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Persistent path homology is stable: a factor-two bound between path complex distance and bottleneck distance.

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2026-08-01 00:26 UTC pith:37LXDZNW

load-bearing objection Clean framework, but the main stability theorem for general path complexes has a load-bearing gap in the homotopy argument. the 3 major comments →

arxiv 2607.26226 v1 pith:37LXDZNW submitted 2026-07-28 math.AT cs.CGmath.CO

Stability of persistent path homology of path complexes

Chris Kapulkin , Kyle Koyanagi This is my paper
classification math.AT cs.CGmath.CO MSC 55N3155N35
keywords path homologypersistent homologystabilitypath complexesbottleneck distancehypergraphssequence hypergraphsdigraphs
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper shows that persistent path homology, an invariant of directed structures that records how topological features appear and disappear across scales, is stable: if two filtered path complexes are close in a metric that compares when they admit each path, then their persistence diagrams are close in bottleneck distance, with a factor of 2. The proof constructs an interleaving of the two persistence modules from any correspondence between vertex sets whose path-entry times agree within epsilon, then applies the algebraic stability theorem. The same covering framework that builds filtrations on path complexes is instantiated to deduce stability for hypergraphs, sequence hypergraphs, and directed graphs, the last recovering an existing stability theorem for digraphs. The value is that measured, noisy directed data can be analyzed with persistent path homology without fear that small perturbations produce large changes in the output.

Core claim

The central claim is Theorem 3.19: for filtered path complexes P and Q, in every homological degree k, the bottleneck distance between their degree-k persistence diagrams is at most twice the path complex distance between P and Q. The path complex distance is half the infimum, over all vertex correspondences, of the maximum difference between the entry times of corresponding paths. The proof shows that any correspondence whose path-distortion is less than epsilon is an epsilon-path-correspondence, which induces an epsilon-interleaving of the persistence modules, and algebraic stability converts that interleaving into the bottleneck bound.

What carries the argument

The construction that carries the argument is the covering framework: a weight on the generating cells of an object (paths, hyperedges, ordered subsequences, directed edges) is extended to all cells by declaring the weight of a cell to be the cheapest total weight of a covering family, i.e., the algebraic path problem in the min-plus semiring. Lemma 2.3 proves the extension is idempotent, so the sublevel sets form a filtration that is already complete. Instantiating this framework for path complexes, hypergraphs, sequence hypergraphs, and digraphs reduces each stability statement to Theorem 3.19 via a distance comparison.

Load-bearing premise

The proof of the key step (Proposition 3.15) assumes that the target path complex is closed under concatenation of allowed paths along a shared vertex and contains the one-vertex path (x,x) for every vertex; the definition of a path complex guarantees neither, and the paper's own Remark 3.7 notes that path complexes need not be closed under concatenation.

What would settle it

Build two filtered path complexes that are within epsilon in path complex distance but whose persistence diagrams are farther than 2epsilon apart, or, more locally, find a filtered path complex violating the concatenation property and two subordinate maps of an epsilon-path-correspondence that are not homotopic, which would break the interleaving construction. The paper's own example in Remark 3.7 is a natural starting point.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If two weighted hypergraphs are close in hypergraph distance, their persistent path homology persistence diagrams are close with factor 2 (Theorem 4.12).
  • The same statement holds for sequence hypergraphs (Theorem 5.10) and for digraphs (Theorem 6.9), recovering the known digraph stability result.
  • Any weighted object that admits a covering relation and a functor to path complexes inherits stability automatically, provided a distance comparison can be proved.
  • Because the induced weight is idempotent, the resulting filtrations are unchanged by repeating the completion process; no iterative refinement alters them.

Where Pith is reading between the lines

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

  • If the concatenation-closure gap in the homotopy argument (Proposition 3.15) is real, the theorem as stated may fail for arbitrary filtered path complexes; however, the applications use generated filtrations whose sublevel path complexes are closed under concatenation, so the gap may not affect the practical cases. Testing whether Theorem 3.19 holds for all filtered path complexes or only for gene
  • The covering construction is the metric completion of a weighted structure in the enriched-category sense, so the same idempotent extension could plausibly yield stability for other hypergraph homology theories or bring Vietoris–Rips filtrations under the same umbrella.
  • Since the distance equality d_path = d_N holds for digraphs but not for hypergraphs in general, the factor of 2 is likely not optimal for hypergraphs; sharper bounds may exist for specific densities.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper introduces a covering framework in which a weight on the generating cells of a combinatorial object is extended to all cells, and uses it to prove a stability theorem for persistent path homology of filtered path complexes (Theorem 3.19). From this theorem it derives stability for hypergraphs (Theorem 4.12), sequence hypergraphs (Theorem 5.10), and digraphs (Theorem 6.9), the last recovering a result of Chowdhury and Mémoli. The proof of Theorem 3.19 follows the standard correspondence/interleaving template: an ε-path-correspondence between filtered path complexes yields an ε-interleaving of their path homology persistence modules, and algebraic stability gives the bottleneck bound.

Significance. If Theorem 3.19 were established, the covering framework would be a genuinely useful unification: Lemma 2.3 is clean and correct, the distance comparisons in Sections 4–6 are transparent, and proving stability once for path complexes would indeed cover several widely used objects. The paper is also candid about an important structural fact, Remark 3.7, namely that path complexes need not be closed under concatenation. However, that very remark exposes a fatal gap in the central proof: Proposition 3.15 is false for arbitrary path complexes, so the main theorem is not established. The significance of the paper is therefore conditional on a repair that, as explained below, cannot be achieved by a local correction within the current definitions.

major comments (3)
  1. [§3, Lemma 3.14] The proof assumes p=(x,x)∈P^δ. Definition 1.1 only guarantees the length-0 path (x), and Definition 3.1 likewise ensures (x)∈P^0. No axiom forces (x,x) into P^δ; the example in Remark 3.7 is a filtered path complex with no loops. Consequently, applying ε-path-multivaluedness to (x) only gives (y)∈Q^{δ+ε}, not (y1,y2)∈Q^{δ+ε}. Concretely, take P^δ={x} and Q^{δ+ε}={a,b} with no 1-paths. Then C(x)={a,b} is ε-path-multivalued in degree 0, but Lemma 3.14 would conclude (a,b)∈Q^{δ+ε}, which is false.
  2. [§3, Proposition 3.15] The one-step homotopy α is not well-defined for general filtered path complexes. A mixed cylinder path is mapped to the concatenation (f1(x0),...,f1(xk),f2(xk),...,f2(xm)). To show this is an allowed path of Q^{δ+ε}, the proof uses Lemma 3.14 for the transition edge and then says the ε-path-multivalued property 'places the whole image' in Q^{δ+ε}. But that property applies only to paths of P^δ, and the duplicated sequence (x0,...,xk,xk,...,xm) is not generally an allowed path of P^δ. Moreover, even if the two segments and the transition edge are each allowed, their concatenation need not be allowed because path complexes are not closed under concatenation, as Remark 3.7 explicitly states. The statement is also false: with P^δ={x}, Q^{δ+ε}={a,b}, and C={(x,a),(x,b)}, both subordinate maps f1(x)=a and f2(x)=b are path-complex maps, but they induce distinct maps on H0, so no canonical HPath
  3. [§3, Theorem 3.19 and Corollary 3.20] Since Proposition 3.15 is false, Proposition 3.18 and Theorem 3.19 are unsupported. Corollary 3.20 is the bridge used by all later sections, so the hypergraph and sequence-hypergraph results in Sections 4 and 5 inherit the same obstruction: the associated path complexes need not contain loops or be closed under concatenation. Even if the digraph case in Section 6 can be salvaged, because digraph path complexes built from reflexive relations do contain loops and are concatenation-closed, the general stability theorem for path complexes—and therefore the claimed stability for hypergraphs and sequence hypergraphs—does not follow from the arguments given.
minor comments (3)
  1. [§3, Definition 3.1] The symbol P^0 is used both for the stage-0 path complex and for the common vertex set. This is confusing; a separate notation such as V_P would clarify the statement and proofs.
  2. [§3, Definition 3.11] The path distortion is defined as a supremum over extended real differences. If both l_P(π1(σ)) and l_Q(π2(σ)) are ∞, the expression |∞−∞| is undefined. A convention should be stated, or the distance should be restricted to paths with finite entry weight.
  3. [§4 and §5] The phrase 'when no confusion arises, we denote the generated filtered path complex again by P' is used repeatedly. It would be helpful to make the completion notation G↦G uniform and to state explicitly in each section which objects are completed before stability is applied.

Circularity Check

0 steps flagged

No circularity: the stability theorem is derived from first principles; the main proof gap is a missing hypothesis, not a circular reduction.

full rationale

The derivation chain is not circular. Theorem 3.19 is proved by the standard CdSO14/BM24 route: a correspondence with small path distortion is shown to be an eta-path-correspondence, Proposition 3.18 turns it into an eta-interleaving, and algebraic stability converts the interleaving into the bottleneck bound. The path-complex distance is defined from entry weights before homology is computed; the bottleneck distance appears only as the target inequality, so the conclusion is not built into the input. The covering framework of Section 2 is an idempotent closure operator proved from the stated axioms (Lemma 2.3), and Sections 4-6 are genuine deductions from Theorem 3.19 via Corollary 3.20 plus explicit distance comparisons (Propositions 4.11, 5.9, 6.8). Self-citations ([CDK+24], [CK24]) are background references and are not load-bearing. There is, however, a serious correctness gap that is not circularity. In Proposition 3.15 the homotopy check uses Lemma 3.14, whose proof assumes "Let p=(x,x) in P^delta," while Definition 3.1 only guarantees length-0 paths in P^0; and the mixed cylinder image requires closure under concatenation, which Remark 3.7 explicitly denies: "a path complex need not be closed under concatenation." This makes Theorem 3.19 unproved as stated for arbitrary filtered path complexes, but it is a missing hypothesis in a proof, not a reduction of the theorem to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The only load-bearing axioms beyond standard TDA facts are the two ad hoc assumptions about path-complex closure (concatenation and loops), both false in general. No free parameters are fitted; all weights are induced by inf/sup constructions. No new entities (particles, forces, dimensions) are introduced.

axioms (5)
  • ad hoc to paper Path complexes are closed under concatenation of allowed paths along a shared vertex.
    Used in Proposition 3.15 to assert that the image of a cylinder path is allowed. Not in Definition 1.1, and Remark 3.7 explicitly says it is false.
  • ad hoc to paper The 1-path (x,x) is allowed in every path complex for every vertex x.
    Used in Lemma 3.14 and Proposition 3.15 to justify the transition edge (f1(x), f2(x)). Not guaranteed by Definition 1.1.
  • standard math The isometry theorem between bottleneck and interleaving distances holds for pointwise finite-dimensional persistence modules.
    Invoked in Theorem 3.19 to conclude d_B ≤ d_I. Standard result, cited as [CdSGO16].
  • standard math Algebraic stability: an ε-interleaving implies d_B ≤ ε.
    Used in Theorem 3.19. Standard, cited as [CdSGO16].
  • standard math Pointwise finite-dimensional persistence modules decompose into interval modules.
    Needed for persistence diagrams to be defined. Cited as [CB15].

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read the original abstract

We show stability of persistent path homology of path complexes. As a consequence, we deduce the stability of persistent path homology of hypergraphs and of sequence hypergraphs, and recover the known stability result for digraphs, originally due to Chowdhury and M\'emoli.

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