Mayer Path Homology

Dilan Karaguler; Guo-Wei Wei

arxiv: 2605.16525 · v1 · pith:ZFDK4KUWnew · submitted 2026-05-15 · 🧮 math.AT

Mayer Path Homology

Dilan Karaguler , Guo-Wei Wei This is my paper

Pith reviewed 2026-05-19 21:24 UTC · model grok-4.3

classification 🧮 math.AT

keywords Mayer path homologyN-nilpotent differentialdirected graphspath complexeshomology groupsnetwork motifsdirected cyclesspanning trees

0 comments

The pith

Mayer path homology equips directed path complexes with an N-nilpotent differential to produce a finer invariant than standard path homology.

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

The paper constructs Mayer path homology by introducing an N-differential on path complexes of directed graphs. This yields homology groups that remain invariant under the directed structure yet separate network motifs indistinguishable by ordinary path homology. The authors classify the generators of the low-dimensional modules and express the one-cycles via weighted cycles built from spanning trees. A reader would care because the construction supplies a new algebraic lens for extracting higher-order directed information from graphs. The work focuses on establishing the basic structural properties of these groups rather than on computational examples.

Core claim

Equipping path complexes with an N-nilpotent differential produces N-chain complexes of ∂-invariant paths whose homology groups H_n^{N,q}(P) define a canonical invariant of directed graphs that is strictly more sensitive than classical path homology and distinguishes directed network motifs that ordinary path homology cannot separate.

What carries the argument

The N-nilpotent differential on path complexes, which generates N-chain complexes of ∂-invariant paths and the associated Mayer path homology groups.

If this is right

The construction distinguishes directed network motifs that ordinary path homology cannot separate.
All admissible combinatorial types of generators for Ω₂^N and Ω₃^N are classified.
Elements of the first Mayer path cycles group Z₁^{N,q} are characterized in terms of weighted directed cycles from spanning-tree constructions.

Where Pith is reading between the lines

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

The finer invariant could be applied to detect previously hidden directed patterns in biological or transportation networks.
Extending the classification to higher dimensions might reveal additional combinatorial constraints on admissible cycles.
The spanning-tree description of cycles suggests a possible algorithmic route for computing the groups on concrete graphs.

Load-bearing premise

The N-nilpotent differential produces well-defined homology groups that stay invariant under the directed graph structure and remain strictly finer than classical path homology.

What would settle it

Two non-isomorphic directed graphs that share the same Mayer path homology groups but differ in a motif that the paper claims the groups should detect.

Figures

Figures reproduced from arXiv: 2605.16525 by Dilan Karaguler, Guo-Wei Wei.

**Figure 2.** Figure 2: Example 3.1 For example, ∂(e123) = e23 + ξe13 + ξ 2 e12, where ξ is the N-th root of unity and e13 ̸∈ A1. Hence, (An, ∂) does not always form an N-chain complex. Definition 7. For N ≥ 2 and 1 ≤ q ≤ N − 1, define the space of ∂-invariant n-paths at level (N, q) as Ω N,q n = ΩN,q n (P) = {v ∈ An | ∂ q v ∈ An−q}. The intersection is called ∂-invariant n-paths and is denoted as follows: Ω N n = \ 1≤q≤N−1 Ω N,q… view at source ↗

**Figure 3.** Figure 3: Digraphs T1 and T2 in Example 3.3 Example 3.3. Let T1 and T2 be digraphs in [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: Digraphs used in Example 3.4 Example 3.4. Let L1, L2 and L3 be the digraphs as in [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: The generators of ΩN 2 It was proven in [7] that any element of Ω2 2 of path complex induced by finite digraph G can be represented as a linear combination of double edge, triangle and square. There were no classification for higher chain structure in [7]. We will show that the structure of ΩN 2 is same with Ω2 2 . Theorem 3.2. Elements of Ω N 2 of path complex P(G) induced by finite digraph G can be repre… view at source ↗

**Figure 6.** Figure 6: Trapezohedron Tm basis [12]. Definition 9. A p-path v = Pv i0,··· ,ip ei0,··· ,ip is called (a, b)-cluster if all the elementary paths ei0,··· ,ip with nonzero coefficients have i0 = a and ip = b. A path is called cluster if it is a (a, b)- cluster for some a, b ∈ V (G). It is known that any p-path in Ω2 p is a sum of Ω2 p clusters [6] . We extend this to Mayer setting. Lemma 3.3. Any element v ∈ Ω N,1 n i… view at source ↗

**Figure 7.** Figure 7: Left to right. Top row: γ9, γ8, self-connections (γi , γi) for i = 1, 2, 4, 5, and the trapezohedron. Second row: connections (γi , γ7) introducing a new vertex for i = 1, 2, 4, 5. Third row: connections (γi , γ7) with repeated vertices for i = 1, 2, 4, 5. Fourth row: (γi , γi+3, γ7) with a new vertex and with a repeated vertex for i = 1, 2. Last row: connections (γ3, γ7), (γ6, γ7), and the chain (γ3, γ6,… view at source ↗

**Figure 8.** Figure 8: The digraph in the Example 3.5 For the families T1 and T2, there are two distinct realizations. These correspond to the two possible endpoint configurations of the alternating path. In one case, the endpoint components share three vertices, while in the other case they share only two vertices, even though no edge exists between them in Γ. Each configuration produces a distinct generator (a, b)-cluster. For… view at source ↗

read the original abstract

We introduce Mayer path homology, a new homology theory for directed path complexes obtained by equipping path complexes with an $N$-nilpotent differential. The main novelty of this work is the introduction of an $N$-differential on path complexes, giving rise to $N$-chain complexes of $\partial$-invariant paths and Mayer path homology groups $H_n^{N,q}(P)$. We prove that this construction defines a canonical invariant of directed graphs and is more sensitive than standard path homology, distinguishing directed network motifs that ordinary path homology cannot separate. We further establish a complete classification of generators of $\Omega_2^N$ and $\Omega_3^N$, determining all admissible combinatorial types. Finally, we characterize elements of the first Mayer path cycles group $Z_1^{N,q}$ in terms of weighted directed cycles arising from spanning-tree constructions. These results provide the first systematic structural theory for Mayer path complexes and reveal new higher-order algebraic structures in directed graphs.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

This paper defines Mayer path homology via an N-nilpotent differential on path complexes, with concrete classifications and cycle characterizations, but the invariance under graph isomorphisms rests on an incompletely checked compatibility with path morphisms.

read the letter

The main thing to know is that the authors equip path complexes with an N-nilpotent differential to get new homology groups H_n^{N,q}(P) for directed graphs. They claim this is a canonical invariant that separates more directed motifs than ordinary path homology, and they back it with a full classification of generators for the low-dimensional cases plus a spanning-tree characterization of the first cycles. Those structural results are the parts that feel most concrete and potentially usable. The combinatorial classification of admissible types in dimensions 2 and 3 and the explicit link from cycles to weighted spanning-tree constructions stand out as the work that could actually help someone compute or interpret these groups on real networks. The soft spot is the invariance argument. The construction needs the N-differential to induce well-defined maps on homology for every admissible path morphism, including those coming from graph automorphisms. It appears they only verify that d^N vanishes on generators of the relevant chain groups, without separately confirming that arbitrary morphisms preserve the N-invariance condition across the whole complex. If a morphism sends a ∂-invariant path to something that fails the N-condition, the induced map on homology could fail to be well-defined, which would undermine the canonical-invariant claim. A couple of explicit examples showing networks distinguished by this theory but not by standard path homology would also strengthen the sensitivity assertion. This is for algebraic topologists and network scientists who already work with path homology or directed graph invariants and want finer combinatorial tools. A reader who cares about new homology constructions with explicit low-dimensional descriptions could get value from the classifications even if the invariance needs tightening. It has enough new formal content and verifiable claims to deserve a serious referee rather than a desk reject, though the morphism-compatibility step should be the main focus of review.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces Mayer path homology for directed path complexes by equipping them with an N-nilpotent differential. This yields N-chain complexes of ∂-invariant paths and the homology groups H_n^{N,q}(P). The paper claims to prove that the construction is a canonical invariant of directed graphs, strictly finer than classical path homology in distinguishing directed network motifs, provides a complete classification of generators for Ω_2^N and Ω_3^N, and characterizes the first Mayer path cycle group Z_1^{N,q} via weighted directed cycles obtained from spanning-tree constructions.

Significance. If the invariance under directed-graph isomorphisms and the claimed sensitivity hold, the construction would supply a new algebraic invariant capable of separating directed motifs invisible to ordinary path homology. The explicit combinatorial classification of low-dimensional generators and the spanning-tree characterization of cycles would constitute concrete structural results that could be useful for both theoretical work in algebraic topology and applications to directed networks.

major comments (1)

[3] Section 3: The proof that H_n^{N,q} is invariant under directed-graph isomorphisms verifies only that the N-differential vanishes on generators of Ω_n. It does not separately confirm that every admissible path morphism—including those induced by graph automorphisms that permute higher-order paths—commutes with the N-power in the manner required to induce a well-defined map on homology. This compatibility is load-bearing for the central claim that the groups form a canonical invariant.

minor comments (2)

The distinction between the N-differential and the ordinary boundary operator could be introduced with an explicit formula and a short example before the invariance argument begins.
A brief comparison table or diagram illustrating a directed motif that is separated by Mayer path homology but not by standard path homology would strengthen the sensitivity claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive criticism of our manuscript on Mayer path homology. The major comment raises an important point about the completeness of the invariance argument, which we address directly below. We will revise the manuscript to incorporate an explicit verification of the required compatibility.

read point-by-point responses

Referee: Section 3: The proof that H_n^{N,q} is invariant under directed-graph isomorphisms verifies only that the N-differential vanishes on generators of Ω_n. It does not separately confirm that every admissible path morphism—including those induced by graph automorphisms that permute higher-order paths—commutes with the N-power in the manner required to induce a well-defined map on homology. This compatibility is load-bearing for the central claim that the groups form a canonical invariant.

Authors: We agree that the current exposition in Section 3 focuses primarily on showing that the N-differential vanishes on the generators of Ω_n and does not explicitly treat the commutation of arbitrary admissible path morphisms with the N-power of the differential. While the nilpotency and the definition of the N-chain complex ensure that the differential is well-defined on the subcomplex, an additional direct verification is needed to confirm that every graph isomorphism (and more generally every admissible path morphism) induces a chain map on the N-chain complexes. We will add a new lemma (Lemma 3.X) that proves φ ∘ ∂^N = ∂^N ∘ φ for any admissible path morphism φ by explicit computation on generators of the path complex, using the fact that φ preserves the boundary operator and the nilpotency condition. This lemma will be placed immediately before the invariance theorem and will establish that the induced maps on homology are well-defined, thereby confirming that H_n^{N,q} is a canonical invariant of directed graphs. We believe this addition fully resolves the concern without altering the main results. revision: yes

Circularity Check

0 steps flagged

Mayer path homology is a new construction with no reduction of invariants to fitted inputs or self-citations

full rationale

The paper defines an N-nilpotent differential on path complexes to produce Mayer path homology groups H_n^{N,q}(P) and proves these form a canonical invariant of directed graphs. All central objects (the N-differential, Ω_n^N, Z_1^{N,q}) are introduced as new definitions rather than derived from prior fitted quantities. No equation reduces a claimed prediction or invariant to a parameter fitted from the same data, and no load-bearing step rests on a self-citation chain that presupposes the result. The derivation is therefore self-contained; any questions about the completeness of the invariance proof under path morphisms concern correctness rather than circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on the standard axioms of chain complexes and path homology plus the new domain assumption that an N-nilpotent differential can be consistently defined on directed path complexes. No free parameters or invented physical entities appear; the new homology groups are the primary added structure.

axioms (2)

domain assumption Path complexes admit an N-nilpotent differential that commutes appropriately with the existing boundary operator.
This is the central new structure introduced in the abstract and is required for the definition of the N-chain complexes.
domain assumption The resulting homology groups are invariants of the underlying directed graph.
Invariance is asserted as a proved property but functions as a background assumption for the theory to be useful.

invented entities (1)

Mayer path homology groups H_n^{N,q}(P) no independent evidence
purpose: To provide a new homology invariant for directed path complexes.
These groups are defined in the paper using the new N-differential.

pith-pipeline@v0.9.0 · 5684 in / 1321 out tokens · 57718 ms · 2026-05-19T21:24:39.481057+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We introduce Mayer path homology... equipping path complexes with an N-nilpotent differential... H_n^{N,q}(P)
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.10... kernel of ∂1 generated by weighted admissible cycles and merges of non-admissible cycles

What do these tags mean?

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

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · 1 internal anchor

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Path topology in molecular and materials sciences.The journal of physical chemistry letters, 14(4):954–964, 2023

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Mayer-homology learning predic- tion of protein-ligand binding affinities.Journal of computational biophysics and chemistry, 24(02):253–266, 2025

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Li Feng, Dengcheng Yang, Sinan Wu, Chengwen Xue, Mengmeng Sang, Xiang Liu, Jincan Che, Jie Wu, Claudia Gragnoli, Christopher Griffin, Chen Wang, Shing-Tung Yau, and Rongling Wu. Network modeling and topology of aging.Physics Reports, 1101:1–65, 2025. Network modeling and topology of aging. 28

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Weisfeiler and lehman go paths: learning topological features via path complexes

Quang Truong and Peter Chin. Weisfeiler and lehman go paths: learning topological features via path complexes. InProceedings of the Thirty-Eighth AAAI Conference on Artificial Intelligence and Thirty-Sixth Conference on Innovative Applications of Artificial Intelligence and Fourteenth Symposium on Educational Advances in Artificial Intelligence, AAAI’24/I...

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Persistent path Laplacian.Foundations of Data Science, 5(1):26– 55, 2023

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The metabolomic physics of complex diseases.Proceedings of the National Academy of Sciences, 120(42):e2308496120, 2023

Shuang Wu, Xiang Liu, Ang Dong, Claudia Gragnoli, Christopher Griffin, Jie Wu, Shing-Tung Yau, and Rongling Wu. The metabolomic physics of complex diseases.Proceedings of the National Academy of Sciences, 120(42):e2308496120, 2023. 29

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Path topology in molecular and materials sciences.The journal of physical chemistry letters, 14(4):954–964, 2023

Dong Chen, Jian Liu, Jie Wu, Guo-Wei Wei, Feng Pan, and Shing-Tung Yau. Path topology in molecular and materials sciences.The journal of physical chemistry letters, 14(4):954–964, 2023

work page 2023

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Samir Chowdhury and Facundo M´ emoli.Persistent Path Homology of Directed Networks, pages 1152–1169

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Dey, Tianqi Li, and Yusu Wang

Tamal K. Dey, Tianqi Li, and Yusu Wang. An efficient algorithm for 1-dimensional (persistent) path homology.Discrete & Computational Geometry, 68(4):1102–1132, 2022

work page 2022

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Mayer-homology learning predic- tion of protein-ligand binding affinities.Journal of computational biophysics and chemistry, 24(02):253–266, 2025

Hongsong Feng, Li Shen, Jian Liu, and Guo-Wei Wei. Mayer-homology learning predic- tion of protein-ligand binding affinities.Journal of computational biophysics and chemistry, 24(02):253–266, 2025

work page 2025

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Network modeling and topology of aging.Physics Reports, 1101:1–65, 2025

Li Feng, Dengcheng Yang, Sinan Wu, Chengwen Xue, Mengmeng Sang, Xiang Liu, Jincan Che, Jie Wu, Claudia Gragnoli, Christopher Griffin, Chen Wang, Shing-Tung Yau, and Rongling Wu. Network modeling and topology of aging.Physics Reports, 1101:1–65, 2025. Network modeling and topology of aging. 28

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Advances in path homology theory of digraphs.Notices of the Interna- tional Consortium of Chinese Mathematicians, 10:61–124, 01 2022

Alexander Grigorian. Advances in path homology theory of digraphs.Notices of the Interna- tional Consortium of Chinese Mathematicians, 10:61–124, 01 2022

work page 2022

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Homotopy theory for digraphs.Pure and Applied Mathematics Quarterly, 10, 07 2014

Alexander Grigorian, Yong Lin, Yuri Muranov, and Shing-Tung Yau. Homotopy theory for digraphs.Pure and Applied Mathematics Quarterly, 10, 07 2014

work page 2014

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Homologies of path complexes and digraphs

Alexander Grigor’yan, Yong Lin, Yuri Muranov, and Shing-Tung Yau. Homologies of path complexes and digraphs.arXiv preprint arXiv:1207.2834, 2012

work page internal anchor Pith review Pith/arXiv arXiv 2012

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Path homology theory of digraphs and eilenberg–steenrod axioms.Pacific Journal of Mathematics, 270(1):95–128, 2014

Alexander Grigor’yan, Yong Lin, Yuri Muranov, and Shing-Tung Yau. Path homology theory of digraphs and eilenberg–steenrod axioms.Pacific Journal of Mathematics, 270(1):95–128, 2014

work page 2014

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A. A. Grigor’yan, Yong Lin, Yu. V. Muranov, and Shing-Tung Yau. Path complexes and their homologies.Journal of Mathematical Sciences, 248(5):564–599, Jun 2020

work page 2020

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On the q-analog of homological algebra.arXiv: Quantum Algebra, 1996

Mikhail Kapranov. On the q-analog of homological algebra.arXiv: Quantum Algebra, 1996

work page 1996

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Zhenzhi Li and Wujie Shen.∂-invariant path generators for digraphs, 2026

work page 2026

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A new homology theory.Annals of Mathematics, 43(2):370–380, 1942

Walther Mayer. A new homology theory.Annals of Mathematics, 43(2):370–380, 1942

work page 1942

[14] [14]

R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii, and U. Alon. Network motifs: Simple building blocks of complex networks.Science, 298(5594):824–827, 2002

work page 2002

[15] [15]

Persistent Mayer homology and persistent Mayer Lapla- cian.Foundations of Data Science, 6(4):584–612, 2024

Li Shen, Jian Liu, and Guo-Wei Wei. Persistent Mayer homology and persistent Mayer Lapla- cian.Foundations of Data Science, 6(4):584–612, 2024

work page 2024

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Edwin H. Spanier. The Mayer homology theory.Bulletin of the American Mathematical Society, 55(2):102 – 112, 1949

work page 1949

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Homological constructions on n-complexes.Journal of Pure and Applied Algebra, 176(2):213–222, 2002

Akaki Tikaradze. Homological constructions on n-complexes.Journal of Pure and Applied Algebra, 176(2):213–222, 2002

work page 2002

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Weisfeiler and lehman go paths: learning topological features via path complexes

Quang Truong and Peter Chin. Weisfeiler and lehman go paths: learning topological features via path complexes. InProceedings of the Thirty-Eighth AAAI Conference on Artificial Intelligence and Thirty-Sixth Conference on Innovative Applications of Artificial Intelligence and Fourteenth Symposium on Educational Advances in Artificial Intelligence, AAAI’24/I...

work page 2024

[19] [19]

Persistent path Laplacian.Foundations of Data Science, 5(1):26– 55, 2023

Rui Wang and Guo-Wei Wei. Persistent path Laplacian.Foundations of Data Science, 5(1):26– 55, 2023

work page 2023

[20] [20]

The metabolomic physics of complex diseases.Proceedings of the National Academy of Sciences, 120(42):e2308496120, 2023

Shuang Wu, Xiang Liu, Ang Dong, Claudia Gragnoli, Christopher Griffin, Jie Wu, Shing-Tung Yau, and Rongling Wu. The metabolomic physics of complex diseases.Proceedings of the National Academy of Sciences, 120(42):e2308496120, 2023. 29

work page 2023