1]])
values
stH. insert (current_e, filtraticn=e_val val)
41 stH sill_aakee__lcrraiooooooooeeeee
42 d_dgms stH. set..persistenee (min (min_persistenceee porsistenno_diimmxx-reee
43 print (H_dgms)
print (G_ dgms dgms)
The output would say that for graphs have the persistence diagrams of the form [(1, (3.0, inf)), (1, (4.0, inf)), (0, (1.0, inf)), (0, (1.0, 4.0)), (0, (2.0, 4.0)), (0, (2.0, 3.0)), (0, (2.0, 2.0)), (0, (3.0, 3.0)), (0, (3.0, 3.0)), (0, (4.0, 4.0))..
Now we show that they have the same backward persistence diagrams. At the initial time, they have the same number of connected components and cycles, there are no distinctions between G and H. In the first step, we contract the (C3(G') or ICs(H) (which is the closed star containing the unique vertex labeled with the value 4). This is connected subtree for both G and H, so the contraction does not produce non-trivial persistence pairs. Then we contract IC2(G)) (or 1C2(H)) which would kill cycle on both sides. Since the birth time of the cycles are the same, we just mark one the (0, -) to die at the time. But by the time we contract IC, (G) (or (C)(H), we will kill done after the last step.
the remaining cycle on both sides, which marks another one, and Thus, we see that they will have the same backward persistence diagram.
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Now to that G and H have different FB- TB-persistence, note that the birth time of the two cycles when contracting (G) (or (H) are different. For G, the cycle being contracted is born when the degree 3 vertices are spawned. For H, the cycle being contracted is born when the degree vertices are spawned. This would create a different persistent pair and hence FB-persistence can tell them apart.
Proof of Proposition 2. Clearly FB- TB-persistence is an example of hourglass persistence, so hourglass has at least as much expressivity as FB. Let G, be graphs constructed the NetworkX library, with filtration function being degree-based vertex-level filtrations,
nx. from fron_edgeelsst (C0,, 1), (1, 2), (2, 3) (1, 4) (4, 5)])
nx nx.fron_edgelist (1 1), (1, 2), (1, 3) (3, 4) (4, 5)])
See Figure 3(d) for visualization of the two graphs above. We wish to show they have the same FB-persistence but different hourglass persistence.
this case, we note that the two graphs would have the same FB-persistence diagram with respect to f. Indeed, at 1, all the vertices labeled I are spawned for both G and H. which gives three copies of the form (1, -). At f = 2, the induced subgraph on vertices labeled and appears. We note here that both G and H, by the elder's 's rule, the two vertices are both killed, thus both
diagrams have the form
(1,-), (1, and (2, 2).
Finally, at = 3, the vertex labeled 3 appears which makes both G and H connected at the step. The vertex labeled 3 dies at the same time, and two of the tuples labeled (1, also dies. This gives
(1. (1. (1,
(2, and (3.3).
Now enter the contraction steps. Indeed, first contract IC2(G) (or the corresponding version for H), evidently we are contracting isomorphic connected subtrees on G and H, so the process would neither create non-trivial cycles non-trivial steps. Then contract (C)(G) (or the corresponding version for H), for H this is subtree, so the contraction does not create new non-trivial tuples. For G. one might be tempted think that this is contracting disconnected
graph. However, since we contracted (C2(G) already, the two edges are actually connected here and forms a tree, so this also does not incur change.
Finally, at the end, we remark the unique remaining tuple to die at co.
(1,, (1, (1, (2. (2, and (3.3).
Remark: Note also that in the proof here we computed the persistence tuples in function time as opposed to combinatorial time (see Section 6 for discussion), but this does not matter terms of expressivity.
Finally, to see why hourglass persistence can tell them apart, we see that if we spawn IC₁ (G) (resp- IC, (H)) first in the sequence, then they would incur two connected components for but only one for H, so hourglass persistence can tell them apart.
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B PROOFS FOR SECTION 4
We split the discussion for the proofs for Section 4 into three parts:
1. In Appendix B.1, we establish a lemma that will be helpful in streamlining the proof of Proposition 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 graph filtration. We will also introduce some concepts helpful in understanding the proof of Proposition 6.
2. In Appendix B.2, we prove Proposition 3, Proposition 5, and Proposition 6 (i.e., the pressivity parts of the section).
3. In Appendix B.3, we will explain the two algorithms for Proposition 4 and prove it.
B. KEY LEMMA AND THE USUAL PERSISTENT PAIRS INTERPRETATION FOR GRAPH FILTRATIONS
Before we give a proof of Proposition 6, we first note that the description of death times for con- nected components holds Proposition 7 would use, so we might as well extract it out as a formal lemma.
elevated generality that both the proofs of Proposition 6 and (later)
For completeness, we will also give self-contained proof of how Definition 3 recovers the usual way to compute persistent pairs for graph filtrations (for instance, see Section 4 of Horn et al. (2021) Algorithm 1 of Immonen al. (2023)).
The lemma is as follows.
XN) be a sequence of inclusions and contractions of graphs (i.e., the set-up of hourglass persistence). Note that by necessity the first step X-1 + Xo has to include in graph. The persistence pairs' for Ho(X.) can be computed algo- rithmically as follows the method of which we called "keeping track of one vertex representative per component":
Lemma 1. Let X. -
1. In the filtration step X-1 Xo, we pick vertex per connected component in Xo and mark tuple (0.-) corresponding to that. We fix one of these vertices to be called the supernode (denoted *), the sense that any time we compare vertex y with * to decide which one to kill off. we always kill off v.
2. For i 2 0, if the step X; + Xi+1 is a filtration, then one treats this in algorithmic time as procedure to spawn every new vertex that appeurs 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 vertices and w, mark the vertex born later to die at time i and pick u as the representative of the new component. (Since we do not focus on trivial deaths,
discard them after this step). the two vertices born at the same time, we randomly pick I to kill off unless one of them is the supernode *, in which case we always kill off the other vertex.
3. For i 2 O, if the step X; + Xit1 is contraction, say contracting a subgraph G. We loop through each connected component X, of G. we find the vertex Vj representing the component X₁ belongs in and kill v, unless (1) it got killed already in some index j <j when looping through the X; or (2) it is the supernode *.
4. After i == N.r for every tuple not dead yet, we mark it to die at time co.
Remark 2. Note that the statement of Lemma I does not require XN to terminate with only the a special case in this lemma.
point x left, so the case of hourglass persistence in Definition A more specific case of Lemma I is when the maps all inclusions. This will recover the usual procedure to compute the O-dimensional PH of a graph filtration (as we just ignore Step 3).
Proof of Lemma Indeed let recall from Remark how one way compute persistence dia- grams are done. Let us interpret how the remark tells us how to compute the quantities here explic- itly:
Recall here we accounting for trivial deaths
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1. We pick non-zero vector v of (Xx)),, and consider the sequence of linear subspaces generated by the iterated image of in (X.) until it becomes O.
2. Then we remove this sequence of linear subspaces off of (..)). there is still non-zero vector w in (xo), we pick w and repeat the process (if the linear map sends w outside. Otherwise, we choose a non-zero vector from what is left in H& (X2) (if it exists) and look at its iterated images ahead, and so on.
Let us also recall the following standard fact in algebraic topology suppose (say) cellular complexes, then the induced map f. : (A; + Ho(B; exactly as follows- recall Ho(A; 2/2) and Ho (B; are respectively isomorphic to the direct sum of copies over 2/2 indexed over their path-connected components. For each path component A, let LA, be the unique non-zero vector representing that component. The continuous image of path-connected component is path-connected, so s(A,,) lies in path-connected component of B, say By, then the map described above sends f- (1A.) 1B,-
-upposs f A + B is a map of can be described
Remark 3. This description might deceptively suggest that f. is not the zero map on all of (o(AA but this is clearly not the case, it just happens that we chose a conveniens basis on both sides such that f. is not zero the chosen basis vectors. The subtlety in the persistence calcu- lation is that we want to choose the basis in the Ho(-) of the next space consistently with the basis of the previous one to nicely compute the persistence pairs.
By the remark in the line above, though, we see that follows that there exists a non-zero vector en E Ho (Xo) (as long as Xo not empty, by requirement) such that the successive images of are never 0 when passing through Ho(X)), thanks to the convenient choice of basis, and can in fact come from representing some connected component. We then choose v here to represent the supernode *. In the next step of the remark, we then remove the subspace generated by successive images of * off Ho(Xo), Xo), and if the next vector we pick lands this subspace, we consider it dead. Note that on the level of Ho, vector 1 E Ho representing component could land in this subspace (X(++) if and only if (C) belongs in the component of the supernode at time i+ 1. Thus, we see that asking vector representing Ic to die if it enters this subspace corresponds exactly to the property of the supernode.
Now we proceed by induction and suppose the i-th vector U; we pick following the remark (1) represents a component, (2) gives the correct persistence pair according to the outline of this lemma, and (3) has been picked according to the rules of Remark 1. Now we wish to show we can pick the i + 1-th vector 0++1 such that they still satisfy (1) and (2). Now we would stop if we have out of vectors to pick, so there is at least some non-zero vector in the complement of the subspace generated by vectors U1, m., i2 and their iterated images (call this W, for convenience). So there exists some vector U1+1 that represents a component C' and say U,+1 E Ho(()).
By assumption, 2411 cannot be in the image of any previous vector, so the component C' must be a new component that appears from X,-1 + Xy (by the deseription of what the correspondent map in Ho is like). Now we have two descriptions of how C gets killed:
1. C' is killed in the sense of this lemma if and only if when (a) it merges with an earlier component during filtration or (b) subgraph of C' was asked to be contracted in the future.
2. Remark says that C' is killed if and only if the iterated image of lc gets landed in W..
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 on Ho(-) behaves, we see that l can land into W; at time j' > j if and only if there is some 0 < k < such that f o ... (C) and f ... o fy (Cx) represents the same connected component, where C2 is the component component uk represents at the time it was born.
Without loss, in W2 and le to be the first 2/ whose component C' merges with. Then it follows that in the step fy-1: : Xy-1 Xz,, C dies by mergining into fy-1 o ff(C)). If f--1 -1 is filtration, this can
choose j' to be the first time step such that (ffy-1) o (f))) is
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only happen if some edge is spawned between them. If fy-1 is a contraction, this only happen = 0 (i.e., it goes into the supernode). Thus, we see that these are exactly the two conditions proposed by the scenario in the lemma. Furthermore, the birth and death times proposed by the lemma and Remark agree.
We then induct repeatedly and conclude the proof.
As noted earlier, Lemma recovers the usual algorithm to compute persistence pairs for graph filtrations on the Oth dimension. We now explain how to obtain the one for the 1st dimension from Definition This will follow from the following more general fact.
Lemma 2. Let X be (fiitee) cell complex of dimension D and X. = X-1 Xo + X, = X be filtration of X by cellular sub-complexes, then the persistence pairs of H (X may be computed as follows:
1. Art 0, we instantiate dim HD(Xo) many tuples of the form (0, o0).
2. For f. O, we instantii dim Hp(X,) dim HD(Xt-) many tuples of the form (1, oc).
3. We end t n.
We first look how Lemma 2 specializes for graphs, before proving it. When D = 1, 1- dimensional cell complex is the same as graph, and the lemma recovers the usual way to calculate 1 1-dimensional PH's.
Indeed, H-(-) of graph corresponds exactly its list of independent cycles. For completeness, by "independent cycles", we mean the following interpretation:
Definition 12. Let C, C" be two cycles of the graph G. The C C" as the XOR of E(C) and E(C") (i.e., their symmetric difference). For a list of cycles Di, .i... Dn. the XOR span of the list is the following set:
(Di, X D; R... Di 18, i2,..., ik S. {1,...,n]}}
Let C1 C2..... C2 be list of cycles G. The list is list of independent cycles if for any Cr C2 is not contained in the XOR span of Cry..., Ci-1,, Cit1, C
Definition 13.
The following is well-known interpretation of Hi(-) of graph and can be interpreted as the definition.
Fact: The maximal list of independent cycles G forms basis for HG))).
The usual algorithm for 1st dimensional persistence diagrams of graph filtration keeps track of vertex representatives for the components, and they mark the birth-time of cycles of the form (1, oo) filtration time for each edge drawn from a component itself at time Proposition 8. Lemma 2 recovers the usual algorithm described above.
Proof. Let us start from the algorithm side and work to Definition 3. Indeed, each (t, oo) corre- sponds to an edge that is born in G, (the subgraph at time t). Choose Ce to be cycle in Ge containing the edge we choose this for every such edge arising above.
By Lemma 2, it suffices for us to check that for each time step Т, HG,),) has basis being CEE where the collection of cycle-creating edges born in time S T. We first check that this is linearly independent. Indeed, appeared earlier than 62 in the algorithm, (even if they are born at the same filtration time, there is an ordering of which one is born first in algorithmic time). Under this total ordering, we rewrite the elements of into el, e
choose total ordering S E' such that e eg if e
Clearly the list {Car is independent, since C., is a non-trivial cycle. Suppose by induction (Cay..., Cot is independent, wish to show adding Case into the list remains independent.
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Indeed, the XOR span of any sublist of [Cel, C,t) } is always contained in the union of edges of Cali C.+'s 's, which does not contain the edge e because of the total ordering we picked. Thus, C,est, which contains e++1 by construction, is not in the XOR span. Thus, we have verified this is a list of independent cycles.
Now, to show that this is a maximal list. We observe that removing the set E from G, will turn Ge into a forest. This implies that there are no additional eycles left, which concludes the proof.
Now we will prove Lemma 2.
Proof of Lemma 2. This lemma follows from the fact that the induced map Hx,)) + (X(+)))) by inclusion has to be injective for all Indeed, this comes from the long exact sequence in ho- mology for the pair (X++1, X₂) (see Theorem 2.16 of Hatcher (2002)), since HD+1(X++1 (X++1,X, X₂) is evidently zero as X1+1 and Xt are both most D-dimensional. Since this is injective, Remark tells us that a chosen component according to the steps of the remark can never merge into pre- existing component, so all the vectors picked survive to oc. Once we move from X; to X141, the new pairs that are created are exactly dim H DD((X+) dim HD(X,) many pairs of the form (t+ 00). This concludes the proof.
B.2 PROOFS FOR THE EXPRESSIVITY PARTS OF SECTION 4
Proof of Proposition 3. As defined in Section of Yan et al. (2022), the extended persistence of filtration function ff is equivalent to the persistent pairs associated to the persistence module:
0= H- +..-. .G.). - H(G) H(G, GT)) + +... G*) + H(G,G-
Here, G. means {I G f(x) S a} and G" means fo E G f(z) a}. and И denotes either Ho or Hy. Here we note that the paper Yan et al. (2022) wrote
H(G (9),, but some prefer the convention that H(-) of the empty set is 0, opposed to the empty set. One motivating reason that the empty set is not a vector space. This does not affect the persistence pairs produced since they start at the non-zero parts.
Although a is indexed over the entire extended real numbers [-o0, +ox], since ʃ takes only finitely many values (or, if f is real valued function taken graph G, viewed as non-discrete topolog- ical space, the topological changes only occur when a vertex is spawned), the persistence module reduces to finite length persistence module of the form
0 + (Ga)) + + H(G..,) = H(G) + (F +... + HH, G
where ag ... < an the sequence of filtration values of f. By Proposition 2.22 of Hatcher (2002), there is morphism of persistence modules of the form:
H(Goo)
(Ga.))
H Gan)
H(G, Gov)
0 (Gao))
H(Ga..)
HG//Go))
HGG/G
where the second row is analogous to the construction of (f.g) persistence we did. Observe we can rewrite Go as (ar E G f(x) < -a;). By and the successive union the intermediate complexes arising in the definition of FF persistence up to step is exactly the same as G"i, so the persistence module of the second row arises exactly from the (f, FB persistence.
Proposition 2.22 of Hatcher (2002) also implies that the morphism above is an isomorphism of persistence modules if H is H, (-), so they will have the same I-dimensional persistent diagrams.
If H is Ho(),, then the maps (G, HG//G*) injects onto direct summand (G/G%)) of (T(G/G")) such that H (G/G%) HGG/ On the other hand, there is very explicit interpretation on what the generator of the Z/2 summand is it is exactly the image of the chosen supernode (see Lemma 1). The difference here is that the supernode in the second row becomes the
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only vertex feature that does not die, of the form (0, oo). and the supernode in the first row becomes the vertex feature of the form (0, d) where achieves the maximum death time among all vertices of the graph (concretely, it is the first i (from to 0) such that G/G is a connected graph). By the decision rule we imposed on the supernode, we see that the first and second row agree on all O- dimensional persistence pairs except for the supernode, but clearly we can invert from one another.
This concludes the discussion that they have the same expressivity.
Proof of Proposition 5. We show that the they have the same extended persistence but different FB persistence.
Indeed, consider the pair of graphs G and H from Figure with the filtration function being the vertex-based filtration function induced by the height function on the vertices. For convenience, we also redraw the two graphs below as:
y = 3
y = 2
==1 1
G
H
In the forward time, we observe that the respective filtrations for G and H only change twice as 0 = G-1 C Go C Gi = G and 0 = H-1 C Ho C H1 = H, where Go and Ho are identical. The same number of vertex births/deaths and cycle births occur at y 3, and hence G and H cannot be told apart in forward time. If we apply the backward contraction with respect to -f (i.e., extended persistence), then y 3, the subgraph being contracted the same, and they both only kill 1 connected component. No changes happen until we get back to y 0, but by then the entire remaining graphs are contracted, and no differences are detected. Thus, we conclude that forward with respect to f backward with respect to -f cannot tell apart G and H.
On the other hand, we observe that F'B-persistence can clearly tell them apart. This is because IC₂ (G) has a cycle and (H) does not, so contracting G by (GI(G) kills a cycle but contracting H by (C)(H) does not.
Proof of Proposition 6. From a similar proof to that of Theorem 1, we know that L-FB has at least the same expressivity as forward PH with f. Note that the explicit description of how to compute (f. F-FB persistence means that it is actually a function of the filtraiton function f. so the explicit description holds then forward PH with f is least as expressive as f) persistence, which will show they have the same expressive power.
Thus, it suffices for us to verify this explicit description. The case for connected components is resolved by Lemma already, so we will examine how to compute the case of cycles.
By a similar procedure to the explaination in Theorem (essentially due to elder's rule), the cycles that appear the contraction steps for J)FBB persistence are exactly the cycles (that are not born at the beginning) in the backward persistence diagram that contracts the intermediate complexes in the order (C, f), (G, ..., (G, Note that a cycle can be born in the contraction stage if and only if we are being asked to contract two components that belong the same connected component of the entire graph G (note that, without loss in algorithmic time, we can always contract two components
time).
Since the intermediate complexes being contracted are in the same order they appear in the filtration, that the birth of cycles in the contraction steps corresponds exactly the appearance of
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connected components in filtraiton that will be merged to connected components that are born earlier. The death time of such cycle also corresponds to when the two components actually merge. Perhaps one way to see this is to note that in the simplicial quotient interpretation (see Section 5), this fills in a list of triangles to the disjoint base point 84 on top of a path connects the two vertex representatives. Prior to the filling, the two vertex representatives would each have edge e+ (which is a cycle since they are in the same component for the entire graph G). The filling of the triangles here would kill the cycle. This proves the case for cycles born in contraction.
Finally, for the eycles born in filtration, we would like to predict their death time in contraction. Recall from the proof of Proposition 8 that there is an explicit description of the compatible basis of Hz where G denotes the sequence of inclusions of subgraphs induced by the filtration function f. The basis of H1 (G) correspond to cycle-creating edges, and an explicit ordered basis of independent cycles C₁ < ... < CN can be found by choosing cycles C2 that contain the cycle-creating edges e; in inductive way (see the proof of Proposition 8 for more details). The upshot is that this would give a linearly independent list of cycles because the cycle-creating edge C contains is not contained in Cy for i.
In the contraction step, observe that necessary condition for the list [CI...., C.. degenerate (i.e., become linearly dependent) is when some cycle creating edge e; is contracted. Furthermore, e, is the last edge in the cycle C to be contracted, the cycle C: would die right there. The diffcult with general (f. - persistence is that, due to the arbitrariness of g. often times the cycle creating edge e; not the last edge being contracted in C, In the case of (f, f) persistence, however, we observe that we can always choose to die last in algorithmic time. Thus, this shows that each cycle C2 born at time dies exactly time + (when the same cycle is being asked to be contracted with respect to f).
B.3 ALGORITHM TO COMPUTE ( FORRFF-AOOARDBAKKKKAA. FILTRATIONS
In this section, we will prove Proposition 4 and explain the algorithms behind in the case when f is vertex-based. We also remark that similar algorithm exists in the arbitrary case when f is not vertex- based, just one also has to carefully permute the edge based values. We chose to present the case when vertex-based for simplicity.
Before we start explaining the algorithm, we first explain why (f1 exists in general. Indeed, this will follow immediately from the following more general fact.
Lemma 3. Let G be graph and G. (0 G- C Go C G2 ... G.,) be strict inclusion of subgraphs on G, then there exists a filtration function f G + R with filtratio values ag < < Oxz
such that f (--((--000) a:l) = Ga.-
Proof. For each vertex or edge x E G. we define fa) be minge (0.1 (o. I Gj. Since G. an inclusion of subgraphs, an edge cannot appear earlier than the vertices that support it, so it follows that f is filtration function. This concludes the proof.
From now on in this subsection, assume that f is vertex-based for simplicity. Let first look at the algorithm for FB-persistence specifically and see why it is only linear time.
Proposition 9. Let f be a filtration function, then there exists filtration function f such that the sequence of topological maps in (f, ( respect to f. Furthermore, Algorithm 3 computes Io in with a runtime of O(V + GG).
the same as the ones in FB-persistence with
Proof. To show that Algorithm 3 correctly computes f it suffices for us to show that IC; fa) is exactly 1Cn-; f). Indeed, the intermediate complexes produced for the pair (G, f) are "upward closed" in the sense that the edges of (C;(G: f) all have the same value a; but the value of vertices are in general only S a,
If we want IC;(G; f) to be the first intermediate complex that appears in the filtration f we would want to relabel all of its vertices the maximal value an, which is precisely what the algorithm
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Algorithm 3 Computing the Backward Filtration Function
Input + Output: The graph G and filtration function f The function f
1: vertex <- [v -OG for each E V(G))
IInitialize f on vertices with values in -c.
2: for edge = (v, w) in (G) do
3: Pvertex [u] max(f(e), (eettee))) fvertex(u]).
4: for vertex e in ()) do
5:
if I Peggexl
00 then
6: Svertex 00) - f(v)
Mark isolated vertices to their value under f-
7:
(fvertex: Tedge) t (map(æ + -; fvertex); map(æ + =x; fedge)).
8: return (fvertex) fedge)
Since ʃ is vertex-based, we only need to do this on vertices and we can modify the edges later.
does.
If want IC; f) to be the second intermediate complex that appears in the filtration fo, would want all vertices that have not been marked on already to be marked 0n-1- This amounts to a maximality comparison in the for loop in the algorithm.
If we want (G; f) be the third that appears, we similarly want to label all vertices not marked ans 1 yet to be marked as such, so repeating this process yields the correctness of the algorithm.
The algorithm has runtime O([V] [GI) since only requires looping through the vertex set and the edge set linearly.
Now we move on to Proposition 4. In fact, Algorithm before is special case of the algorithms for this. In order create backward-filtrtioon function that contracts in permutation the intended order specified by a permutation we observe that Algorithm 3 would actually hold if the max function is operated with respect to an "ordering given by o" as opposed to the natural ordering of the reals.
Instead of changing the ordering system of the reals in our algorithm, we will instead change the filtration function f inside the parameter of the max function to this ordering. To do this, we define the following variant f:
Definition 14. Let f G + be a filtrationn function with filtration values a1 a2 < ... < an Let a be permutation of the list {1,..., n.., we define as the function
a. ʃ G + R e((x)) (o(s) if f(x) Oii
where x is either a vertex or an edge.
Algorithm Computing the 7-Bakwward Filtration Function
Input + Output: The graph G, function f. permutation - + The function fr
1: g < (reor-1)
2: Viertex +- [v =00: for each E V(G))
& Initialize f on vertices with values in -o.
3: for edge = (v, w) in (G) do
4: Feettex [v] +- max(g(e), (Tetexlul))
5: for vertex in (G) do
6: if F [v] = 0o then
7:
Svertex 0) < g(v)
Mark isolated vertices to their value under f.
8: FTertex + map(2 + -I; fTertex).
9: fédge + -mppp1 I + -x; gedge):
> Gedge is the function g on E(G).
return (frortexs fedge)-
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In practice, permutation can be realized as a dictionary with entry being and output being (i). We can use this to make an associated dictionary whose entry a, and output is az(i)- Depending on how the filtration function is represented list of filtration values the values are not sorted already. We can then produce a f by a linear sean through the entries of and swap out its output using the associated dictionary.
data structure, this may require sorting of the
slight modification of Algorithm now produces Algorithm 4. In the special when T re (the reverse list permutation), this will recover Algorithm 3. This gives f2 as requested in Propo- sition 4, but note that plugging into т also gives the desired fi in the proposition. A slight modification of the proof for Algorithm will ensure the correctness of the algorithm here. Because we possibly may need to sort list, this would incur a worst-case runtime of O(N log N), where N = (V))) + [(
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C PROOFS FOR SECTION 5
Proof of Proposition 7. Let Y be a simplicial complex, and let Y, be a sequence of inclusions and contractions in some arbitrary order, of the form:
0 Y - Yo
Ym Ym+ = *.
Using the simplicial quotient method, we adjoin a disjoint base point et to and also form a sequence of inclusions Z. of the form
4 Zo - 71 + - 7m - Zm+1-
Taking (-) gives the sequence
Hz 0 (Z) - (21) +... Hz( - Hx((mm+)).
Fix l 0, we also let (Y.) be the k-th persistent module for this from Definition 3.
Now for each Z, let C, be the closed star of the vertex e4 (i.e., the union of all simplices containing V4 and their faces). Observe that Y; is isomorphic to the quotient Z:/C;- We also note that C, is actually contractible- indeed, in the standard geometric realization of Zz, there is an explicit straight line homotopy based on line segments from the simplicial link of 4 to e4 from construction, which shows that Cj's 's are contractible.
a general fact that for a simplicial complex K and contractible subcomplex K", the quotient map K + K/K' is homotopy equivalence (see Proposition 0.17 of Hatcher (2002)).
(Z))
(z))
H (m))
H& (i(m+m))
1
0
HA Zo/Coo
(z(Z1/C1)
H (Zm/ Cm)
HA (1+})) =
!
=
+
(Y,)
(Y1)
H (Ym)
Hx({)))=
Here Hz(2m+i) 0 because Cm+1 Zm++1 in this case. Thus, we have an isomorphism of persistence modules between Hz and HY.)),,, so they have the same persistence diagrams.
To see what goes wrong in the case lo = O, we observe that the placement of the node 84 can affect the birth / death time of vertices since v4 is now the oldest node, so the birth times of the vertices would shift. This can be remedied, for FBB-ersistence or (a, T. -perssssttece where we instead place "4 spawn after filtration finishes and before contraction begins. However, this does not work for hourglass persistence, since we can contract before all the filtration finishes.
Thus, would like consider a different method, as described in the proposition. For simplicial complex K, we write (K)² to denote the I-skeleton of the simplicial complex K (i.e., the union of all vertices and edges). On the level of I O, observe that the inclusion of the 1-skeleton of X induces a map of persistence modules of the form:
0 Ho(()))²)
- Ho((())²)
Ho((Ym)))
Ho((+))
0
(Y)))
(Y)
Ho (Y.))
Ho({+))
Here each verticial arrow is an isomorphism because the inclusion map (Y,) Y, is a bijection on connected components. Thus, we have an isomorphism of persistence modules and hence the top and bottom row have the same persistence diagrams. What this means is that O-th dimensional persistent homology for the simplicial complex is equivalent to the 0-th dimensional persistent homology for its restriction to the I-skeleton. Thus, this reduces to the scenario in Lemma 1, and this concludes the proof.
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D PROOFS FOR SECTION 6
Proof of Theorem 2. We split the proof of stability into two parts the case where k > 0 and the case where 0.
Let X be simplicial complex and Hx. (, and (x. g') be the two persistence modules in (J.g)-BB persistence and g'', persistence respectively.
For k > 0, and when X is simplicial complex, we use the simplicial quotient interpretation of the persistence modules here (in the sense of Section 5). By Proposition 7, we see that the persistence pairs associated to Hx f. is the same as the persistence pairs associated to the persistence homology of a simplicial filtrations of a simplicial complex Z. which we will write the filtration function as h. Similarly, for Hx.,,, f'. g'), we will get (possibly) different simplicial filtration of the same simplicial complex Z, which we will write the filtration function as h'.
Thus, we see that d H (X., f,g), (x( g') d (Z. h), H& (..Z,h' h')) is the bottle- neck distance of the persistence diagrams coming from two filtration functions and h.. By the classic bottleneck stability of Cohen-Steiner al. (2006), we have that
dB(Hx(Z. h), (Z., h')) S |/h- h'lloc.
Let us unwrap the construction of I and h here. Indeed, recall Z is the simplicial cone of X. For (f.g). the function h agrees with f when restricted to the base of the simplicial cone (which is copy of X). Every other simplex must contain the disjoint basepoint v4, and is obtained by joining 2+ to a base simplex in X. We write all such simplices as (o, 84). In this case, h(a,, e+) is defined to be max(f) (()) Finally, one also specifies that (v++)) to be born before all the other persistence
values, say 0 for uniformity (this does not affect the persistence pairs of dimension higher than 0). There is similar description for h', and hence we see that
||/ - h'llso = max(I1f - f' 1lg g' + (max(f)) max(f'))ll
$ Ilf - fl + llg g" + (max(f) - max max(f'))
< ||ʃ - ~l + Ilg g mma + max(X/) max(
Remark: Note that there is no coefficient 2 here. For brevity, in the main paper, we added a 2 because that is the bound we will get for k
When X is a (regular) cell complex, we evidently can still take the cone of cell complex, which has cell decomposition according to the subcomplex of the base. The arguments above would still hold in this case. This is because the original Bottleneck stability Cohen-Steiner et al. (2006) was proven for all triangulable spaces and tame functions on them. Although not every cell complex is trianguable, it turns out every regular cell complex is trianguable (see Theorem 3.4.1 of Fritsch & Piccinini (1990)). The filtration is clearly still tame on the triangulation as the proof constructs the triangulation one skeleton at time. It follows that the bound above applies through.
The case of = 0 is different because the persistence tuples differ. First of all, by the same argument in the proof of Proposition 7, we can reduce this to the case where X is graph. We will still work in the simplicial quotient perspective. The reader might wonder there are some graphs that are technically not simplicial complexes (i.e., has self-loops or multiple edges), does the simplicial quotient also work for them? The answer is yes because for such graphs, one can always discretize further by labeling the mid-point of exceptional edges (i.e., self-loops and multi-edges) as a vertex of the graph. This then becomes simplicial complex and makes no difference in filtration since we will be spawning the midpoints at the same time as the edges, so we can ignore these trivial deaths.
Thus, we without loss reduce to the case where X is simplicial graph, and we can try to apply a variant of the simplicial quotient method. Indeed, we observe that, if we move the adjoined formal basepoint 04 to be spawned after we have filtrated the entire graph X first using f and before we start the contraction using g. then the 0-th dimensional persistence diagram with the pair representing v4 removed is the 0-th dimensional persistence diagram from the 0-th dimensional (f. (l.g))-een persistence diagram. Indeed, this is because, U4. being the last vertex spawned, will die immediately when the first vertex in g appears and asks to be contracted (which will not change the other outputs).
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To go further, (2024)) and adapt it to our scenario. Indeed, let Z be the simplicial cone of X, and h and h' be the filtrations corresponding to the pairs (f.g) and (f ".g).. But also, because k = 0, we can again restrict to the I-skeleton Z' of Z, which is now graph.
proof strategy of bottleneck stability (ex. Skraba & Turner (2021), Schnider
working with
For each simplex (either an edge or a vertex) x E Z', we consider a linear interpolation function he(v) (1 t)h(ar) + (x:) with values in 1]. We can divide the interval [0. 1] into finite collection of intervals (4, 41 [1,, fel,.., [b,, In+1] with l0 0 and bx+1 = 1 such that, for every pair of simplices and y and [ti, ts+1), we either have that (x(x) he(y) for all t or hx(z) & hy) for all t.
Since the relative ordering of simplices are not changed in this interval, the O-dimensional persis- tence diagrams for hr, and h44 are the same in combinatorial time. Now choose any bijection of the function time persistence diagrams for PHБ h4,)) and PH (Z'; (ag (by which we denote their forward time 0-diemnsional persistence diagrams) such that x(b,, = (b++1, de+1) if and only if both tuples represent the same tuple in combinatorial time. Note that due to the placement of 0t, this will necessarily pair the two pairs associated to 4+ together (call them (b+, d4) and (V,,d", Notably the restriction π of ı to the tuples excluding the one corresponding to u4 is still bijection.
For any two filtrations d, of Z' (the 1-skeleton of the cone of X). We write
d'g (PH) (Z'; o), PH (Z'; h+;r)) )) as the term
inf bijections PHE (Z (b- (odd)))rngggwe (M [1(b, d) sp(b, d)lloo
Observe that when @ = and eb = It, one has that
FB B
Ho (X, Ho(X, l'g'))) = d', (PH, (Z';h) hZ); PH (Z):
Now d'g can fail the non-degeneracy condition for being metric, but it will still have triangle inequality!
Now each tuple in the persistence diagram here actually comes from a vertex-edge pair (v, e), where e makes birth to the tuple and e kills it. Choosing Ti as the bijection here, we have
(PH. (Z'; he,), PH (Z': hear))
< [l(b, d) - TI (b, (b.d)l
< max [1(he, (v), (v), he,(e)) (heets (v), heem (e) (v.e)eX
Definition of गi and associating pairs
max oEX
Il, (v) 1e, (v) + max [|,,, (e) her (e)ll
eEX
I (f++1 - ta)llf - f' + max -
[/,, (c) - her (e)llo
xEX
For the second term, there are two possibilities for the edge either it comes from the filtration or it comes from the contraction. Thus, we have that
max (Mr.(e) hemm (e)llo S maxx((4a) - -411S5 1' x
(t++1 - t4)ll( - g') (max(f) max max(f')lloo)
< (1.+1 (4( 5'1bo 11(g g')l mmxx(/) (fal(
into the sum En d' (PH. (Z': hv,), PH (Z'; hy,..)) and using the bound above, we have that
Now
decomposing
(PHH (Z': h), PH. (Z':k()
d BH f.g) Ho N.f f. f''')) 2 - f'l + 11(e o')llos + max(ff max(fa))
FB
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E MORE DISCUSSIONS ON COMPARISON WITH OTHER METHODS
E. ZIGZAG FILTRATION
In the definition of (.gg)=FB-persistenee the filtration function g is specifying the order of subgraphs being contracted. This is different from the use case of zigzag filtration Carlsson & de Silva (2010), where they are looking at a sequence of insertions and removals of graph, but the removal process is fundamentally not continuous, their persistence diagrams do not follow linear time (whereas our set-up does).
To give an example, consider G the path graph on 2 vertices, H1 the emptyset, H2 sub- graph of G which discrete with vertices. Then the following diagram that occur in zigzag filtration
H1 C CG3 H2
Observe that there is no map from to H2 that continuous, since we cannot split G into components. In general, the arrows in zigzag filtration go in different directions, and they are all inclusion maps. the other hand, the class of diagrams we always consider (f.g)-IB persistence is sequence of continuous maps
G G3 +
which are inclusions followed by contractions. Here the arrows all go in the same direction, but they can be either inclusions or contractions (this even more apparent in the setting of hourglass persistence).
More technically, zigzag filtration is considering quiver spaces the path graph Pn with any possible orientations but with inclusions, whereas we are considering quiver spaces on P. oriented in uniform direction with both inclusions and contractions. Generally, we expect zigzag persistence and 9FB persistence to be incomparable, and their methods can complement each other.
E.2 BIPATH FILTRATION
In Aoki al. (2025), the authors introduced bipath persistence built bipath filtrations, which quiver spaces on the Hasse quiver B,,m. which is the form
Similar to the case of zigzag filtration, the filtrations that appear in bipath filtration on Page of Aoki et al. (2025) is again using inclusions only, whereas our settings consider the usage of contractions along with inclusions. Furthermore, bipath filtration considers quiver spaces on the Hasse quiver Br with inclusions, but we study quiver spaces on P, oriented in uniform direction with both inclusions and contractions. Therefore, we expect that the two methods general do not subsume one another but are complementary ideas.
E.3 EFFICIENCY COMPARISON
For persistence diagrams (PDs) of dimension > 0 on simplicial complexes, hourglass persistence itself can be computed using the "simplicial quotient" trick mentioned in Section of the paper using Proposition 7. Proposition 4 reduces the question to computing inclusion-based PH on simplicial complex, which has a standard runtime dominated by the runtime to perform matrix reduction algorithms (see Otter al. (2017)). For PDs of dimension 0, this looking at PDs of just underlying I-skeleton (i.c., a graph). In this case, there is a way to optimize the calculation with union-find with runtime dominated by On log n m)) where is the number
be reduced to
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of edges and m is the number of vertices (see Horn al. (2021) for more details). Although Horn et al. (2021) only discuss the runtime in the filtration steps, tracking vertex representatives in the contraction steps would also have the same runtime.
When focusing FB- FB-persistence and (f:g) (Bgg)-FB-i-) TTB--ersistence, though, expect the algo- rithms presented in Section 7 (which are not using the "simplicial quotient" trick) to be faster empirically. In general, the ability reduce the size of object using contractions (ie. using the cellular complex extension in Section 5 instead) would lead to decrease in the memory complexity. We therefore expect the cellular-base based methods of hourglass persistence, (f.g)- FB persistence, etc. to have more efficient algorithms in the future.
E.4 SCALING ON LARGE GRAPIIS
Interleaving the filtration and contraction steps avoids the need to filtrate the entire graph before starting contractions, which allows the total size of the graph that appears in its entire lifetime to be bounded.
This can improve over the runtime on general simplicial complexes. Let K be simplicial complex with total simplicies. To give a heuristic/informal estimation from practitioner perspective the typical runtime of inclusion-based PH on K scales approximately O(z²) as it requires matrix reduction algorithm.
Suppose we bound the threshold traction framework can implement a specific instance of hourglass persistence follows: as soon as we reach more than d simplicies given step, we contract everything to point, and so on. Then we believe the runtime should roughly to be ( (d))).
simplicies, and practitioner equipped with the con-
For practical purposes, it may also be beneficial the threshold is exceeded. If the contractions are done in multiple stages, then we believe a similar runtime analysis holds. We are excited the possibilities that this framework can allow our community to address the problem of PH scaling on large graphs/simplicial complexes.
contract everything
point when
Remark 4. Edge contractions have been used in graph pooling (Diehl et al., 2019; Limbeck et al., 2025) independently from persistent homology. We spectulate whether or approach be applied with edge contractions in graph pooling.
F THE USE OF LARGE LANGUAGE MODELS
We used large language models to aid the writing process of the introduction, to help check gram- mar / suggest edits for the main paper, and double-check algorithmic discussions arising from the paper.
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Reviews:
[{"review_id": "3a0nQRFDwG", "reviewer": "Reviewer_WkSg", "summary": "The paper proposes a new persistent homology approach that combines forward and backward passes in creating persistent homologies.", "questions": "None.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper has deep mathematical discussion describing the properties of the forward backward persistence. \n\nThe mathematical descriptions are well discussed. As a person who applied TDA for various application settings, I followed most of the discussion very carefully.", "weaknesses": "As a practitioner of topological data analysis (TDA), it is not clear to me whether the proposed forward–backward approach offers practical utility. The experimental evaluation appears limited, as it compares the method with only a single PH  approach used in GNN). Consequently, it is difficult to assess whether this work provides a substantive contribution beyond a theoretical exercise.\n\nMoreover, based on my experience, many PH-based methods struggle to scale to large graphs. In this case, the computational cost of the proposed approach seems likely to increase at least twofold, without a clearly demonstrated benefit to justify the added complexity.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "CMgIkYXzUP", "reviewer": "Reviewer_7Wbe", "summary": "This paper introduces a new family of topological descriptors for graphs, simplicial complexes, and cellular complexes, termed hourglass persistence, which interleaves inclusion and contraction operations to capture richer topological signals than traditional persistent homology (PH). The authors generalize beyond forward PH by defining backward PH, forward-backward (FB) persistence, and an expressive hourglass framework that allows arbitrary interleavings under a causal constraint. They provide theoretical separations demonstrating increased expressivity over classical PH and extended persistence, introduce a unifying (f,g)-FB perspective, and establish stability guarantees. Practical algorithms for computing these descriptors are developed, and empirical results show improvements over baseline PH and RePHINE on specific benchmarks.", "questions": "1. How does the runtime and memory footprint scale for hourglass persistence relative to standard PH and extended PH? Can you provide empirical cost comparisons?\n\n2. In practice, how are inclusion and contraction orders chosen? Are learnable schedules feasible, and if so, how do they interact with stability guarantees?\n\n3. Can you provide case studies showing specific structural motifs uniquely captured by backward or hourglass persistence?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper defines backward persistence and hourglass persistence, expanding the PH toolkit beyond inclusion-based filtrations.\n\n2. The paper proves the expressiveness and the stability of the proposed framework.\n\n3. The paper provides practical algorithms to compute inclusion–contraction PH with cycle-basis tracking and supernode maintenance.", "weaknesses": "1. While mathematically rigorous, the intuition behind when hourglass persistence most benefits learning tasks could be elaborated.\n\n2. Empirical validation is limited to small graph-classification datasets (NCI109, PROTEINS, IMDB-BINARY, NCI1); scalability to large-scale benchmarks (e.g., OGB, ZINC) remains untested. In addition, the improvement compared to baselines is not significant.\n\n3. Although extended persistence is mentioned, the empirical study does not include an extended-persistence baseline, and adding such a comparison would clarify practical differences\n\n4. Computational overhead vs. benefit is not quantified; contraction bookkeeping may introduce non-trivial runtime/memory costs. Please include the efficiency analysis between FB-persistence with traditional PH and extended PH.\n\n5. There are some missing related works, e.g., [1] also provides theoretical results on the expressiveness of PH.\n\n[1] Yan et al. \"Enhancing graph representation learning with localized topological features.\" JMLR 2025.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "18hV4ShDY7", "reviewer": "Reviewer_UtXw", "summary": "This paper proposes a novel filtration for feature extraction based on persistent homology. Unlike conventional approaches that employ forward filtration, the proposed method leverages a backward filtration. This design enables the method to achieve effective performance on graph datasets.", "questions": "- Please include an ablation study comparing **forward-only**, **backward-only**, and **(f,g)-FB persistence** variants to quantify the contribution of the backward filtration (e.g., absolute/relative gains, with statistical significance).\n    \n- Please clarify how (f,g)-FB persistence differs from a simple concatenation/union of features derived from two filtrations. A theoretical distinction and an empirical comparison (which can be incorporated into the ablation) would be helpful.\n    \n- Either explain why zigzag filtration cannot achieve the same effect as the proposed method, or demonstrate empirically that your approach outperforms zigzag-based baselines.", "limitations": "", "rating": 2, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "This paper proposes a new type of filtration from a novel perspective of persistent homology. While conventional approaches have primarily relied on forward filtration, the authors construct an interesting framework based on a backward filtration. As a result, the proposed method achieves improved performance in the experiments.", "weaknesses": "The main concerns are as follows:\n\n- The central claims of this paper are the use of backward filtration and the introduction of (f,g)(f,g)-FB persistence. While the proposed method combines forward and backward (f,g)(f,g)-FB persistence, the actual contribution of the core idea — the backward filtration itself — remains unclear. Moreover, it is not evident how (f,g)(f,g)-FB persistence fundamentally differs from simply combining features derived from two filtrations. In general, combining multiple heterogeneous features often leads to performance gains, so the improvement shown here may not be specific to the proposed formulation.\n    \n- Although the notion of backward filtration is not identical, similar ideas have been explored in approaches such as Zigzag filtration [1]. The distinction between these existing methods and the proposed one is insufficiently clarified.\n    \n- The combination of two filtrations has also been proposed in prior work [2]. While the meaning differs due to the reversed filtration direction, the proposed combination seems to follow a rather straightforward idea, making it difficult to consider the novelty as significant.\n    \n- Since this conference focuses on machine learning and deep learning, introducing a new idea in persistent homology alone is not sufficient to meet the scope. It is therefore crucial to demonstrate the effectiveness of the approach in machine learning applications, but the current experimental evidence is not sufficient. In addition, Theorem 1 appears to be merely the authors’ claim and does not constitute a theorem in a rigorous sense. A theorem should be written in a mathematically precise manner that clearly states in what sense the proposed method is effective.\n\n[1] G. Carlsson et al., Zigzag Persistence, Foundations of Computational Mathematics, Volume 10, pages 367–405, (2010)\n\n[2] T. Aoki et al., Bipath persistence, Japan Journal of Industrial and Applied Mathematics, - Volume 42, pages 453–486, (2025)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "K2WsQ4FFhg", "reviewer": "Reviewer_rVVx", "summary": "This paper proposes Hourglass Persistence (HP), a novel topological descriptor that generalizes traditional persistent homology (PH) through alternating inclusion and contraction steps. This leads to a more expressive and metrizable topological representation, which is further extended to Hourglass Persistence, where inclusions and contractions can interleave multiple times.", "questions": "1. Please provide additional experimental evidence.\n2. Please include a comparison with zigzag persistence.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "1. Conceptual novelty and clear motivation.\n    \n    The idea of introducing contraction-based backward persistence and merging it with inclusion-based persistence provides a genuinely new perspective beyond traditional or extended PH.\n    \n2. Strong theoretical grounding.\n    \n    The paper gives clear definitions for backward, forward–backward, and hourglass persistence, proving expressive power hierarchies (Proposition 1–2) and linking them to extended PH via the general ((f,g))-FB formulation. Stability and metrizability are discussed rigorously, addressing key limitations of standard PH.\n    \n3. Generalizability.\n    \n    The framework is elegantly extended from graphs to higher-dimensional topological spaces, including simplicial and cellular complexes, with appropriate handling of quotient and contraction operations.\n    \n4. Algorithmic contribution.\n    \n    The paper provides clear computational procedures using union–find and cycle basis bookkeeping, making the method implementable and efficient.", "weaknesses": "1. Limited experimental validation\n- Although Hourglass persistence appears highly competitive, the current experiments are not strong enough to convincingly demonstrate its superiority. Is combining Hourglass persistence with neural networks truly the best application scenario?\n- The authors do not provide sufficient experimental details. From the description, it seems that the PH module in RePHINE was simply replaced with the forward–backward persistence. The authors should better clarify and justify this design choice.\n- Could the authors include an ablation study for the Backward-only variant?\n- Since Hourglass persistence introduces additional computational overhead, could the authors report runtime or provide a complexity analysis?\n2. Related work\n    \nIn my view, the idea most closely related to Hourglass persistence is zigzag persistence [1], which also models both the appearance and disappearance of complexes during the filtration process. Could the authors discuss this connection in more detail?\n   \n[1] Carlsson G, De Silva V. Zigzag persistence[J]. Foundations of computational mathematics, 2010, 10(4): 367-405.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)