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arxiv: 2601.04559 · v2 · submitted 2026-01-08 · 🧮 math.AT · math.CO· math.DS

Asymmetrically Weighted Dowker Persistence and Applications in Dynamical Systems

Tobias Timofeyev , Christopher Potvin , Benjamin Jones , Kristin M. Kurianski , Miguel Lopez , Sunia Tanweer This is my paper

Pith reviewed 2026-05-16 16:59 UTC · model grok-4.3

classification 🧮 math.AT math.COmath.DS MSC 55N31
keywords Dowker persistencepersistent homologydynamical systemsdirected networksperiodic orbitstopological data analysisattractorscactus graphs
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The pith

Asymmetric Dowker persistence on binned directed networks distinguishes periodic from non-periodic cycles via one-dimensional characterization.

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

The paper bins dynamical time series data into coarse-grained weighted directed networks that retain both positions and sequential order. It computes the persistent Dowker homology of these asymmetric networks to extract topological features that include temporal directionality. A central analytic result fully characterizes the one-dimensional Dowker persistences of periodic versus non-periodic cycles, showing how asymmetry encodes the distinction. The work also proves that homologies of graph wedge sums are determined by the component homologies and extends the characterization to cactus graphs of arbitrary weights and orientations, yielding diagrams that remain stable under noise yet reflect specific dynamical regimes.

Core claim

By binning dynamical data into coarse-grained weighted directed networks and computing persistent Dowker homology of the asymmetric network, spatial and temporal information is encoded. A full characterization is given of the one-dimensional Dowker persistences for periodic and non-periodic cycles. Homologies of graph wedge sums are described in terms of the wedge component homologies, and the characterization is generalized to cactus graphs with arbitrary edge weights and orientations. This produces a persistence framework robust to noise and sensitive to dynamical structure.

What carries the argument

Persistent Dowker homology of asymmetrically weighted and directed networks obtained by binning the time series.

If this is right

  • Periodic and non-periodic cycles produce distinct one-dimensional Dowker persistence barcodes.
  • The homology of a wedge sum of graphs reduces to the homologies of its individual components.
  • The one-dimensional characterization extends to cactus graphs with arbitrary weights and orientations.
  • The resulting persistence diagrams remain stable under noise while distinguishing dynamical features.

Where Pith is reading between the lines

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

  • The directed binning step could serve as a general preprocessing pipeline for applying Dowker persistence to other ordered data such as event sequences or flow data.
  • Higher-dimensional Dowker homology computed on the same networks might reveal additional structure in the attractors beyond the one-dimensional case.
  • The asymmetry-based distinction suggests the method could be tested on experimental time series from physics or biology to classify observed regimes without knowing the governing equations.

Load-bearing premise

Binning the dynamical data into a coarse-grained weighted directed network preserves the essential topological and sequential features of the underlying attracting set.

What would settle it

A periodic orbit and a non-periodic orbit that, after identical binning, produce identical one-dimensional Dowker persistence barcodes would disprove the claimed full characterization.

read the original abstract

By their nature it is difficult to differentiate chaotic dynamical systems through measurement. In recent years, work has begun on using methods of Topological Data Analysis (TDA) to qualitatively type dynamical data by approximating the topology of the underlying attracting set. This comes with the additional challenges of high dimensionality incurring computational complexity along with the lack of directional information encoded in the approximated topology. Due to the latter fact, standard methods of TDA for this high dimensional dynamical data do not differentiate between periodic cycles and non-periodic cycles in the attractor. We present a framework to address both of these challenges. We begin by binning the dynamical data, and capturing the sequential information in the form of a coarse-grained weighted and directed network. We then calculate the persistent Dowker homology of the asymmetric network, encoding spatial and temporal information. Analytically, we highlight the differences in periodic and non-periodic cycles by providing a full characterization of their one-dimensional Dowker persistences. We prove how the homologies of graph wedge sums can be described in terms of the wedge component homologies. Finally, we generalize our characterization to cactus graphs with arbitrary edge weights and orientations. Our analytical results give insight into how our method captures temporal information in its asymmetry, producing a persistence framework robust to noise and sensitive to dynamical structure.

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.

Referee Report

2 major / 2 minor

Summary. The paper proposes binning dynamical trajectories into coarse-grained weighted directed networks and computing their asymmetrically weighted Dowker persistence to differentiate periodic from non-periodic cycles in attractors. It supplies an analytical characterization of the one-dimensional Dowker persistence for such cycles, first via a description of homologies of graph wedge sums and then by generalization to cactus graphs with arbitrary edge weights and orientations; the resulting persistence is claimed to be robust to noise while sensitive to temporal structure.

Significance. If the central analytical results hold and extend to the networks arising in the dynamical-systems application, the work would supply a directed, noise-robust TDA invariant capable of distinguishing cycle types that standard undirected persistence cannot separate, thereby offering a concrete advance in the qualitative classification of high-dimensional attractors.

major comments (2)
  1. [analytical results on cactus graphs and §5 (dynamical applications)] The full characterization of one-dimensional Dowker persistence for periodic versus non-periodic cycles is established only for cactus graphs (via the wedge-sum decomposition and its generalization). The dynamical-systems application, however, obtains arbitrary weighted directed networks by binning; no reduction, approximation, or stability argument is supplied showing that the persistence of these general networks is controlled by, or coincides with, the cactus case.
  2. [§5 and numerical examples] The abstract asserts that the method is 'robust to noise and sensitive to dynamical structure,' yet the manuscript provides no quantitative controls or sensitivity analysis for the binning parameters (number of bins, weighting scheme) that determine the input network; without such controls it is unclear whether the observed cycle-type separation survives changes in coarse-graining.
minor comments (2)
  1. [preliminaries] Notation for the asymmetric weight function and the precise definition of the Dowker complex should be stated once in a single preliminary section rather than re-introduced piecemeal.
  2. [figures] Several persistence diagrams in the figures lack explicit axis labels or scale bars, making direct comparison of birth-death intervals difficult.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and indicate the revisions we will make to strengthen the connection between the analytical results and the dynamical applications.

read point-by-point responses

  1. Referee: [analytical results on cactus graphs and §5 (dynamical applications)] The full characterization of one-dimensional Dowker persistence for periodic versus non-periodic cycles is established only for cactus graphs (via the wedge-sum decomposition and its generalization). The dynamical-systems application, however, obtains arbitrary weighted directed networks by binning; no reduction, approximation, or stability argument is supplied showing that the persistence of these general networks is controlled by, or coincides with, the cactus case.

    Authors: We appreciate the referee's observation on the scope of the analytical characterization. The full one-dimensional Dowker persistence results are indeed established rigorously for cactus graphs, which we use to distinguish periodic from non-periodic cycles via the wedge-sum decomposition and its generalization to arbitrary weights and orientations. For the dynamical-systems examples, the binning procedure produces general weighted directed networks on which we compute the persistence directly; the cactus characterization is presented to provide theoretical insight into why the asymmetry captures temporal structure. We acknowledge that no explicit reduction, approximation, or stability theorem is supplied to control the general case by the cactus case. In the revised manuscript we will add a clarifying paragraph in §5 noting this distinction, explaining that the numerical examples illustrate the method on networks arising from typical attractors, and stating that a full stability result remains an open direction for future work. revision: partial

  2. Referee: [§5 and numerical examples] The abstract asserts that the method is 'robust to noise and sensitive to dynamical structure,' yet the manuscript provides no quantitative controls or sensitivity analysis for the binning parameters (number of bins, weighting scheme) that determine the input network; without such controls it is unclear whether the observed cycle-type separation survives changes in coarse-graining.

    Authors: We agree that quantitative sensitivity analysis is necessary to support the robustness and sensitivity claims. In the revised manuscript we will augment §5 with additional experiments that systematically vary the number of bins and the weighting scheme over representative ranges. These will include tables or plots showing that the separation between periodic and non-periodic cycles persists for moderate changes in coarse-graining parameters, thereby providing the requested controls while preserving the existing numerical examples. revision: yes

Circularity Check

0 steps flagged

No significant circularity; independent mathematical characterizations

full rationale

The paper's core derivation consists of explicit proofs: first characterizing 1D Dowker persistence on wedge sums, then generalizing to cactus graphs with arbitrary weights and orientations. These steps are presented as self-contained mathematical arguments that do not reduce to fitted parameters, self-definitions, or prior self-citations. The dynamical application (binning trajectories into a weighted directed network) is a separate preprocessing choice whose output is then fed into the already-proven persistence framework; no equation or claim equates the general-network persistence back to the cactus case by construction. The framework extends established Dowker homology without load-bearing self-citations or smuggled ansatzes, so the claimed differentiation of cycle types rests on the independent proofs rather than circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces a methodological extension rather than new physical entities or many fitted parameters; it relies primarily on standard mathematical properties of persistent homology.

axioms (1)
  • standard math Standard properties of Dowker complexes and persistent homology for directed graphs
    The method depends on established foundations of topological data analysis for directed structures.

pith-pipeline@v0.9.0 · 5548 in / 1318 out tokens · 57871 ms · 2026-05-16T16:59:37.985948+00:00 · methodology

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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 begin by binning the dynamical data, and capturing the sequential information in the form of a coarse-grained weighted and directed network. We then calculate the persistent Dowker homology of the asymmetric network... full characterization of their one-dimensional Dowker persistences... generalize our characterization to cactus graphs with arbitrary edge weights and orientations.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Proposition 15... nontrivial equivalence class in the H1 homology group of Dδ(G), for max i ω(i,i+1) ≤ δ < min i≠j max{ω(i,j),ω(j,i)}.

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