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Computing singular simplicial homologies of digraphs and quivers

T0 review · 4 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper proves that three singular simplicial homologies of quivers—directed graphs with loops and multiple edges—are isomorphic to the homologies of smaller flag complexes, making their persistent homology computationally tractable.

desk verdict Plausible and useful quiver homology paper, but the two main isomorphism theorems (5.3 and 6.4) are not proven as written because the chosen chain map h is not shown to be a chain map and Lemma 4.3 is applied beyond its stated domain. read the letter →

arxiv 2509.02065 v2 pith:7WTQZOZB submitted 2025-09-02 math.AT

Matthew Burfitt , Jie Wu , Stephen S.-T. Yau , Shing-Tung Yau This is my paper · ORCID
classification math.AT MSC 55U1005C2055N31
keywords singularsimplicialhomologyquiversdigraphsdirectedflagcomplexpersistentacycliccarriertheoremquiverhomotopyreduced
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to make three singular simplicial homology theories for quivers practical to compute. It proves that each of the three homologies $H^{\Delta,m}$, $H^{\Delta,h}$, and $H^{\Delta,i}$ is isomorphic to the homology of a smaller, flag-type complex: the reduced directed flag complex, the partial directed flag complex, and the generalized directed flag complex, respectively. Because the isomorphisms are natural, persistent homology of a filtered quiver can be read off from a filtration of the smaller complex. These homologies have theoretical advantages over the widely used directed flag complex—strong homotopy invariance and, in the m-case, stability—but previously had no efficient computational route. The paper also supplies algorithms and complexity bounds showing the smaller complexes are at least as convenient as the directed flag complex to build.

What carries the argument

The mechanism is a pair of simplification functors plus a classical chain-homotopy criterion. The reduced digraph $\bar R(G)$ collapses all loops and multiple edges to a single edge, and its reduced directed flag complex $\bar F(G)$ carries $H^{\Delta,m}$. The partially reduced quiver $\tilde R(G)$ collapses only multiple edges incident to loop-bearing vertices, and with a loop-maximal total order $<$ it produces the partial directed flag complex $\tilde F_<(G)$, a $\Delta$-set carrying $H^{\Delta,h}$. The acyclic carrier theorem—a criterion that makes two chain maps homotopic when each simplex is carried into an acyclic subcomplex—is the tool that converts the evident algebraic comparison maps $g$ and $h$ into chain homotopy

What would settle it

Take the quiver on two vertices with a double edge (and, for the h-case, a loop at one vertex), compute $H^{\Delta,m}_*(G)$ and $H^{\Delta,h}_*(G)$ directly from the singular definitions, and compare with $H_*(\bar F(G))$ and $H_*(\tilde F_<(G))$. One mismatch would disprove Theorems 5.3 and 6.4 as stated; alternatively, exhibit a choice of the maps $s_{v0,...,vn}$ for which $h$ fails to commute with $\partial$, showing the published proof is incomplete.

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Extended reading notes

Core claim

The central claim: for every quiver $G$, $H^{\Delta,m}_*(G) \cong H_*(\bar F(G))$, $H^{\Delta,h}_*(G) \cong H_*(\tilde F_<(G))$, and $H^{\Delta,i}_*(G) \cong H_*(F(G))$, each isomorphism natural in the relevant morphisms. Here $\bar F$ is the reduced directed flag complex after collapsing loops and multiple edges; $\tilde F_<$ is the partial directed flag complex that keeps loop information; $F$ is the directed flag complex generalized to quivers. The proofs use acyclic carriers to make the singular chain complexes chain-homotopy-equivalent to the smaller simplicial chains. Fewer generators and naturality make persistent $H^{\Delta,m}$ and $H^{\Delta,h}$ computable by filtering the smaller complexes, and $H^{\Delta,h}$ is shown invariant under weak loc

Load-bearing premise

The load-bearing premise is that the arbitrarily chosen singular simplex for each vertex set in the proofs of Theorems 5.3 and 6.4 can be chosen compatibly with face maps, so that the map $h$ is actually a chain map; the paper asserts this by a 'similar argument' without proving it.

Editorial extensions

If this is right

  • H^{Δ,m} of any quiver is computable from the reduced digraph: loops and multiple edges do not affect it, so persistent H^{Δ,m} can be computed on a much smaller flag complex.
  • H^{Δ,h} is invariant under weak local strong h-homotopy, so degenerate loops and certain multiple-edge structures can be removed before computation without changing the output.
  • For loopless quivers H^{Δ,h} equals H^{Δ,i}, and for digraphs without double edges all three homologies coincide; the exceptional behavior comes precisely from loops, multiple edges, and double edges.
  • The algorithms in the appendix compute filtered versions of F, \bar F, and \tilde F_<, so persistent homology packages can be applied directly to weighted directed networks.
  • The computational reductions are natural, so persistence barcodes for H^{Δ,m} and H^{Δ,h} respect filtered quiver maps, not just individual quivers.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial: the collapse results imply H^{Δ,m} is blind to whether interactions are multiple or looped; for applications where multiplicity encodes real information, persistent H^{Δ,h} may be the better choice, since its partial complex keeps loop data.
  • If the missing verification that h is a chain map cannot be supplied, the main isomorphisms might still be true for a different choice of chain map or a slightly different carrier; checking this directly is a concrete first step.
  • A similar acyclic-carrier reduction may work for cubical singular homologies of quivers, provided the face-compatibility issue is solved at the level of choices.
  • The partial directed flag complex depends on a loop-maximal total order; practical implementations must handle the sign and order bookkeeping in the chain functor automatically, or users may see different barcode conventions for the same quiver.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper develops computational reductions for three singular simplicial homology theories of quivers introduced in [33]. For the map-type homology H^{Δ,m}_* it constructs the reduced digraph \bar R(G) and the reduced directed flag complex \bar F(G), and claims H^{Δ,m}_*(G) ≅ H_*(\bar F(G)) with naturality. For the homomorphism-type homology H^{Δ,h}_* it introduces local strong h-homotopies, a partially reduced quiver \tilde R(G), and a partial directed flag complex \tilde F_<(G), and claims H^{Δ,h}_*(G) ≅ H_*(\tilde F_<(G)) with naturality. For inclusion-type homology H^{Δ,i}_* it observes the isomorphism with the directed flag complex F(G). The paper also gives algorithms for computing persistent homology of filtered quivers via these smaller complexes and analyzes their complexity. The main conceptual contribution is that the three singular simplicial homologies can be computed from flag-type complexes that are typically much smaller than the singular chain complexes, with naturality suitable for persistence.

Significance. If the main isomorphisms hold, the paper provides a genuinely useful computational bridge between singular simplicial quiver homologies and ordinary simplicial homology of smaller flag complexes, with persistent homology applications. The naturality claims are central for applying the results to filtered data. The paper also contains explicit algorithms and a demonstration implementation, and the complexity analysis identifies where the proposed reductions improve over direct computation. The novel local strong h-homotopy invariance statement for H^{Δ,h}_* is potentially valuable. However, the proofs of the two principal isomorphisms contain a load-bearing gap: the map h from the smaller complex back to the singular chains is chosen non-constructively and is not proved to be a chain map. Because the acyclic carrier argument requires h to be a chain map, the main theorems are not established as written. The statements appear plausible and likely repairable, but the repair is a substantive technical step rather than a typographical fix.

major comments (4)
  1. [§5.2, proof of Theorem 5.3] The chain map h: C_*(\bar F(G)) → C^{Δ,m}_*(G) is defined by choosing, for each abstract simplex {v_0,...,v_n}, an arbitrary singular simplex s_{v_0,...,v_n} realizing that vertex set. The proof then asserts, without proof, that h is a chain map 'by a similar argument'. A chain map requires ∂h({v_0,...,v_n}) = h∂({v_0,...,v_n}), i.e. the chosen representative of each face must be the corresponding face of the chosen representative of the parent simplex, up to the fixed sign. The arbitrary choices in the definition do not guarantee this. Since h is then used in the acyclic carrier theorem to compare h∘g with id, and since the acyclic carrier theorem only applies to chain maps, the chain homotopy equivalence, and hence the isomorphism H^{Δ,m}_*(G) ≅ H_*(\bar F(G)), is not proven as written. The argument needs either an explicit face-compatible selection of representatives or a different pr
  2. [§6.2, proof of Theorem 6.4] The same gap occurs for h: C_*(\tilde F_<(G)) → C^{Δ,h}_*(G). The sentence 'which is a chain map for the same reasons as g in the case of singular simplex inclusions' repeats the problematic step: for each s ∈ \tilde F_<(G)_n the proof chooses a singular simplex inclusion f_s inducing s, and these choices are independent across simplices. No argument is given that the chosen representative of a face is the face of the chosen representative of the parent. The subsequent acyclic carrier argument for h∘g and id therefore requires an unproved property. This load-bearing issue must be fixed before Theorem 6.4 can be accepted.
  3. [§5.2, proof of Theorem 5.3; §4, Lemma 4.3] The acyclicity of C^{Δ,m}_*(G_{f(0),...,f(n)}) is derived from Lemma 4.3, which is stated only for digraphs. In Theorem 5.3 the subquiver G_{f(0),...,f(n)} is a full subquiver of an arbitrary quiver and may contain loops and multiple edges; it need not be a digraph. Since this acyclicity is the carrier used in the acyclic carrier theorem, the proof needs either a quiver version of Lemma 4.3 or an explicit justification that the digraph statement extends. The argument of Lemma 4.3 looks adaptable to quivers, but the adaptation is not supplied in the manuscript.
  4. [§5.2, equations (12) and the naturality proof] The naturality claim for \bar F uses the rule \bar F(ϕ)({v_0,...,v_n}) = {ϕ(v_0),...,ϕ(v_n)}. If ϕ collapses two vertices of a simplex, the righthand side is not an n-simplex of the target simplicial complex, so this does not define a chain map C_n(\bar F(G_1)) → C_n(\bar F(G_2)) in the usual degree-preserving sense. The proof of the commutative diagram implicitly uses the convention that such a simplex is sent to 0. This convention should be stated explicitly, otherwise the naturality claim is not well-defined as written. The same convention issue appears in the definition of the chain-level partial flag functor in equation (20).
minor comments (6)
  1. [Abstract] Typo: 'principle underlying structure' should be 'principal underlying structure'; likewise 'demision zero' in the introduction should be 'dimension zero'.
  2. [§2.5] Typo: 'see Prostitution 3.1' should be 'see Proposition 3.1'.
  3. [§6.2, paragraph before Definition 15] The text says 'from Remark 18'; this should be 'Remark 6.1'.
  4. [Throughout] The notation for the homomorphism-type homology is inconsistent: H^{Δ,h}_* is standard, but Theorem 4.1 and Theorem 6.5 write H^{h,Δ}_* or H^{h,Δ}. Please unify.
  5. [Example 5.2] Typo: 'tipples' should be 'triples'.
  6. [§6.2 and Appendix A.3] Minor wording issues: 'fuctoriality' should be 'functoriality'; 'known complexity' should be 'known complexity'; 'strongly contactable' and 'strongly contractable' are used inconsistently.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: main isomorphisms are proved by explicit chain maps and acyclic carriers; only minor self-citation to prior work [33].

full rationale

I walked the claimed derivation chain. Equation (5) identifies H^{Δ,i}_* with H_*(F(G)) because F(G) is defined using exactly the images of singular simplex inclusions; this is a definitional reformulation, not a fitted prediction or a theorem derived from itself. Theorem 5.3 and Theorem 6.4 are proved by constructing explicit chain maps g and h and applying the acyclic carrier theorem. The input to those proofs is the homotopy invariance of H^{Δ,m} and H^{Δ,h} (Theorem 2.3), quoted from [33]; although [33] has overlapping authors, it is a separate published result and is not the conclusion of the present paper, so citing it is not circular. I also checked for renamed known results: the paper explicitly attributes the directed Vietoris–Rips identification to [41] and the ordered-tuple-complex identification to [50], and these remarks are not load-bearing. The real weakness is not circularity: in §5.2 (Theorem 5.3) and §6.2 (Theorem 6.4) the map h is defined by arbitrary choices of singular simplex per vertex set and is asserted to be a chain map "by a similar argument" without proving face-compatible choices; that is a missing proof/correctness gap, not a logical circle, because the theorem's conclusion is not assumed in the definition of h. Hence no circular step is exhibited; at most the paper is one point on the self-citation scale.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper's new theorems rest on prior homotopy invariance results from [33] and on several chain-homotopy arguments that involve choices of representatives; no data-fitting parameters are present.

assumptions (6)
  • domain assumption Strong homotopy invariance of H^Δ,m and H^Δ,h
    Quoted from [33, Theorem 5.8 and 5.17] and used in Corollary 5.2, Theorem 5.3, and Lemma 4.4. This is a prior result by overlapping authors; if it fails, the new isomorphisms fail.
  • standard math Acyclic carrier theorem
    Used in Theorems 5.3 and 6.4 to show chain homotopy equivalence; standard result from Munkres.
  • ad hoc to paper Lemma 4.3 strong contractibility of full subquivers on images of simplices
    Stated for digraphs but applied in Theorem 5.3 to arbitrary quivers with loops and multiple edges; the extension to quivers is not proved.
  • standard math Axiom of choice for choosing singular simplex representatives
    Used in defining the h maps and in notes that infinite quivers require choice.
  • domain assumption Existence of loop maximal total order
    Definition 14; used to construct \tilde{F}_<; existence is clear for any quiver.
  • ad hoc to paper Lemma 4.5 conditions for acyclicity of C^Δ,h_*(G_f)
    Used in the acyclic carrier step of Theorem 6.4; the proof of Lemma 4.5 depends on Lemmas 4.2 and 4.4.

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Pith. "Pith review of Computing singular simplicial homologies of digraphs and quivers." pith.science (2026). https://pith.science/paper/7WTQZOZB

@misc{pith2026250902065,
  author       = {Pith},
  title        = {Pith review of: Computing singular simplicial homologies of digraphs and quivers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7WTQZOZB}},
  note         = {Machine review of arXiv:2509.02065}
}
read the original abstract

The dynamics of large complex systems are predominately modeled through pairwise interactions, the principle underlying structure being a network of the form of a digraph or quiver. Significant success has been obtained in applying the homology of the directed flag complex to study digraphs arising as networks within numerous scientific disciplines. This homology of directed cliques enjoys relative ease of computation when compared to other digraph homologies, making it preferable for use in applications concerning large networks. By extending the ideas of singular simplicial homology to quivers in categories of different morphism types, several new singular simplicial homology theories have recently been constructed. Computationally efficient homologies for quivers have in general not previously been seriously considered. In this paper we develop further the homotopy theory of quivers necessary to derive functors that realise isomorphisms between the singular simplicial quiver homologies and the homologies of certain spaces. The simplicial chains of these spaces arise in a conveniently compact form that is at least as convenient as the directed flag complex for computations. Moreover, our constructions are natural with respect to the isomorphisms on homology making them suitable for applications in conjunction with persistent homology for practical use. In particular, for each of the singular simplicial homologies considered, we provide efficient algorithms for the computation of their persistent homology.

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    M. Zia, B. Jones, H. Feng, and G. Wei,Persistent directed flag laplacian (pdfl)-based machine learn- ing for protein–ligand binding affinity prediction, Journal of Chemical Theory and Computation 21(2025), no. 8, 4276–4285, PMID: 40186580. A Algorithmic implementations In this...

    2025

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.