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arxiv: 2506.08618 · v5 · pith:MYQUONCPnew · submitted 2025-06-10 · 💻 cs.LG · cond-mat.mes-hall· cond-mat.other· cs.AI· cs.CV

HSG-12M: A Large-Scale Benchmark of Spatial Multigraphs from the Energy Spectra of Non-Hermitian Crystals

Xianquan Yan , Hakan Akg\"un , Kenji Kawaguchi , N. Duane Loh , Ching Hua Lee This is my paper

Pith reviewed 2026-05-21 23:47 UTC · model grok-4.3

classification 💻 cs.LG cond-mat.mes-hallcond-mat.othercs.AIcs.CV
keywords Hamiltonian spectral graphsnon-Hermitian crystalsspatial multigraphstopological fingerprintsalgebra-to-graph linkgraph neural networkscondensed matter physicscharacteristic polynomials
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The pith

Spectral graphs from non-Hermitian crystal Hamiltonians serve as universal topological fingerprints for polynomials, vectors, and matrices.

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

The paper presents Poly2Graph, an automated pipeline that converts the energy spectra of one-dimensional non-Hermitian crystal Hamiltonians into spatial multigraphs embedded in the complex plane. It uses this tool to release HSG-12M, a dataset of roughly 16.7 million such graphs drawn from 1401 distinct characteristic-polynomial families. The central claim is that these graphs function as unique topological signatures that directly encode algebraic objects, creating a new bridge between polynomial algebra and graph structure while preserving geometric edge distinctions that standard benchmarks discard.

Core claim

Hamiltonian spectral graphs formed by the complex-plane energy spectra of non-Hermitian crystals act as universal topological fingerprints of the underlying characteristic polynomials, vectors, and matrices, thereby establishing a direct algebra-to-graph correspondence that retains multiple geometrically distinct trajectories as separate spatial edges.

What carries the argument

The Poly2Graph pipeline that losslessly maps 1-D crystal Hamiltonians to spatial multigraphs by extracting and connecting features from spectral potential data.

Load-bearing premise

The automated extraction process captures every geometric and topological feature of the spectra without introducing artifacts or selection biases that would change the resulting multigraph structure.

What would settle it

Discovery of two algebraically distinct polynomials or matrices that produce identical spatial multigraphs under the same mapping rules would falsify the universal fingerprint claim.

Figures

Figures reproduced from arXiv: 2506.08618 by Ching Hua Lee, Hakan Akg\"un, Kenji Kawaguchi, N. Duane Loh, Xianquan Yan.

Figure 1
Figure 1. Figure 1: Number of graphs v.s. number of classes in HSG-12M compared to other graph-classification datasets. HSG-12M is the only large-scale multigraph (i.e. unlike sim￾ple graph that only allows one edge between any node pair) dataset, with exceptional class diversity even exceeds all other simple graph datasets. T-HSG-5M holds temporal spatial multigraphs. Table A3 lists comprehensive comparison. In this work, we… view at source ↗
Figure 2
Figure 2. Figure 2: Poly2Graph pipeline. (a) Starting from a 1-D crystal Hamiltonian H(z) in momentum space—or, equivalently, its characteristic polynomial P(z, E) = det[H(z) − EI]. The crystal’s open-boundary spectrum solely depends on P(z, E). (b) The spectral potential Φ(E) (Ronkin function) is computed from the roots of P(z, E) = 0, following recent advances in non-Bloch band theory [81, 83, 84]. (c) The density of states… view at source ↗
read the original abstract

AI is transforming scientific research by revealing new ways to understand complex physical systems, but its impact remains constrained by the lack of large, high-quality domain-specific datasets. A rich, largely untapped resource lies in non-Hermitian quantum physics, where the energy spectra of crystals form intricate geometries on the complex plane -- termed as Hamiltonian spectral graphs. Despite their significance as fingerprints for electronic behavior, their systematic study has been intractable due to the reliance on manual extraction. To unlock this potential, we introduce Poly2Graph: a high-performance, open-source pipeline that automates the mapping of 1-D crystal Hamiltonians to spectral graphs. Using this tool, we present HSG-12M: a dataset containing 11.6 million static and 5.1 million dynamic Hamiltonian spectral graphs across 1401 characteristic-polynomial classes, distilled from 177 TB of spectral potential data. Crucially, HSG-12M is the first large-scale dataset of spatial multigraphs -- graphs embedded in a metric space where multiple geometrically distinct trajectories between two nodes are retained as separate edges. This simultaneously addresses a critical gap, as existing graph benchmarks overwhelmingly assume simple, non-spatial edges, discarding vital geometric information. Benchmarks with popular GNNs expose new challenges in learning spatial multi-edges at scale. Beyond its practical utility, we show that spectral graphs serve as universal topological fingerprints of polynomials, vectors, and matrices, forging a new algebra-to-graph link. HSG-12M lays the groundwork for data-driven scientific discovery in condensed matter physics, new opportunities in geometry-aware graph learning and beyond.

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 manuscript introduces Poly2Graph, an open-source pipeline for automating the extraction of Hamiltonian spectral graphs (spatial multigraphs) from the energy spectra of non-Hermitian 1-D crystals. It releases the HSG-12M dataset containing 11.6 million static and 5.1 million dynamic graphs across 1401 characteristic-polynomial classes, distilled from 177 TB of spectral data. The work benchmarks popular GNNs on this dataset to highlight challenges in learning spatial multi-edges and claims that spectral graphs serve as universal topological fingerprints of polynomials, vectors, and matrices.

Significance. If the extraction pipeline proves accurate and the fingerprint claim is substantiated, the work supplies a large-scale benchmark for geometry-aware graph learning that retains metric embeddings and multi-edges, addressing a clear gap in existing graph datasets. The scale, open-source release, and potential applications to condensed-matter physics constitute notable strengths for data-driven discovery.

major comments (2)
  1. [§3] §3 (Poly2Graph pipeline description): the pipeline is presented as high-performance and lossless for mapping spectra to spatial multigraphs, yet no quantitative validation, error rates for root extraction or trajectory tracking, or comparison against manual baselines is provided. This directly undermines the artifact-free extraction assumption required for the universal fingerprint claim.
  2. [§5] §5 (universal topological fingerprint claim): the assertion that spectral graphs uniquely encode topological features of characteristic polynomials lacks a formal bijectivity argument or demonstration that distinct polynomials cannot produce isomorphic multigraphs after root clustering and edge retention. Without this, the algebra-to-graph link remains unproven.
minor comments (2)
  1. [Abstract] Abstract: clarify whether the 11.6M static and 5.1M dynamic graphs are disjoint or include overlap, and specify the exact total number of unique graphs.
  2. [§4] Figure captions and §4 (benchmarks): ensure all GNN performance tables include standard deviations across seeds and explicit ablations isolating the effect of spatial multi-edges versus simple-graph baselines.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which help clarify the presentation of the Poly2Graph pipeline and the supporting evidence for our claims. We address each major point below.

read point-by-point responses

  1. Referee: [§3] §3 (Poly2Graph pipeline description): the pipeline is presented as high-performance and lossless for mapping spectra to spatial multigraphs, yet no quantitative validation, error rates for root extraction or trajectory tracking, or comparison against manual baselines is provided. This directly undermines the artifact-free extraction assumption required for the universal fingerprint claim.

    Authors: We agree that explicit quantitative validation strengthens the lossless extraction claim. The pipeline relies on standard, deterministic numerical methods (companion-matrix eigendecomposition for roots and nearest-neighbor sorting for trajectories) whose accuracy is well-established for the polynomial degrees considered. Nevertheless, we will add a dedicated validation subsection in the revised manuscript that reports (i) maximum absolute errors versus high-precision symbolic solvers on a stratified sample of 10,000 polynomials, (ii) trajectory-tracking agreement with manual inspection on representative low-degree cases, and (iii) a comparison against an independent root-clustering baseline. These additions will directly support the artifact-free assumption. revision: yes

  2. Referee: [§5] §5 (universal topological fingerprint claim): the assertion that spectral graphs uniquely encode topological features of characteristic polynomials lacks a formal bijectivity argument or demonstration that distinct polynomials cannot produce isomorphic multigraphs after root clustering and edge retention. Without this, the algebra-to-graph link remains unproven.

    Authors: The mapping is constructed so that each characteristic polynomial determines a unique multiset of roots whose continuous trajectories in the complex plane are retained as distinct multi-edges; this construction is information-preserving by design. We therefore view the resulting spatial multigraph as a faithful topological encoding. We acknowledge, however, that a formal proof of injectivity (i.e., that non-isomorphic polynomials cannot yield isomorphic multigraphs) is not supplied. In revision we will (a) articulate the supporting reasoning from the pipeline’s deterministic construction, (b) report an empirical collision check across all 1401 classes in HSG-12M, and (c) qualify the wording from “universal” to “faithful topological fingerprints” while noting the scope of the current evidence. revision: partial

Circularity Check

0 steps flagged

Dataset construction and benchmarking paper; fingerprint claim asserted via procedural pipeline without reducing to self-fit or self-citation chain

full rationale

The paper centers on introducing the Poly2Graph pipeline to generate the HSG-12M dataset of spatial multigraphs from non-Hermitian crystal spectra, with the universal fingerprint claim presented as an observed consequence of this automated mapping across 1401 polynomial classes. No load-bearing step reduces by the paper's equations or self-citation to a fitted parameter or prior author result that is itself unverified; the construction is procedural and the central result remains an empirical dataset rather than a closed derivation. This yields a minor score for the general self-referential nature of any new pipeline but no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central contribution rests on the assumption that automated spectral-to-graph conversion preserves all relevant topological information and that the resulting multigraphs are faithful representations of the underlying polynomials and matrices.

axioms (1)
  • domain assumption The energy spectra of 1-D non-Hermitian crystal Hamiltonians can be reliably mapped to spatial multigraphs without loss of geometric distinction between trajectories.
    This premise underpins the entire dataset construction and the claim of universal fingerprints.

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