HSG-12M: A Large-Scale Benchmark of Spatial Multigraphs from the Energy Spectra of Non-Hermitian Crystals
Authors: Xianquan Yan
Hakan Akgün
Kenji Kawaguchi
N. Duane Loh
Ching Hua Lee
Venue: ICLR 2026 Poster (2026)
Keywords: graph-level learning
spatial network
multigraph
dataset generator
large-scale dataset
condensed matter physics
non-Hermitian physics
topological physics
AI4Science
Toeplitz matrix
Primary area: datasets and benchmarks

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 $\textit{Hamiltonian spectral graphs}$. 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 $\textbf{Poly2Graph}$ (https://github.com/sarinstein-yan/Poly2Graph): a high-performance, open-source pipeline that automates the mapping of 1-D crystal Hamiltonians to spectral graphs. Using this tool, we present $\textbf{HSG-12M}$ (https://github.com/sarinstein-yan/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 $\textit{spatial multigraphs}$—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.

HSG- A LARGE-SCALE BENCHMARK OF SPA- TIAL MULTIGRAPHS FROM THE ENERGY SPECTRA OF NON- HER-HERMITIAN CRYSTALS
Xianquan Yan Hakan Akgün' Kenji Kawaguchi N. Duane Loh Ching Hua Lee't
Department of ( Physscs., *Computer Science, "Biological Sciences), National University of Singapore "NUS Centre for Bioimaging Sciences, National University of Singapore
ABSTRACT
AI is transforming scientific research by revealing new ways to understand com- plex physical systems, but its impact remains constrained by the lack of large. high-quality domain- datasets. A rich, largely untapped resource lies in non-Hermitian quantum physics, where the energy spectra of crystals form intri- cate geometries on the complex plane plane-termed termed as Hamiltonian spectral graphs. Despite their significance as fingerprints for electronic behavior, their systematic study has been intractable due the reliance manual extraction. To unlock this potential, we introduce Poly2Graph's : peghoppeformnnce, open-source pipeline that automates the mapping of 1-D crystal Hamiltonians to spectral graphs. Us- a dataset containing 11.6 million static and 5.1 million dynamic Hamiltonian spectral graphs across 1401 characteristie- polynomial classes, distilled from 177 TB of spectral potential data. Crucially, HSG-12M the first large-scale dataset of spatial multigraphss-grapph embed- ded in a metric space where multiple geometrically distinct trajectories between two nodes are retained as separate edges. This simultaneously addresses 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 practi- cal utility, we show that spectral graphs serve as universal topological fingerprints of polynomials, vectors, and matrices, forging 12M lays the groundwork for data-driven scientific discovery in condensed matter physics, new opportunities in geometry- graph learning and beyond.
ing this tool, we present HSG-12M "
algebra-to-gue to-graph link. HSG-
INTRODUCTION
The integration of AI into scientific research is transforming how complex physical systems understood (Carleo et al., 2019). However, this transformation is often hindered by a shortage of high-quality, domain- -ppecic datasets, particularly in physical sciences. Recent breakthroughs in protein folding (Jumper et al., 2021; Varadi et al., 2024), materials discovery (Merchant et al., 2023; et al., 2025), and many-body physics (Yang et al., 2024; Torlai et al., 2018) underscore how well- curated scientific datasets can unlock Al's full potential, enabling discoveries that would otherwise remain inaccessible.
A rich resource lies in the Hamiltonian spectral graph, fascinating and diverse object emerging from recent advances in non-Hermitian physics. Recent advances have shown that the energy spec- trum of one-dimensional crystals under open boundary conditions forms arcs and loops the complex energy plane. These spectral loci can be naturally represented as spatial graphs embedded in the two-dimensional complex plane (C-plane). Moreover, these spectral graphs (Tai & Lee, 2023;
muanelohênzr wtw.uu....ddsgggg....e..ee
cheiichema.eede edu..g
'ttpps: : //gi thub. wttp://t.//o/tsttieieeeoo
y2Graph
httpps: //gi thub. com/ sa sari rirriniiiiie in HSG-11 12M
TTo he precise, it is 1-D crystal (lattice) Hamiltonian, under open boundary conditions (OBC), in the ther- modynamic limit (í.e. large-size limit, the length of the lattice oc).

---

Lin et al., 2023; Xiong & Hu, 2023; Wang al., 2024) serve fingerprints with far more intricate structures than conventional topologcal signatures for electronic behavior (e.g., Z/Zz invariants, Chern number (Hasan & Kane, 2010)). Figure 2&A4 show examples of these graphs, featuring a kaleidoscope of nontrivial edge geometries and multiplicities that form diverse patterns beyond existing graph datasets.
Despite their theoretical significance, spectral graph extraction has traditionally relied on manual plotting and visual inspection-an approach limited toy examples and small-scale investigations. In the absence of any automated workflow or large curated dataset, its systematic studies have re- mained out of reach.
To overcome the reliance on manual inspection, we developed Poly2Graph: an open-source pipeline that combines algebraic geometry, non-Bloch band theory, and morphological image processing to fully automate spectral graph extraction. By delivering unprece- dented speed and memory efficiency, Pol Poly2Graph enabled us to in- troduce HSG-12M (Hamiltonian Spectral Graphs, 12 Million). This dataset distills 177 TB of spectral potential data into 12M spatial multigraph representations (256 GB), spanning 1401 characteristic polynomial classes. Each graph is derived from the energy band structure of a crystal Hamiltonian, encoding the complex energy spec- trum's 's full geometry'. condensed matter physics, energy band structure a fundamental concept, key to understanding insulators, conductors, phase transitions, electron dynam- ics, and system symmetries. We additionally provide 5.1M temporal spatial graphs capturing continuous deformations of spectral graphs.
THHSG-5M HSG 12M
140]
10'
Natural Sciences
Skilctse (Kicttis)
MALNET
Compute: Science
Networks
Simple Gmphs
11.6 Graphs
of Classes
10
Spatial Multigraphs
5.IM
Grophh
RUNDIT-MULI--II
MNIIST
10'
ENZYYMES
REDDIT-MMUTT-SK
COLLAB
MUTAA
DBLP vl
MUV
PROTEINS
Toxtee
molpse
10
10
10'
10
10
10
10"
Graphs
Figure I: Number of graphs number of classes in HSG-12M compared to other graph-classification datasets. HSG-12M 12M is the only large-scale multigraph (i.e. unlike simple graph that only al- lows one edge between any node pair) dataset, with exceptional class diversity that even exceeds all other simple graph datasets. T- HSG-5M holds temporal spatial multigraphs. Table A4 lists com- prehensive comparison against 45 other datasets.
Moreover, the Hamiltonian spectral graph is inherently spatial mulrigraph- type of graph re- source fundamentally different from existing datasets. Graph representation learning (Hamilton et al., 2017; Corso et al., 2024) has emerged yet a critical limitation persists: virtually all public benchmarks treat data as simple graphs, allowing at most one edge between any node pair (Hu et al., 2020b; Ranveer & Hiray, 2015; Freitas et al., 2021). Even when source data contains multi-edges, these are typically aggregated into a single attributed edge, discarding crucial spatial information. Due to the absence of such datasets, the development of methodologies for spatial multigraphs has been severely hindered.
powerful paradigm for modeling structured data,
In contrast, many real-world networks are spatial multigraphs, i.e., graphs embedded in metric space, where entities may connect through multiple distinct geometrically meaningful paths. Such spatial graphs geometric networks (Barthélemy, 2011; Guo al., 2021) naturally arise in ban street networks (Kujala et al., 2018; Boeing, 2019), biological neural networks (Weiner et al., 2010; Di Martino et al., 2014). protein structures (Anand & Huang, 2018: Guo et al., 2020), and beyond (Bullmore & Sporns, 2009; Caldarelli, 2007). When the properties of interest include both connectivity topology and connection geometry, collapsing intrinsically distinct multi-edges results in critical information loss.
In mathematical terms, the energy spectrum refers to the set of eigenvalues of the Hamiltonian matrix. Within this work, energy band structure can be considered the same as energy spectrum.
2

---

HSG- 12M addresses this critical gap in graph representation learning, being the first large-scale database of spatial multi-graphs, grounded in non-Hermitian physics. Furthermore, the temporal component establishes the first large-scale temporal (dynamic) spatial graph dataset for graph-level tasks. Our benchmark with popular GNNs expose new challenges in learning spatial multi-edges at scale.
In summary, this work introduces a large-scale spatial multigraph dataset and methodology at the intersection of condensed-matter physies and graph representation learning. Our key contributions include:
1. Open-source, High-pen performance, End-to-End Automated Pipeline. We release Poly2Graph that can map arbitrary ID Hamiltonians to spectral graphs, providing the first automated tool to study spectral graphs with high speed and efficiency. Poly2Craph not only enables us to produce HSG-12M, but also empowers researchers to generate custom spectral graph datasets, vastly expanding the possibilities for future study.
2. Large Scale & Exceptional Class Diversity. 11.6 million static and 5.1 million dynamic spatial multigraphs spanning 1401 classes, distilled from 177 TB of spectral potential data. HSG-12M the first large-scale multigraph dataset for graph-level tasks (Figure A6) with class diversity exceeding all simple graph datasets.
3. Novel Graph Type & New Challenges. Spatial multigraphs simultaneously capture connection topology with edge multiplicity preserved and geometry of space. This first large-scale collection introduces new challenges for developing gcometry-awas graph learning algorithms.
multiedges & nodes the embedding
4. New Domain, Chysics-groundee Universal Relevance. Spectral graphs are firmly grounded in theories of Mer-Hermitian quantum physies, introducing an abundant database from and for an entirely new domain. Physically, spectral graph encapsulates information about quantum state dynamics and topology. Hamiltonian symmetry class, response strength, quantum sensing capa- bility, and more. Thus our database paves the way for accelerating discovery of exotic phases, enabling rational design of materials with desired quantum properties.
Additionally, we identify Hamiltonian spectral graph as new class of topological object de- serving attention in its own right-ins in section we show that vectors, matrices, and polynomials, be they real complex, admit spectral graphs their topological fingerprint, bridging graph and ubiquitous algebra objects.
POLY2GRAPH: AUTOMATING SPECTRAL GRAPH EXTRACTION
Poly2Graph is reegrperformacce and the first end-to-end automated pipeline that converts an arbi- trary one-dimensional crystal Hamiltonian into its spectral graph representation. It operationalizes the mathematical construction reviewed in appendix B by integrating non-Bloch band theory, alge- braic geometry, and morphological image processing.
Full algorithmic details are deferred to appendix C. Here we highlight the design choices that make Poly2Graph 10ax faster and more memory-efficier than the best available code (empirical bench- mark in appendix C.5), therehy enabling the construction of HSG-12M.
From Hamiltonians to Characteristic Polynomials. Poly2Graph initializes with either a Bloch Hamiltonian matrix H or its characteristic polynomial. For a s-band tight-binding crystal chain, the Bloch Hamiltonian reads
9 H(z) Σ TT j=ーP
= kE(TTT)). T₃ C*xs
(I)
Its open-boundary spectrum solely depends on the roots of the Laurent characteristic polyeetommmtiiioyyoiiiiii
9
P(z, E) det [H EII
Σ
(E) n.
We choose an energy domain C C in the complex energy plane (the minimal square) that encloses the entire spectral graph G. By default, Poly2Graph estimates Ω by diagonalising small real-space
3

---

(a)
9,27 Goote
a
r=i
0-12-
Oge7
P(s, B) = det (:)) BL] Σ (.
log (oggla(())
Σ logla,(E)].
(())
(E))
(diii)
(b)
(Ronkin function)
1 ==100.5
(
-1-
Imte)
Re(E)
-2-
1.00
Re(F)
2.80
(dii)
(Density of States)
1.0.1
(50d)
im(()
Sie Ro(E)
Re(E)
(di)
0.5
0.1
1.0
15-
05-
06-
-03
-10-
(d3)
(d2)
(d1)
(d)
Re(E)
Figure 2: Poly2Graph pipeline. (a) Starting from crystal Hamiltonian Hx) in momentum ppace-or, equivalently, its characteristic polynomial P(z, = det (z) ET). The crystal's 's open-boundary spectrum solely depends on P(z, E). (b) The spectral potential &b E) (Ronkin function) is computed from the roots of P(2,E) E) Wang et al., 2024). (c) The density states p(E) obtained
following recent advances in non-Bloch band theory (Tai & Lec, 2023; Xiong & Hu, 2023;
Laplacian of (EE). (d) The spectral graph extracted from p(E) via morphological computer-vistte pipeline. Varying the coefficients of P(z, E) produces diverse graph morphologies in the real domain (dl (d(3) and imaginary domain (di)-(diii).
Hamiltonian with 40 unit cells, though users may optionally specify custom domain and resolution. The resultant domain Ω is discretized into a grid of complex energy values. In HSG- 12M, we used default resolution of 256 (initial) 4 (adaptive enhancement) 1024 points along each axis.
For each sample energy E E Ω2 we solve the roots (x of P(z, E) E) and then sort them by magnitude 2 (E)l < 22 [F) < S (xpty This is the computational roots of an enormous batch (by default 1024" x 10%) of
(treating E as constant)
bottleneck in naive approaches-solvieee polynomials for every grid point is extremely expensive. To tame this bottleneck, we implement custom, optimized root-solver based on Frobenius companion matrices and parallel eigen-solvers with auto-backes backend detection for optional GPU acceleration, cutting wall-tima from hours to milli- seconds.

---

Spectral Potential & Density-of-States (as 2D Images). With the roots {2)(E) computed, leverage non-Bloch band theory (Tai & Lee, 2023; Xiong Hu, 2023; Wang et al., 2024) and reliably compute the spectral potential" as:
(E)) = - log (B)
Σ impt
log| (x
(3)
where Gg (E) is the leading coefficient of the characteristic polynomial. The Laplacian of this po- tential yields the Density of States (DOS):
(L) 1 (E. 2%
(4)
where / = ReE + om Physically, B quantifies the number of eigenstates per unit area energy in the complex plane. p > region traces out spectral graph (Figure 2c). Geometrically, since DOS is defined as the second derivative (curvature), the spectral graph in other words corresponds to the "ridges" of the spectral potential landscape (Figure 2b).
In addition, we exploit inherent symmetries in special polynomials. For example, the complex conjugate root theorem guarantees that if P(z, E) has purely real coefficients, its spectral graph symmetric about the real axis; similarly, purely imaginary coefficients produce symmetry about the imaginary axis. By calculating only the relevant half-plane and mirroring the results, we reduce computation time by up to 50% for qualifying polynomials.
Image-to-Graph Routine. To extract the spectral graph from the DOS image, we binarize the DOS and apply skeletonization to obtain a one-pixel-wiid graph skeleton.
However, we face a resolution resolution-computttion tradcoff: insufficient resolution results in lost topolog- ical features (small loops, adjacent nodes, etc). while uniform high-resolution caleulation across the entire energy domain Ω is prohibitively expensive, especially since the spectral graph typically occupies only small fraction of this area.
We resolve this challenge with a two-stage adaptive resolution approach:
1. Coarse identification: We first compute the DOS on a moderately grid (256 x 256), threshold binarize the image, and perform morphological dilation with 2 x 2 disk. This gen- erates a conservative binary mask that envelops the spectral graph while excluding approximately 95-99% of non-contributive region.
Refined calculation: Within only the masked region, we subdivide each pixel into an m X m grid (default 4), recalculating the spectral potential and DOS this higher resolution. This targeted approach achieves an effective resolution of 1024 x 1024 while computing just 1-5% of the grid points.
The high-resolution DOS is then re-binarized and subjected iterative morphological thinning op- erations (Lee et al., 1994) until a one-pixel-wide skeleton remains, preserving topological features ready be distilled into graph representation.
For the final graph extraction, we analyze this skeleton to identify three point types: (I) junction nodes where three or more paths intersect, (2) leaf nodes where paths terminate, and (3) edge points along continuous segments. The output is an Net workX MultiCe MultiGraph object. Crucially, each edge stores its complete geometrie information as an ordered sequence of (Re( E),, Im(E)) coordi- nates, preserving not just connectivity but the exact shape of each spectral curve.
Quality Assurance and Limitations. We validated Poly2Graph on hundreds of characteristic poly- nomials, by visually checking that the spectral graph from Poly2Graph agrees with the energy spec- trum from exact diagonalization. close to the junction nodes whose surrounding edges have extremely low DOS (see appendix C.6). Poly2Graph will attempt to mitigate such cases by merging nearby nodes and contracting edges shorter than predefined tolerance.
rare complicated cases, numerical instabilities can still arise
'The spectral potential is also known the Ronkin function, an algebro-geometric property of P(z, F)) (Wang al., 2024)
5

---

Open-Source Release and Broader Impact. Poly2Graph' is released under the MIT licence. See tutorial in appendix G. Poly2Graph establishes turn-key mechanism for translating linear operators into machine-learning-reee
graphs, bridging condensed matter physics and graph representation learning. The same principle extends to any vector, matrix, and univariate/bivariate polynomial, opening new "algehra-as-grapp" perspective, broadening the applicability of Poly2Graph wide range of other areas (section 5, appendix F).
3 HSG-12M DATASET DESCRIPTION
The speed and memory efficiency of Poly2Graph make large-scale spatial multigraph research prac- tical for the first time. Figure A&A illustrate the scale of HSG- 12M, showing #graphs vs. #classes and #graphs vs. total #nodes relative to other graph classification datasets. To our knowledge, HSG- 12M is not only the largest dataset by number of graphs and classes but also the only large-scale spatial multigraph dataset available.
Moreover, each graph class corresponds a particular Hamiltonian family (hopping pattern). A well- trained graph neural network could therefore potentially serve surrogate model to predict ma- terial structure from desired spectral graph, thereby facilitating inverse design materials with targeted properties-cn e.g. design of acoustic metamaterial, electrical circuit, or photonic crystal with desired spectral response.
In Table A4 provide comprehensive comparison with existing graph datasets and benchmarks. Most prior popular graph-classification datasets are non-spatial, simple graphs. A few are spatial, e.g., some superpixels and molecular graphs have node coordinates in 2D 3D, but their edges re- main an abstract connection defined by adjacency. HSG-12M uniquely provides spatial multigraphs, where the intricate geometric structure of multi-edges carries essential information that cannot be simplified without loss. The most relevant resource, OpenStreetMap (Boeing, 2019) is much smaller, less diverse, and lacks associated MI. tasks in comparison.
Furthermore, while temporal graph datasets exist (Huang et al., 2023), they typically focus on node/edge-level tasks or involve small numbers of graphs and classes. Our T-HSG-5M represents the first large-scale collection of dynamic graphs for graph-level tasks, capturing the continuous evolution of spectral graphs over Hamiltonian parameters.
Data Format and Accessibility. To maximize accessibility and flexibility, we release HSG-12M un- der a permissive CC BY 4.0 license. The dataset publicly available via Dataverse (Yan al., 2025a). Users download the full dataset select specific subsets using the code provided at ht https: // //gittuub.bb/ com/ barinstein sesteiinyyan.. an/ /HSG- 12M. In companion, we release an auxil- iary package HSG- 2M2 for effortless data handling, benchmark reproduction, custom featurization and dataset generation, interactive tutorial, and more.
The dataset comprises 1401 separate Python npz files, each containing graphs from class with relevant metadata. Raw files use NetworkX Multi BultiGraph format, preserving full node and edge geometry- ① Node atrriulee: complex coordinates, spectral potential, and density of states. ? Edge attributes: edge length (also serving as weight), coordinate sequences along the edge, average spectral potential and average DOS over the edge.
We provide this descriptive format because representation learning on spatial multigraphs remains nascent, with no agreed-upon standard for representing continuous multi-edge geometry. Rather than imposing particular featurization, we encourage researchers to explore various approaches, e.g., treating edge curves as sequences, computing summary features like curvature, or developing novel and more sophisticated neural network-based representations. Moreover, the attribute-rich format here aids interpretability and relevant to researchers interested the underlying physics rather than solely ML.
That said, for convenience, we propose our own featurization scheme and include a conversion API that transforms raw data into PyTorch Geometric (PyG) datasets for graph classification benchmark- ing. Particularly, to manage the inhomogeneity of edge coordinates and make the spectral graphs compatible with standard GNN input, reference conversion uses fixed-sized, irrection-ignorant edge summary features (appendix E) length, the straight-ling distance between start and end nodes, middle point coordinates, average spectral potential, and average DOS along the edge.

---

Key statisties the HSG dataset variants, #Graphs: number of graphs; #Classes: number classes; Ratio: the #Graphs of the largest class/ / #Graphs of the smallest class; Temporal: whether the graphs are tem- poral. All other five datasets are derived from HSG-I2M: thus all datasets are spatial and irreducibly multi- graph. HSG-topol ogy contains non-isomorphic graphs in each class and is the only imbalanced dataset; T-HSG-5M is the temporal spectral graph collection; the rest four teal-colored datasets balanced, static datasets.
Table 1:
Name
#Graphs
#Classes
Ratio
Temporal
198,744
24
1.0
HSG-two-band
2,277.275
275
1.0
HSG-three-band
9,125,662
1102
1.0
HSG-topology
1,812.325
1401
660.2
T-HSG-5M
5,099.640
1401
1.0
HSG-12M
11,601,681
1401
1.0
Dataset Construction. Graphs are grouped by different Hamiltonian families (i.c. characteristic polynomial classes) as detailed in appendix B. We systematically sample polynomial classes while respecting mathematical symmetries to avoid spurious abundance. For instance, if a polynomial exhibits z-reciprocity -ie. P(z) 2 P(1/2) this reciprocal transformation physically means flipping the crystal chain from left to right, which leaves the spectrum unchanged and yields the same spectral graph.
Specifically, we start from base polynomial with hopping range p and energy bands:
P( + P +
(S)
We then set the degree of E* k 10.1...., 1} for each 2" : 1....... 1}.
Subsequently, we assign two free coefficients (a, b) to two chosen monomials x : E {-p + 1,...,- -....,-1,,,, 1,.,,,,..:4 1] excludinng 2 since varying the constant term only raise lower the entire spectral potential landscape, no effect exerted on the spectral graph.
For example, a two-band polynomial with p 3 and q may take the form:
E), - + + (a 2 E + 23, a.beC.
(6)
Under such a sampling scheme, we iterate over all combinations for one-band three-band poly- nomials, with hopping ranges varied from four six. This range has well covered all realistic 1D tight-binding crystals (typically p + q S 4 and less than three bands).
After removing symmetric redundancy, we collect 24 one-band classes, 275 two-band classes, and 1102 three-band classes, amounting to total of 1401 unique classes.
Finally, we vary the two free coefficients from 10 5ů to 10 5i respectively, with 13 real and imaginary values, yielding (13 x 7)2 8281 samples per class.
Dataset Variants. We provide six datasets tailored to different research needs.
HSG-one-band: Small-to-medium scale, the collection of all one-band polynomials, balanced subset with 198,744 graphs across 24 classes. These graphs in this subset display simpler patterns ideal for rapid prototyping and algorithm validation.
HSG-two addd and HSG- band: Medium-to-lare scale, the collection of all two-band and three-band polynomials respectively, balanced datasets with increasing complexity, containing 2.3M and M. graphs across 275 and 1,102 classes, respectively.
HSG-12M: The complete dataset spanning all 1,401 classes with balanced sampling, totaling 1.6M static graphs, designed for large-scale challenge.
HSG-topology: An imbalanced subset preserving only topologically distinct (i.e. non- isomorphic) graphs within each class. This filtered dataset removes isomorphic duplicates, resulting in highly skewed class distributions (max class size ratio 660.2), useful for analyzing spectral graph topology diversity and benchmarking graph algorithms imbalanced datasets.

---

T-HSG-5M: Our temporal multigraph collection capturing continuous spectral graph evolution. As shown in figure 2d, varying either the real or imaginary part of a coefficient in the characteristic polynomial continuously morphs the geometry of the spectral graph: at certain transition points, one can observe the graph topology changes discontinuously. For each class, we collect all sequences of the variation in real (or imaginary) parts of free coefficient, adding up 5.1M temporal graphs across 1401 classes. T-HSG-5M is suitable for evaluating temporal graph-level tasks such as temporal extrapolation and classification on early sequences. Functionality select any desired sequence or subset is provided in the package.
4 BENCHMARKING RESULTS
To assess the capabilities of existing graph learning methods on the new challenges introduced by our HSG datasets, particularly their spatial nature, edge multiplicities, class imbalance, and scale, we benchmark popular GNNs and discuss the implications.
Baseline Models. We benchmark eight popular graph neural networks (GNNs)-GCH (Kipf & Welling, 2017), GAT (Veličković et al., 2018), GATv2 (Brody et al., 2022), GIN (Xu et al., 2019), GINE (Hu et al., 2019), MF (Duvenaud et al., 2015). CGCNN (Xie & Grossman, 2018), and Graph- SAGE (Hamilton et al., 2017).
Experimental Setup. Full details supporting reproducibility are provided in appendix E.2 including data preprocessing and split; model architecture and hyperparameters: optimizer setting and learning rate schedule; hardware and trainer specifics. particular, to ensure fair comparison, for each dataset we tune each model's convolution hidden dimension equalize the total number of learnable parameters, and we cap the training budget by max epochs and max steps.
Evaluation Metrics. Given the high class diversity, we report Top-1 accuracy, Top- 10 accuracy (rel- evant for scenarios where multiple plausible answers are acceptable), and Macro-averaged Fy-score (which weights every class equally and exposes performance minority classes), Additional eval- uation including test loss, Top-5 accuracy, peak GPU memory utilization, throughput are reported in Tables A5-A8.
Results and Analysis. The graph-level classification results are presented in Table 2. Seed vari- is small across variants, indicating stable training. Additional results and analysis are in ap- pendix E.4. Several observations emerge:
Performance degrades with task difficulty. Test metrics decay monotonically from simpler to harder dataset, consistent with increasing graph size, richer multi-edge geometry, more complex isomor- phisms, and growing class diversity. Memory usage likewise grows with complexity (e.g.. SAGE ~0.067 MB/graph).
Edge altributes malter. Edge-aware GINE consistently outperforms edge-agnostic GIN (e.g.. on HSG-12M. Accuracy 0.460+00025 vs. 0.063+0051). reflecting that multi-edge spatial geometry (length, straight-line distance, midpoint, average potential/DOS) carries irreducible signal.
Top-k is high high-promising signal for inverse design. On the full dataset which covers all realistic cases, despite moderate Top-1 accuracy, Top-10 10 accuracy high (e.g., on HSG-12M: SAGE 95. 95.2%, CGCNN 94.8%) and retrrsaturated on easier subsets (99%+ on one-band), enabling retrieval of a small candidate set of Hamiltonian families for downstream expert screening.
GraphSAGE excels under limited budget and compurable parameter county. Under matched pa- rameters (544 variance) and fixed budget metax_eppchs=lle 100, max mex__steps=1000, GraphSAGE attains the best performance all subsets, indicating either stronger inductive bias for spatial multi- graphs or better sample/compute efficiency than more expressive baselines under tight budgets.
Take-away. With our proposed edge summaries, popular GNNs already capture substantial discrim- ination; nonetheless, the Top-1 1 vs. Top-10 gap and degradation at high class diversity suggest that richer edge geometry encoding (e.g.. curvature/torsion, higher-order moments, splinc/Bezier encod- ings, or polyline sequences) could potentially improve performance, especially on the more chal- lenging datasets.
8

---

Table Graph-leve classification results on HSG dataset variants. three random seeds; best result in Bold.
Test metrics shown as t over
Model
Test Metric
one-band
two-band
three-band
topology
HSG-12M
GCN
Accuracy
211
.78
.337
.397..00
.365
Macro Fj
204
4.55.011
.313
.3924.011
.3494.02
Top-10 10p-10 Acc.
.999
816+.011
.825.006
881
GAT
Accuracy
6677 2.002
#.00
.3441.012
.4341.015
.365 2.100
Macro Fr
.67 2,03
.499 2017
.327+
.431:.01
.37
Top-10 Acc.
-998..0
9922 004
.825
.855
.846
GATv2 [6]
Accuracy
2.44
.444
282
.401
.351
Macro
2635
2.00..05
265
.397..001
,300
Top-10 Acc.
.977.001
2.14
.765
.833..005
8835
GIN [109]
Accuracy
.799_.00s
.343
050
.095..09
.63
Macro
.7962.006
.333
.0304.016
.082..060
.042
Top- 10 Acc.
1.000
868
.295
.390
,339
GINE [40]
Accuracy
764_.000
518
.379
.533..017
.460
Macro Fi
.758
.501 +.053
3622
.5314.011
442
Top-10 Acc.
1.000
.872+008
.927
.911
MF|22]
Accuracy
..89
.308
2.00.001
.3484.012
2255..010
Macro Fi
.5722.012
.287
224
.343..012
2714
Top-10 Acc.
.9772000
2838
.754
.7932.010
.791
CGCNN [16]] (106
Accuracy
.7962.008
.585_.029
.71
.5664.016
.511
Macro Fj
.792.2.01
.722
164.1.015
.563..008
5188.003
Top-10 10 Acc.
1.000
995
.23
.940a.005
.48
GraphSAGr [331
Accuracy
.854..m3
.678
.523
.620...a3
.5462.004
Macro Fj
.853 .002
.670 Lz .000
.512
.622
.534
Top-10 Acc.
1.000
.988
940+.006
.958..om
.51
DISCUSSION
Benchmarking and algorithmic opportunities. HSG-12M fills four key gaps at once: the first systematic resource of Hamiltonian spectral graphs, @ is the first large-scale multigraph dataset, @ it is the first spatial multigraph resource, retaining edge multiplicity with rich continuous geometry, and O it provides both static and dynamic multigraph sequences.
These traits open a suite tasks that are under-served by current methods: multi-edge featurization, geometry-awazd message passing, apatio-tempora prediction, etc. Beyond supervised learning, the dataset is sufficiently large and expandable with Poly2Graph to support topology-condittee conditioned gen- eration and pre-training foundation models for rational inverse-design of materials.
Universal Relevance of Spectral Graphs. While HSG-12M rooted in non-Hermitian band theory, its reach extends well beyond condensed-matter physics.
1. Any bivariate Laurent polynomine P(,, E) has spectral graph.
Any univariate polynomial h(z) can be viewed as one-band Bloch Hamiltonia;; and any vector can be treated as symmetrised coefficient list of univariate polynomial.
3. Any matrix be decomposed into product of one-band Hamiltonian matrices (Ye & Lim, 2015), and thus general has a multiser spectral graphs (detailed in appendix F)
Hence polynomials, vectors, and matrices all admit spectral graphs as their topological fingerprints. This establishes universal bridge between algebraic objects and graphs, inviting graph-baser based meth- ods to problems in linear algebra.
Since the characteristics of most scientific systems be expressed as the aforementioned algebraic objects, our approach offers a novel analytical lens. This opens up new avenues for research across numerous fields, direction actively exploring and invite the broader community to join.

---

CONCLUSION
We present Poly2Graph, open-source, gegh-performance pipeline that automatically extracts Hamiltonian spectral graphs, and HSG-12M, a large-scale dataset of 11.6M static and 5.1M dy- namic Hamiltonian spectral graphs. HSG-12M collects physics- grouundd data, offering the first large-scale database for spatial multigraph with irreducible edge multiplicity and spatial geometry. The construction generalizes arbitrary polynomials, vectors, and matrices. Our benchmark popular GNNs indicate the need for methodological advances. We release Poly2Grap and HSG- 12M under permissive licences and invite the community build on this resource for new models, tasks, more comprehensive and carefully designed benchmarks, and insights across machine learn- ing, condensed-matter physies, and beyond.
ACKNOWLEDGMENTS
This research is supported by the Singapore Ministry of Education (MOE) through the Academic Re- search Fund Tier-Il Grants (WBS No. A-8002656-00-00 and Award No. T1 251RES2509). Additional support was provided by the Air Force Office of Scientific Research (Award No. FA2386-24-1-4011
and MOE-T2EP50224-00 0000224-000), the Tier-I grants
REPRODUCIBILITY STATEMENT
(sec- tion 2, appendix C&G), HSG-12M (section 3, appendix D) and an auxiliary dataset pack- age https: ///gggttubb////////////////..hht com
We release Poly2Graph https://git
Saan-yynnHSc-1222 2M under permissive open-source li- cences to facilitate reproducibility and further customization. The packages include detailed docu- mentation and tutorials for easy starting, full reproduction, and extension.
LLM USAGE
We used large language models, such as ChatGPT and Gemini, to aid in the writing of this paper. All generated content has been reviewed and edited by the authors. The authors are solely responsible for any errors or inaccuracies.
ETHICS STATEMENT
This research was conducted an ethical manner, with no direct negative societal impact identified from the implementation of Poly2Graph the creation and benchmark of the HSG-12M dataset. The computation pipeline and dataset are derived from physical and mathematical principles. The authors have no known conflicts of interest or financial incentives beyond standard academic publi- cation.
REFERENCES
H. Altae-Tram B. Ramsundar, A. S. Pappu, and V. Pande. Low data drug discovery with one-shot learning. ACS Cent. Sci., ((4):283-293, 2017.
Namrata Anand and Po-Ssu Huang. Generative modeling for protein structures. In Advances in Neural Infor- mation Processing Systems (NeurIPS), pp. 7505-7516, 2018.
Mare Barthélemy. Spatial networks. Physics Reports, (9911)):1--01,, 2011.
Geoff Boeing. Street network models and measures for every city, county, urbanized area, tract, and zillow-defined neighborhood. Urban Science, 3(28), 2019. ISSN 2413-8851, doi: (0.3390/urbanscc3000000 Karsten M. Borgwardt, Cheng Soon Ong, Stefan Schönauer, S.V.N. Vishwanathan, Alexander J. Smola, and Hans-Peter Kriegel. Protein function prediction via graph kernels. Bioinformatics (Oxford, England), (Suppl 1):14--i54 2005.
Shaked Brody, Uri Alon, and Eran Yahav. How Attentive Graph Attention Networks?, January 2022.
Michael M. Bronstein, Joan Bruna, Taco Cohen, and Petar Veličkovié. Geometric deep lcarning: Grids, groups, graphs, geodesics, and gauges. arXiv preprint arXiv:2104.
2021.
Jérôme Buhl, Jacques Gautrais. Nicholas Reeves, Ricard V. Solé, Sergi Valverde, Pascal Kuntz, and Guy Ther- aulaz. Topological patterns in street networks of self-organized urban settlements. The European Physical Journal 49:513-522, 2006.
Edward Bullmore and Olaf Sporns. Complex brain networks: graph theoretical analysis of structural and functional systems. Nature Reviews Neuroscience, 10:186-198, 2009.
10

---

Landon Butler, Alejandro Parada-Mayorga, and Alejandro Ribeiro. Convolutional learning multigraphs. IEEE Transactions on Signal Processing, 11:933-946, 2023,
Guido Caldarelli, Scale-Free Networks. Oxford University Press, Oxford, 2007.
Giuseppe Carleo, Ignacio Cirac, Kyle Cranmer, Laurent Daudet, Maria Schuld, Nafaali Tishhy, Leslie Vogt- Maranto, and Lenka Zdeborová. Machine learning and the physical sciences. Reviews of Modern Physics, 91(4):045002. December 2019. doi: 10. MMdS/ReeWMd/Phyy...0055000
Nada Chaari. Mohammed Amine Gharsallaoui, Hatice Camgöz Akdağ, and Islem Rekik. Multigraph classifi- cation using learnable integration network with application to gender fingerprinting. Neural Networks, 151:
250-263, July 2022. ISSN 08936080. doi: 10. 10.101 w...00//..neumet.... 20uumet.2022.0..... 2022.03.035
Cai Chen and Yusu Wang. A simple yet effective baseline for non-attributed graph classification. In Interna-
tional Conference Learning Representations (ICIR), 2019.
Chao Chen. Freeway performance measurement system (pems). Technical report, University of California,
Berkeley, 2002.
Gabriele Corso, Hannes Stark, Stefanie Jegelka, Tommi Jaakkola, and Regina Barzilay. Graph neural networks,
Nature Reviews Methods Primers, 4:17, 2024.
Tomasz Danel, Przemystaw Spurek, Jacek Tabor, et al. Spatial graph convolutional networks. arXiv preprint arXiv: 2019.
A. Di Martino, C.-G. Yan, Q. Li, E. Denio, F. X. Castellanos. K. Alaerts, et al. The autism brain imaging
data exchange: Towards a large-scale evaluation of the intrinsic brain architecture in autism. Molecular Psychiatry. 1966)659-6677 2014.
Kexin Ding, Mu Zhou, Zichen Wang, Qiao Liu, Corey W. Amold, Shaoting Zhang, and Dimitri N. Mctaxas. Graph convolutional networks for multi-modality medical imaging: Methods, architectures, and clinical applications, 2022. URL https://arxxtt www.v...oo....ooooooooooo.ooo.rr wow.oorggaaa.o......... 2202 08916. 6.
P. D. Dobson and J. Doig. Distinguishing enzyme structures from non-enzymes without alignments. J. Mol. Biol., pp. 771-783, 2003.
Yuanqi Du. Shiyu Wang. Xiaojie Guo, Hengning Cao. Shujie Hu, Junji Jiang, Aishwarya Varala, Abhinav An- girekula, and Liang Zhao. GraphGT: Machine Learning Datasets for Graph Generation and Transformation. David Duvenaud. Dougal Maclaurin, Jorge aguilera-Iparraguue Rafael Gómez-Boo Bombbaell,, Timothy Hirzel, Alán Aspura-Guzik, and Ryan P. Adams. Convolutional Networks on Graphs for Learning Molecular Fin- gerprints, November 2015.
V. P. Dwivedi, C. K. Joshi, T. Laurent, Y. Bengio, and X. Bresson. Benchmarking graph neural networks. arXiv preprint arXiv:2003.00990
2020.
Vijay Prakash Dwivedi, Ladislav Rampášek, Mikhail Galkin, Ali Parviz, Guy Wolf, Anh Tuan Luu, and Do- minique Beaini. Long Range Graph Benchmark.
Béni Egressy, Luc von Nicderhäusem, Jovan Blanuša, Erik Altman, Roger Wattenhofer, and Kubilay Atasu.
Provably powerful graph neural networks for directed multigraphs. In Proceedings of the 38th AAAI Con- ference on Artificial Intelligence (AAAI), 2024.
Zhening Fang, Mengying Hu, Lei Zhou, and Kun Ding. Geometry-depender skin effects in reciprocal photonic crystals. Nanophotonics, 11(15):34473356, August 2022. ISSN 2192-8614. doi: 10.1515/
nanoph-2022-022 2022-0211. URL http: / aawivv .rgg / abs aba//2/s/000 /2004. 08866.
Scott Freitas, Yuxiao Dong, Joshua Neil, and Duen Homg Chau. A large-scale database for graph reprosentation
learning, 2021.
Song Gao. Geospatial artifiiiaal intelligence (GeoAl), volume 10. Oxford University Press New York, 2021. K. M. Gayvert. N. S. Madhukar. and O. Elemento. A data-driven approach to predicting successes and failures of clinical trials. Cell Chem. Biol., 23(10): 1294-13111 2016.
Zongping Gong. Yuto Ashida, Kohei Kawabata, Kazuaki Takasan, Sho Higashikawa, and Masahito Ueda. Topological phases of non- Hermitian systems. Physical Review X, 803),03177, September 2018. ISSN 2160-3308. doi: 10.1
(etpyysseex...331177.. URL. http://a rxi v .org// aaba/ / 111880200777 07964.
Xiaojie Guo, Sivani Tadepalli, Liang Zhao, and Amarda Shehu. Generating tertiary protein structures via an interpretative variational autoencoder. arXiv preprint, arXiv:2004.071 119, 2020.
Xiaojie Guo, Yuanqi Du, and Liang Zhao. Deep generative models for spatial networks. In Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, KDD '21, pp. 198-208, Virtual Event, Singapore, August 14 18 2021. ACM.
Will Hamilton, Zhitao Ying, and Jure Leskovec. Inductive representation learning on large graphs. Advances in neural information processing systems, 30, 2017.
William L. Hamilton. Graph representation learning. Synthesis Lectures on Artificial Intelligence and Machine Learning, 14(3): 2020.
M. Z. Hasan and C. L. Kane. Colloquium: Topological insulators. Reviews of Modern Physics, 82(4):3045-
3067, 2010. doi: 10. 103//eeMMoPPys..22.... 82..045.
Benjamin Hilprecht and Carsten Binnig. Zero-shot cost models for out-of-thc-boo learned cost prediction, 2022. URL wwt.tttss//axiiii wttp://rgg/ab//////eees
11

---

Emóke-Agnes Horvat and Katharina Anna Zweig. Multiplex Networks, pp. 1430-1434, Springer New York,
New York, NY, 2018. ISBN -79-1-293-- doi: 10.1007/978- 1-1199778--99---111 -11--9993--2222 URL https:
//doi. org/10.100 100.1009778---0
-9--------1111 7131-2255.
W. Hu, M. Fey, M. Zitmik, ..., and Leskovec. Open graph benchmark: Datasets for machine learning on graphs. Proc. NeurIPS. pp. 22118-22132 2020a.
Weihua Hu, Matthias Fey, Hongyu Ren, Maho Nakata, Yuxiao Dong, and Jure Leskovec. OGB-LSC: A Large- Scale Challenge for Machine Learning on Graphs.
Weihua Hu. Bowen Liu, Joseph Gomes, Marinka Zitnik, Pency Liang. Vijay Pande, and Jure Leskovec. Strate- gies for pre-training graph neural networks. arXiv preprint /90Xvv/095./22000 2019.
Weihua Hu, Matthias Fey, Marinka Zitnik, Yuxiao Dong, Hongyu Ren, Bowen Liu, Michele Catasta, and Jure Leskovec. Open graph henchmark: Datasets for machine learning on graphs. In Advances in Neural Information Processing Systems 33 (NeurIPS 2020), 2020b.
Shenyang Huang. Farimah Poursafaci, Jacob Danovitch, Matthias Fey, Weihua Hu, Emanuele Rossi, Jure Leskovec, Michael Bronstein, Guillaume Rabusseau, and Reihanch Rabbany. Temporal Graph Benchmark for Machine Learning on Temporal Graphs, September 2023.
Chukwuemeka Iddianozie and Gavin McArdle. Towards robust representations of spatial networks using graph neural networks. Applied Sciences, I 1115)x6918, 2021, doi: 10.3390/app1 I 1559188.
John Ingraham, Vikas Garg, Regina Barzilay, and Tommi Jaakkola. Generative models for graph-based protein design. In Advances in Neural Information Processing Systems (NeurIPS), pp. 15820-15831, 2019. J. J. Irwin, T. Sterling, M. M. Mysinger, E. S. Bolstad, and R. G. Coleman. ZINC: A free tool discover chemistry for biology. Journal of Chemical Information and Modeling, 52(7):1757- 1768, 2012.
Hosagrahar V Jagadish, Johannes Gehrke, Alexandros Labrinidis, Yannis Papakonstantinou, Jignesh M Patel, Raghu Ramakrishnan, and Cyrus Shahabi. Big data and its technical challenges. Communications of the ACM, 57(7):86-94, 2014.
Jin and Z. Song. Balk-Boundary Correspondence in Non-Ilermitian System in One Dimension with Chiral- Inversion Symmetry. Physical Review B, 99(8):081103, February 2019, ISSN 2469-9950, 2469-9969. doi:
10.1 1..1000yysse w.t..ee........10000.. 103. URL http:/ / /arxivivvvvv.///////aa///aaaa orgg albs//bs/// / 18899.0000 03139.
W. Jin, K. Yang, R. Barzilay, and Jaakkola. Learning multimodal graph-to-graph translation for molecular optimization, 2018
John Jumper, Richard Evans, Alexander Pritzel, Tim Green, Michael Figurnov, Olaf Ronneherger, Koray Tun- yasuvunakool, Russ Bates, Augustin Zídek, Anna Potapenko. Charlic Bridgland, Clemens Meyer, Simon A. A. Kohl. Andrew J. Ballard, Andrew Cowie, Bernardino Comera-Paredees Stig Nikolov, Rishub Jain, Jonas Adler. Tom Back. Stig Petersen, David Reiman. Ellen Clancy. Michal Zielinski, Martin Steinegger, Marta Pacholska, Thomas Berghammer. Sebastian Bodenstein, David Silver, Oriol Vinyals, Andrew W. Se- nior, Koray Kavukcuoglu, Pushmeet Kohli, and Demis Hassabis. Highly accurate protein structure prediction with alphafold. Nature, 596(7873):583-555 589, 2021. ISSN 1476-4687. doi: 10. 10.1038/s41 (1388/411866-11 021 021-03819-2, Kohei Kawabata, Ken Shiozaki, Masahito Ueda, and Masatoshi Sato. Symmetry and Topology in Non- Hermitian Physics. Physical Review X. 944)0010155, October 2019. ISSN 2160-3308. doi: 10.1103/ PhysRewx.9.0410155 URL /ttp:://axiiii /tt:///axvvvv///// org/ aaa////s/////
Will Kay, Joao Carreira, Karen Simonyan, Brian Zhang, Chloe Hillier. Sudheendra Vijayanarasimhan. Fabio Viola, Tim Green, Trevor Back, Paul Natsev, et al. The kinetics human action video dataset. arXiv preprint arXiv: (kKiv:/705.069400 06950, 2017.
Yung Kim, Sonu Verma, Minwook Kyung, Kyungmin Lee, Wenwen Liu, Shuang Zhang, Bumki Min, and
Moon Jip Park. Boundary-Driven Complex Brillouin Zone in Non-Hermitian Electric Circuits, February 2025. URL http: / www.xvvvrrg........ 2202 http//222.111......11
Thomas N. Kipf and Max Welling. Semi-Supervised Classification with Graph Convolutional Networks, Febru-
ary 2017.
Thomas N. Kipf, Ethan Fetaya, Kuan-Chieh Wang. Max Welling, and Richard Zemel. Neural relational infer- ence for interacting systems. In International Conference on Machine Learning, pp. 2688-2697, 2018. Jin Ming Koh, Wen-Tan Xue, Tommy Tai, Dax Enshan Koh, and Ching Hua Lee. Interacting non-Hermitian edge and cluster bursts on a digital quantum processor, March 2025. URL http:://p///iiii orgg /
2503.1 14595.
N. Kriege and P. Mutzel. Subgraph matching kernels for attributed graphs. J. Mach. Learn. Res., 12:291-296 2012.
A. Krizhevsky and G. Hinton. Learning multiple layers of features from tiny images. In Technical Report, University of Toronto, 2009.
Rainer Kujala, Christoffer Weckström, Richard Darst, Milos Mladenovic, and Jari Saramäki. A collection of public transport network data sets for 25 cities. Scientific Data, 5:180089, May 2018. doi: 10. 10.. 2018.89.
Flore K. Kunst, Elisabet Edvardsson, Jan Carl Budich, and Emil J. Bergholtz. Biorthogonal Bulk-Bounde Correspondence in non-Hermitian Systems. Physical Review Letters, 121(2)0268088 July 2018 ISSN
12

---

0031-9007, 1079-7114. doi: 10.1003/PyysReeLeet12...06666 URL http::/aaxiv..rrtteee 1805.06492.
Y. LeCun. The mnist database of handwritten digits. httt:://yamt www.lleeccccnnn....eeeeoe.ee.eee wwwp///bb//nnss///.///c 1998.
Ching Hual Ice and Ronny Thomale. Anatomy of skin modes and topology in non-Hermitian systems. Physical
Review B, 99(20):201 103, May 2019. ISSN 2469-9950, 2469-9969.
doi: 10.1 101///ysseev..99...m 103.
URL http:// /atp/v/axvvv.////////// org/ /1899 00111990122........
Ching Hua Lee. Linhu Li, Ronny Thomale, and Jiangbin Gong. Unraveling non- Hermitian pumping: Emergent spectral singularities and anomalous responses. Physical Review B. 102(8)085151, August 2020. ISSN 2469-9950, 2469-9969. doi: 10.1 103/PhysRevB....m 102.085151 06974.
URL http: http:////xxxvvoovo httpgg//b///////////////
T.C. Lee, R.I., Kashyap, and C.N. Chu, Building Skeleton Models via 3-D Medial Surface Axis Thinning Algorithms. CVGIP: Graphical Models and Image Processing, 56(6))462-478, November 1994, ISSN 10499652. doi: 10. (w00//cgp..199.1100
Huan Li. Hao Zheng. Ting Yue, Zhi Xie, Shanshan Yu. Jun Zhou, Tarun Kapri, Yiming Wang. Zhi Cao, Hong Zhao, Akerke Kemelbay, Jie He. Guojing Zhang. Paul F. Pieters, Eric A. Dailing, James R. Cappiello, Miquel Salmeron, Xiaodan Gu, Ting Xu, Peng Wu, Yuzhang Li, Karl B. Sharpless, and Yi Liu. Machine leamingg-accelerated discovery of heat-resistant polysulfates for electrostatic energy storage. Nature Energy, 10(1):90-100, 2025. ISSN 2058-7546. doi: 10. 16080444656-002---6777--
Linhu Li and Ching Hua Lee. Non-Hermitian pseudo-gaps. Science Bulletin, 67(7):685-660, April 2022. ISSN 2095-9273, doi: 10.1016/j.scibb wtt.:i...222200.111.. URL wtt.ttttp//st..... www.ssseiieeeeeee........ wwt.:/twtttteeteeeeeooeoeeeeeeee science/
7322000172.
Linhu Li. Sen Mu, Ching Hua Lee, and Jiangbin Gong. Quantized classical response from spectral wind- ing topology. Nature Communications, 12(1):5294, September 2021. ISSN 2041-1723. doi: 10.1038/
641467-021-2562--1 URL, http://///
08799.
Rijia Lin, Tommy Tai, Mengjie Yang, Linhu Li, and Ching Hua Lee. Topological Non-Hermitian skin ef- fect. Fronteers of Physics, 18(5):33605, October 2023. ISSN 2095-0462, 2095-0470. doi: 10.1007/
$11467-023-1300-z
nly Loshchilov and Frank Hutter.
Sgdr: Stochastic gradient descent with warm restarts. arXiv preprint
arXiv:1608.08: 03983, 2016.
Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. arXiv preprint 171000111..05101.
2017.
Yao Ma and Jiliang Tang. Deep Learning on Gruphs. Cambridge University Press, 2021.
L F. Martins, A. L. Teixeira, L. Pinheiro, and A. O. Falcao. A bayesian approach to in silico blood-brain barrier penetration modeling. J. Chem. Inf. Model., 1696-666666677 2012.
David Mendez et al. ChEMBL: Towards direct deposition of bioassay data. Nucleic Acids Research, 47(D1):
D930-D940, 2019.
Haiyu Meng, Yee Sin Ang, and Ching Hua Lee. Generalized Brillouin Zome Fragmentation, February 2026.
13275. A. Merchant, S. Batzner, Samuel S. Schoenholz, Muratahan Aykol, Gabin Cheon, and Ekin D. Cubuk. Scaling deep learning for materials discovery. Nature, 624(7990):80 85, 2023. ISSN 1476-4687. doi: 10.1038/
URL http:// / www.v..rr/bbbbee
s41586-023-00735-
C. Morris, F. Bause, M. Neumann, K. Kersting, and P. Mutzel. Tudataset: A collection of benchmark datasets for learning with graphs. CoRR, ahs/2007008663, 2020.
Juan Nunez-Iglesias, Adam J. Blanch, Oliver Looker, Matthew W. Dixon, and Leann Tilley. A new Python library analyse skeleton images confirms malaria parasite remodelling of the red blood cell membrane skeleton. PeerJ. 6:c4312, February 2018. ISSN 2167-8359. doi: 10.7717/peerj.44321 Fivos Papadimitriou. Spatial artificial intelligence.
Phitchaya Mangpo Phothilimthana, Sami Abu-El-Haija, Kaidi Cao, Bahare Fatemi, Mike Burrows, Charith Mendis, and Bryan Perozzi. TpuGraphs: A Performance Prediction Dataset on Large Tensor Computational Graphs.
Jinghui Pi, Chenyang Wang, Yong-Chun Liu, and Yangyian Yan. Geometric origin of self-intersection points in non-Hermitian energy spectra. Physical Review B, 111(16))16507, April 2025, ISSN 2469-9950, 2469- 9969. doi: 10. 1033P0yyssssvB.... 1. 1654407 URL http:/ /wtpiv.vvvvv.////////a.a/aaaaava /rg/ abs/ 2202055022. 04755.
Daniil Polykovskiy et al. Molecular sets (MOSES): A benchmarking platform for molecular generation models. Frontiers in Pharmacology. 11, 2020.
R. Ramakrishnan, P. O. Dral, M. Rupp, and O. A. Von Lilienfeld. Quantum chemistry structures and properties
of 134 kilo molccules. Scientific Data, 1(1):1-7, 2014.
Smita Ranveer and Swapnaja Hiray. Comparative analysis of feature extraction methods of malware detection. International Journal of Computer Applications. 120:1-7 June 2015. doi: (0.5120/21220-360)
A. M. Richard, R. S. Judson, K. A. Houck, and Toxcast chemical landscape: Paving the road to 21st century toxicology. Chem. Res. Toxicol. 12255 2016.
13

---

Ignacio Rodriguez-Iuure and Andrea Rinaldo. Fructal River Basins: Chance and Self-Organization. Cam- bridge University Press, Cambridge, 1997.
Sebastian G Rohrer and Knut Baumann. Maximum unbiased validation (MUV) data sets for virtual screening based on PubChem bioactivity data. Journal of Chemical Information and Modeling, 49(2)169-184 2009, Adrian Runions, Anne M. Fuhrer, Peter Federl, Brendan Lane, Anne-Gaëlle Rolland-L and Przemyslaw Prusinkiewicz. Modeling and visualization of leaf venation patterns. ACM Transactions Graphics (TOG),
24(3):00--71, 2005.
Anwar Said, Roza G. Bayrak, Tyler Dert, Mudassir Shabbir, Daniel Moyer, Catie Chang. and Xenofon Koutsoukos. Neurograph: Benchmarks for graph machine learning in brain connectomics, 2024. URL.
https: /arxi v. wwrorr/abbeee. at//ab//2300 2200 06202.
Amir Shahroudy, Jun Liu, Tian-Tsong Ng, and Gang Wang. Nuu rgh+d: A large scale dataset for 3d human activity analysis. In IEFF Conference on Computer Vision and Pattern Recognition, pp. 1010-1019, 2016. Ruizhe Shen, Tianqi Chen, Bo Yang. and Ching Hua Lee. Observation of the non- non-Hermitian skin effect and Fermi skin on a digital quantum computer. Nature Communicationss 16(1):1340, February 2025. ISSN 2041-1723. doi: 10. 10.1038/s41467-000 025-55535-5553-4 4. URL http: http:://arxivi org/ abs /211 .00143.
G. Subramanian, B. Ramsundar, Pande, and R. Denny. Computational modeling b-serrttase (bace-1) 1) inhibitors using ligand-based approaches. J. Chem. Inf. Model., 56(10): 1936-1999, 2016.
Tommy Tai and Ching Hua Lee. Zoology of non-Hermitian spectra and their graph topology. Physical Review
B, 107(22)2220001, June 2023. ISSN 2469-9950, 2469-9969. doi: 10.1 10330hsseeB..1100 1.7.....0...200001..
Jie Tang. Jing Zhang, Limin Yao, Juanzi Li, Li Zhang, and Zhong Su. Arntminne: extraction and mining of academic social networks. In ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp.
2008.
Giacomo Torlai, Gugliclmo Mazzola, Juan Carrasquilla, Matthias Troyer, Roger Melko, and Giuseppe Carleo. Neural-network quantum state tomography. Nature Physics, 14, May 2018. doi: 10.1038/ -415677-11-0048-5.
Tox21 Data Challenge. Tox21 data challenge 2014. https://tripod nih wtw.th..oooooooooeeo chapse//aaaaeen.g.e..mmmm 2014.
Mihaly Varadi, Damian Bertoni, Paulyna Magana, Urmila Paramval, Ivanna Pidruchna, Malarvizhi Radhakrish- nan, Maxim Tsenkov, Sreenath Nair, Milot Mirdita, Jingi Yeo, Oleg Kovalevskiy, Kathryn Tunyasuvunakool, Agata Laydon, Augustin Zidek, Hamish Tomlinson, Dhavanthi Hariharan, Josh Abrahamson, Tim Green, John Jumper, Ewan Birney, Martin Steinegger, Demis Hassabis, and Sameer Velankar. AlphaFold Protein Structure Database in 2024: Providing structure coverage for over 214 million protein sequences. Nucleic Acids Research, 52(DI):D368-D375 2024. doi: 10.10///aa///
Petar Veličković, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua Bengio. Graph Attention Networks, February 2018.
Anliang Wang, Xiaolong Yan, and Zhijun Wei. Imagepy: an open-source, python-based and platform- independent software package for bioimage analysis. Bioinformatics, 34(18))3238-3244 2018
Aoxi Wang and Chang Qing Chen. Observing non-Bloch braids and phase transitions by precise manipu- lation of the non-Hermitian boundary and size. Communications Physics, 8(1):294, July 2025. ISSN
2399-3650. doi: 1003:///2200002-002222
URL https:/ / www.w/w............... http://ww.ntttt.. nature com/ articles/
Chenyang Wang, Jinghui Pi, Qinxin Liu, Yaohua Li, and Yong-Chun Liu. General theory for geometry- dependent non-Hermitian bands, November 2025a. URL http: / / aarxiv www.viow.g/..o.ooo..oooo.oooooot /otg//b////ta /206/ 22743. Heming Wang, Janet Zhong, and Shanhui Fan. Topology of the generalized Brillouin zone of one-dimensional models, October 2025b. URL http::////// //tp/////vv/////// org horg/abs als /2510 O. 0722
Hong-Yi Yi Wang, Fei Song. and Zhong Wang. Amoeba Formulation of NlocBloch Band Theory in Arbitrary Dimensions. Physical Review X. 14(2)021011, April 2024. ISSN 2160-3308. doi: 10.1103/hhsReexx.144 14. 021011.
Yaoli Wang, Di Zhu, Ganmin Yin, Zhou Huang, and Liu. A unified spatial multigraph analysis for public transport performance. Scientific Reports, 10:9573, 2020. doi: 10. 19.1088/441988-0000 020-66175xx.
M. W. Weiner, P. S. Aisen, Jr. Jack, C. R., W. J. Jagust, J. Q. Trojanowski, L. Shaw, al. The Alzheimer's disease neuroimaging initiative: Progress report and future plans. Alzheimer's 's & Dementia, 603)222-- 2010.
Lingfei Wu, Peng Cui, Jian Pei, Liang Zhao, and Le Song. Graph neural networks. In Graph Neural Nenworks: Foundations, Frontiers, and Applications, pp. 27-37, Springer, 2022.
Zhenqin Wa, Bharath Ramsundar, Evan Feinberg, Joseph Gomes, Caleb Geniesse, Ancesh Pappu, Karl Leswing, and Vijay Pande. MoleculeNet: benchmark for molecular machine learning. Chemical Science,
9():513-530, 2018
Tian Xie and Jeffrey C. Grossman. Crystal graph convolutional neural networks for an accurate and inter- pretable prediction of material properties. Physical Review Letters, 120(14):145301, April 2018 doi: 10. 1ew111/o//sselcctttt 120 145301.
14

---

Yuncheng Xiong and Haiping Hu. Graph Morphology of Non-Heermitian Bands, November 2023.
Yuncheng Xiong, Ze-Yo Xing, and Haiping Hu. Non-Hermitian skin effect in arbitrary dimensions: Non-Bloch band theory and classification, July 2024. URL http: / /arxiv pow.v.ore www.v.owg//....oooo......o.oo /aa/////// / /2407. 700700000 00109966 26.
Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How Powerful are Graph Neural Networks?, February 2019.
Mingtao Xu, Zongping Gong. and Wei Yi. Optimal spectral transport of non- non-Hermitian systems. Physical
Review B. 111221:224305, June 2025. ISSN 2469-9950, 2469-9969. doi: 10. 10.1103/phtm- 2hb2. URL
http:// //axiiv wwtp///g.///...oeeeeeeeeeeeeee/e
15209.
Sijie Yan. Yuanjun Xiong, and Dahua Lin. Spatial temporal graph convolutional networks for skeleton baseed action recognition. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018. Xianquan Yan. Automated metric discovery: Navigating quantum geometry with symbolic regression. In A14X 2025 International Conference, 2025. URL https: / /ooenreviies ew net / forum? i dd-E0fa 4FXUMM.
Xianquan Yan and Jian-Song Pan. Precise identification of topological phase transitions with eigensystem- based clustering. In AI4X 2025 International Conference, 2025. URL ht /ttp:////ppooeoeeeeee/ooo nnt// forum? ? i BB-
Xianquan Yan, Hakan Akgün, Kenji Kawaguchi, Duane Loh, and Ching Hua Lec. HSG-12M https://
doi. .at..org/110 /0/./10....0 2...799100 7910
2025a. URL https: / /tt//ii.oorrrroor wogg/.r//...... /00. 7910/DVV/ PDDSSO.
Xianquan Yan. Hakan Akgün, Kenji Kawaguchi, N Duane Loh, and Ching Hua Lee. Classifying the graph
topology of non-hermitian energy spectra with graph transformer. In Second Workshop on Xai4science: From Understanding Model Behavior Discovering New Scientific Knowledge, 2025b. URL https: // /pperreviie net/ forum?idd- simmmndddsssxxeee
Pinar Yanardag and SVN Vishwanathan. Deep graph kernels. In Proceedings of the 21st ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1365-1374, ACM, 2015. Russell Yang, Jun Wei Tan, Tommy Tai, Jin Ming Koh, Linhu Li, Stefano Longhi, and Ching Hua Lee. De- signing non-Hermitian real spectra through electrostatics. Science Bulletin, 67(18):185--1855 September 2022. ISSN 20959273. doi: 10. (o.1016jjseibb22222 08.00...
Sheng Yang, Fengge Wu, and Junsuo Zhao. MBDS: A multi-body dynamics simulation dataset for graph networks simulators. October 2024. doi:
Zhenyu Yang, Ge Zhang, Jia Wu, Jian Yang. Quan 7., Sheng, Shan Xue, Chuan Zhou, Charu Aggarwal, Hao Peng, Wenbin Hu,, Edwin Hancock, and Pictro Liò. State of the Art and Potentialities of Graph-level Leam- Leamm
ing, May 2023.
Zhesen Yang. Kai Zhang, Chen Fang. and Jiangping Hu. Non-Hermitian bulk-boundary correspondence and auxiliary generalized Brillouin zone theory. Physical Review Letters, 125(22):226402, November 2020. ISSN 0031-9007, 1079-7114 14. doi: 10. 1..110/PhyyReeeeLt..25.....
Ke and Lek- Heng Lim. Every matrix is product of Toeplitz matrices. Foundations of Computational Mathematics, 15(1): 2:12 2015. doi: 10.
Taki Youssef, Elmoukhtar Zemmouri, and Anas Bouzid. STM-GCN: A spatiotemporal multi-graph convolu- tional network for pedestrian trajectory prediction. The Journal of Superconputing. 79(18):20923-2093) December 2023. ISSN 9920-8542, 1573-0484. doi: 10. 1tt1:/112227002-0554442
Miao Zhang, Yue Zhang, Shuai Li, Rui Tian, Tianhao Wu, Yingchao Xu, Yi-an Li, Yuanbang Wei, Hong Gao,
M. Suhail Zubairy, Fuli Li, and Bo Liu. Observation of non non-Hermitian bulk- bolldaboundry correspondence in non-chiral non non-unitary quantum dynamics of single photons, April 2025. URL http: http:///xxvvvv/eee ab5/22044 04882.
Lianmin Zheng, Ruochen Liu, Junru Shao, Tianqi Chen, Jaseph Gonzalez, Ion Stoica, and Ameer Haj-Ali. Tenset: A large-scale program performance dataset for learned tensor com- pilers. In J. Vanschoren and S. Yeung (eds.), Proceedings of the Neural Information Pro- cessing Systems Track on Datasets and Benchmarks, volume 1, 2021. URL https: (ew.eeeeee66f52222
nwurips. saraper_ricee wwtps/ippeeeeeeeeeeeeeeea 8 Pawpcca4---ppeerroooo .ddf.
Jia-Xin Zhong, Jeewoo Kim, Kai Chen, Jing Lu, Kun Ding, and Yun Jing. Experimentally Probing Non- Hermitian Spectral Transition and Eigenstate Skewness. Physical Review B, 112(22):1.20001, December 2025. ISSN 2469-9950, 2469-9969. doi: 10.1 103/xxp9-ng7e. URL http: /ttpp://aaxvvv..o wttpaaawwwv....//////////0 08160.
M. Zitnik, R. Sosic, M. W. Feldman, and J. Leskovec. Evolution of resilience in protein interactomes across the tree of life. Proc. Natl Acad. Sci. U.S.A., 116(10):446 4433, 2019.
15

---

APPENDIX TABLE OF CONTENTS
A Related Work
17
A.1 Graph Representation Learning, Datasets, and Benchmarks.
17
A.2 Graph Learning in Multigraphs.
17
A.3 Graph Learning in Spatial Graphs
17
A.4
Non-Bloch Band Theory
18
Mathematical Background Hamiltonian Spectral Graph
18
B.1
The Hamiltonian and Energy Spectrum in 1D Tight-Binding Systems
18
B.2
Hamiltonian Spectral Graph: Emergent Topology in the Thermodynamic Limit
19
B.3
Theoretical Framework: Non-Bloch Band Theory
19
B.3.1
The Bloch Hamiltonian and Characteristic Polynomial P(z, E)
19
B.3.2
Limitations of Standard Bloch Theory (PBC)
21
B.3.3
Non-Bloch Band Theory and the Generalized Brillouin Zone (GBZ)
22
B.3.4
Characteristic Polynomial Class Cr
22
B.3.5
Recap of Key Concepts (Hierarchy of Abstractions)
23
B.4
The Shortcut Spectral Graph via Electrostatic Analogy
24
B.4.1
Density of States pEE) and the Spectral Potential (
24
B.4.2
Efficient Calculation of ((E)
24
Poly2Graph Pipeline Details
24
C.I Initialization and Input
25
C,2 Accelerated Root Finding
................................
26
C.3 Adaptive Resolution and Image Processing
26
C,4 Graph Extraction and Post-processing
27
C.5 Benchmark Poly2Graph Speed-Up
27
C.6 Caveats: Component Fragmentation
27
D Dataset Details
28
D.I Comparison with 45 Other Datasets
D.1.1
Comparison with Other Datasets in Physical Sciences
28
D.2 Clarification on tgg Dataset Variant
28
Benchmark Details
30
E.1
Data Preprocessing (including Spatial Multiedge Featurization)
30
E,2 Training Configuration
30
E.3
Evaluation Metrics
32
E,4
Additional Benchmark Results and Analysis
32
E.5
Physical Motivation of the Benchmark Task
35
16

---

F
Universality of Spectral Graphs Through Toeplitz Decomposition
35
G Tutorial of Poly2Graph Package
36
G.I Features
36
G.2 Installation
36
G.3 Usage
37
G.3.1 A generic one-hand example (p2g. aa
37
G.3.2
A generic multi-band example (p2g. Spectralrraapph)
G.3.3 Node and Edge Attributes of the Spectral Graph Object
44
G.3.4 A generic multi-band class (pag.chhrPPlyylaass)
45
H Limitations and Future Work
48
A RELATED WORK
A.1 GRAPH REPRESENTATION LEARNING, DATASETS, AND BENCHMARKS.
Graph learning has seen a rapid rise in recent years, driven by advances in graph neural networks (GNNs) (Hamilton, 2020; Bronstein et al., 2021; Ma & Tang, 2021; Wu et al., 2022; Corso et al., 2024) and proliferation of datasets and benchmarks (Ranveer & Hiray, 2015; Freitas et al., 2021; Chen & Wang, 2019; Yanardag & Vishwanathan, 2015; Hu et al., 2020b). HSG-12M addresses critical gaps in existing benchmarks by introducing not only the first large-scale spatial multigraph dataset but also one of the largest known graph machine learning datasets and natural science-based datasets. This work sets a new standard in terms of scale and class diversity.
A.2 GRAPH LEARNING MULTIGRAPISS
In contrast to simple graphs, multigraphs permit multiple edges between the same pair of nodes. Apart from a handful of exploration on multigraph learning algorithms (Butler et al., 2023; Egressy et al., 2024), progress has been hampered by the absence of large-scale data sources.
In many practical settings, multiple edges are typically collapsed into a single edge- often sacri- ficing valuable information. This simplification may be acceptable when edge-level details can be represented as aggregated attributes, as is often the case in heterogeneous graphs (Chaari et al., 2022; Youssef et al., 2023), multi-module models (Said et al., 2024; Ding et al., 2022), or multiplex networks (Horvat & Zweig, 2018).
However, in spatial multigraphs (Barthélemy, 201 1), where edges carry rich geometric information such as distances, directions, or physical observable information, such aggregation results in signif- icant information loss. This critical issue has remained underexplored due the lack of datasets where edge aggregation is inherently infeasible. HSG-12M addresses this gap by providing the first benchmark where capturing both multi-edge relationships and edge geometry is essential.
A.3 GRAPH LEARNING IN SPATIAL GRAPHS.
spatial (or geometric) graph a network in which nodes and edges are spatial entities living in a metric space (Barthélemy, 201 I; Iddianozie & McArdle, 2021). Such networks emerge naturally in domains where spatial embedding is fundamental to structure and function: urban, transportation, and communication networks shaped by physical distances and road geometries (Buhl et al., 2006; Wang et al., 2020); biological systems like neural and vascular networks are constrained by surrounding tissue geometry (Runions et al., 2005; Bullmore & Sporns, 2009); and river networks
TTo our knowledge, this is also the first large-scale multigraph dataset-large-scalee conforms to OGB crite- ria (Hu al.).
17

---

evolve through interactions of gravity and topography (Caldarelli, 2007; Rodriguez-lurbe & Ri- naldo, 1997). In all these cases, spatial graph structure encodes essential information that cannot be inferred from connectivity alone reconstructed from non-spatial data.
Despite growing recognition of spatial information in Spatial and Geo AI (Papadimitriou; Gao, 2021), its importance remains underappreciated in graph learning. Currently, benchmark exists with sufficiently rich geometric structure to exhibit intricate spatial patterns, let alone one of spatial multigraphs. As result, despite many efforts to develop algorithms for spatial graphs (Guo al., 2021; Iddianozie & McArdle, 2021; Danel et al., 2019; Ingraham et al., 2019; Yan et al., 2018), the field has lacked a standardized, large-scale scale testbed.
A.4 NON-BLOCII BAND THEBOYY
Non- Non-Hermitian band theory has developed mature language for classifying complex spectra and their topological phases, including point--line-gap notions, symmetry-baan -based classifications, and as- sociated boundary phenomena (Lee & Thomale, 2019; Kawabata et al., 2019; Gong et al., 2018; Li Lee, 2022). central theme the breakdown (and restoration) of bulk-boundary correspondence via non non-Bloch descriptions, which underpin the non- non-Hermitian skin effect (Kunst et al., 2018; Jin & Song, 2019; Yang et al., 2020; Lin et al., 2023; Xiong et al., 2024). Recent advances further sharpen the geometric viewpoint complex band structures through generalized Brillouin-zone construc- tions and higher-dimensional non-Blocd theory, e.g., the amoeba formulation (Wang et al., 2024; 2025b; Kim et al., 2025; Meng et al., 2026). parallel, there growing interests "spectral ge- ometry" and morphology in the complex plane-from graph-topological characterizations of spectra and band morphology the structure of self-intersections and spectral transitions (Tai & Lee, 2023; Xiong & Hu, 2023; Yan et al., 2025b; Pi et al., 2025; Zhong et al., 2025). Real-space geometry dependence also appears in experimental settings and in general theories (Fang al., 2022: Wang et al., 2025a). Beyond classification, there exists dynamical and response-based viewpoints (Li et al., 2021; Lee al., 2020; Xu et al., 2025). Finally, growing experimental and ture has begun to directly probe these effects (Wang & Chen, 2025; Zhang et al., 2025; Shen et al., 2025; Koh et al., 2025; Yang et al., 2022).
litera-
B MATHEMATICAL BACKGROUND -amillonnia Spectral Graph
B. THE HAMILTONIAN AND ENERGY SPECTRUM ID TIGHT-BINDING SYSTEMS
In physical sciences, is customary to represent and study system through its Hamiltonian matrix. The energy spectrum, which refers to the of eigenvalues of this matrix, reveals the energy band structure-2 central object of study in condensed matter physics. Let us consider generic ID tight-binding Hamiltonian with internal degrees freedom (bands or orbitals) per unit cell:
Σ z,J
where and j index unit cells and hopping lengths, respectively; é is the annihilation operator (vector of length s) for the æ-th unit cell. Each term T; E C*** represents the transition amplitude matrix for a particle hopping from site + j æ. The L norm of the amplitude corresponds the transition probability. These hopping terms can generically be complex.
The matrix representation of this Hamiltonian in real space, Hreal: for which the block at (x. x²) is H (Heal)a,rr Tr--zs - is block Toeplitz matrix- matrix in which each descending block diagonal from left to right is constant:
To
T
T
T-1
Tb
T
T2
T-2
T
Tg
T
real
T-2
T-1
To
(8)
T2
Tg
T
0
T-2
T
To
18

---

2
(a) L-
(b) L 150
(c) DOS in TDL (L->]
Im(E)
0 Re(E)
-2
2 -2
0 Re(E)
2
Re(E)
Figure A3: The emergence spectral graphs. (a)-(b) show the OBC energy spectra with creasing system size L = [50, 150). of the non-Hermitian lattice whose characteristic polynomial is P(z, E) = -
E - 2 + 2t. In the thermodynamic limit (L + o0), the spectra becomes band continuum and the energy loci traces out a planar graph the complex plain, namely the spectral graph. For this particular example, it is a 3-Cayley tree. (c) shows the corresponding density of states when L +
there are L sites in total, H real CLaxLs, In general, T, + Ti (where TI, is the conjugate transpose of J))), which breaks the Hermiticity of the Hamiltonian, i.e., Ht + H. Consequently, the eigenvalues can take on complex values. The energy spectrum is obtained by diagonalizing Heall
B.2 HAMILTONTAN SPECTRAL GRAPH: EMERGENT TOPOLOGY IN THE THERMODYNAMIC LIMIT
For non-Hermitian systems, the energy eigenvalues form intricate patterns in the complex plane. The spectral graph 9 emerges from the energy spectra under open boundary conditions (OBC) in the thermodynamic limit (i.e., the system size L oo). In this limit, the discrete energies become continuous, and their loci trace out a planar graph on the complex plane (Tai & Lee, 2023; Xiong Hu, 2023). Figure A3 illustrates this emergence: the OBC energy spectra for finite system sizes L 50 and L = 150 of a non-H -orreeiii lattice (whose characteristic polynomial, defindd later, P(z, E) =
z + 2 clearly approach Cayley tree increases. Figure A3c shows the corresponding density of states (DOS) in the T, - oo limit.
The spectral graphs of different lattices exhibit kaleidoscope of geometries, including ares, loops, and more exotic shapes resembling stars, kites, braids, and even rockets (Tai & Lee, 2023; Lin al., 2023), as showcased in Figure A4. These structures represent an uncharted band topology, embedding hidden symmetries and graph topological transitions that lie beyond standard homotopy- based frameworks (Hasan Kane, 2010). effect, a new class of topological invariants appears- those tied to the global geometry of the eigenvalue loci.
However, accurately diagonalizing large non-Hermitian matrix Hreal to obtain the OBC spectrum notoriously difficult (Yang et al., 2020), let alone for infinite-sized matrix (i.c. operator). This necessitates a more sophisticated theoretical approach.
B.3 THRORETICAL FRAMEWORK: NON-BLOCH BAND THEORY
The standard approach analyze such systems, guided by non-Bloch band theory, begins with Fourier transformation and the analysis of the resulting characteristic polynomial.
B.3.1
THE BLOCH HAMILTONIAN AND CHARACTERISTIC POLYNOMIAL P(2,
Fourier transforming the real-space Hamiltonian (second quantized form equm-inuation or its matrix form- equation form-equation 8) yields the Bloch Hamiltonian:
H ==)== TT z:=
19

---

1.3
6.5
Figure A4: A Gallery of Spectral Graphs. The top four rows highlight the intricate structures characteristic of spectral graphs. The bottom illustrates the distinct phenomenon we refer to as component fragmentation (Section 5) some nodes in theory should be connected, however its surrounding low density of states limits accurate edge detection, causing certain nodes frag- mented into disjoint nodes, often leading to fragmentation of an otherwise connected component. The phenomena often occurs for high-band and long-range hopping crystals.
20

---

with T C Let the hopping range of H be I-PH: qHl, such that T for & [-PH q8] (z) is a matrix-valued Laurent polynomial of the phase factor el where k is the crystal momentum.
The energy dispersion relation is found by solving the secular equation. The characteristic poly- nomial of the Hamiltonian is defined as:
QP Σ (F)))²
P(z, R) = det H()) FT
(10)
This is finite Laurent polynomial in whose coefficients a, (F) are themselves scalar polynomials in E' of degree S This equation is also known as the energy-momentum dispersion.
is sometimes convenient to clear the negative powers in equation 10 by defining the ordinary polynomial in
P(z, E) PP P(e, F)) E C(2, F),
(11)
whose degree in 2 equals d, := PP + qp.
Degree bounds in z. The highest and lowest degree bounds of the associated P(z, E) satisfy:
pr S 8 x РH: gr S 8 9H:
(12)
with equality holding generically (i.e., unless the leading/minor determinants vanish because of special symmetries parameter fine-tuning). In particular, d, pp qp S s(pH + qH).
Roots z at fixed E. For any fixed E C, the equation P(z, E) = 0 has exactly d, solutions {za (E) (E)) de (equivalently, P(z, E3) has d, roots in z). We will order them by modulus,
221 (E) S [x2 S ... S led, (F)..
Examples.
1. One-band, nearest-neighhor (s= PH = 9H 1). Let
(=)) = 1-12 -- -1
f412, P(a H())
Here pp = Qp = 1 and d₂ = 2. The periodic boundary condition (PBC) dispersion is E(k) = to t t
2. Two-band SSH-type model (s = 2, PH = 9H = 1).
t + t2
H22)
P( = det H (2) = E2=
ti + 122
0
Here pp = Qp (so d2 2), which is strictly smaller than the generic bound s(p + (H) 4.
Interpretation exponents. The exponents of a in H(2) have a direct hopping interpretation (a nonzero T, encodes j-th neighbor hopping). After taking the determinant to form P(z, E), indi- vidual 2 terms no longer correspond to a single hopping distance; rather, they arise from products of matrix entries and thus encode composite hopping paths. This explains why the bounds in equa- tion 12 can be strict: symmetries (e.g.. the chiral symmetry in the SSH example above) can cause cancellations of the leading powers of z in P(,,E).
B.3.2 LIMITATIONS OF STANDARD BLOCH THEORY (PBC)
Under periodic boundary conditions (PBC), Bloch theory applies, utilizing the constraint |z| = (real momentum k)). If (=) 3)) Hermitian on the unit circle, the PBC spectrum consists of real E values forming continuous bands (the usual Bloch dispersion). For non-Hermitian systems, the PBC spectrum typically forms loops or closed curves in the complex energy plane.
However, for non-Hermitian systems under OBC, eigenstates often exhibit the non-Hermitian skin effect (NHSE) (Lin et al., 2023), where a macroscopic number eigenstates localize at the bound- aries. Consequently, the PBC and OBC spectra can be qualitatively different. As evolves the system from PBC to OBC (e.g., by turning off boundary hoppings), the PBC spectrum (loops) often collapses inwards to form the skeletal structure of the OBC spectral graph, depicted in Figure A5.
21

---

oIE)
DSC Spsatur FBC Spectuur
Figure A5: Spectral Collapse & Spectral Potential. PBC spectrum usually appears as eircles and loops; changing to OBC, the spectrum collapses into graph skeleton. The spectral graph resides on the ridges of the potential landscape, (FF).
B.3.3 NON-BLOCH BAND THEORY AND THE GENERALIZED BRILLOUIN ZONE (GBZ.
Since standard Bloch theory is inapplicable under OBC due to the NHSE, non-Bloch band theory employed. This theory introduces the concept of the generalized Brillouin zone (GBZ).
Under OBC, z allowed to leave the unit circle, meaning can complex. The imaginary part of k. Im (A) log |z]. is the inverse decay length (or inverse skin depth), quantifying the localization of skin modes.
Non-Bloch band theory establishes that the OBC spectrum in the thermodynamic limit is determined by those E E C for which the = roots of P(z, E) = 0 satisfy a specific equal-modulus condition. Given the characteristic polynomial P(z, F.) with degrees pp and qr defined in equation 10, the condition for E to be in the OBC spectrum is:
12pr (E)) 1 (E)).
(13)
The corresponding loci of z's the generalized Brillouin zone (GBZ). The spectral graph 9 precisely the set of E' values that satisfy this condition; equivalently, it is the image of the GBZ under the map (z + E: P(z, = 0).
B.3.4 CHARACTERISTIC POLYNOMIAL CLASS Cp
The algebraic structure of P(z, E) plays a crucial role in determining the topology of the spectral graph. To understand this relationship, we must consider P(z, E)) as bivariate polynomial, ex- amining the interplay between the powers of z (representing spatial structure) and the powers of E (representing energy bands). We expand P(2, E) fully as:
qr
P(z, E) = Σ
(E)2² =
Σ
Σ
Cn.m Em z"
(14)
n=-PP
n= m=0
We define the characteristic polynomial class Cp based the monomial support of P(z, E). The support identifies which specific monomials Em are present in the polynomial structure, regardless of the exact values of their coefficients Cn.m-
Formally. we define the support Sp as the set of index pairs (n, m) for which the coefficient Cn,m is structurally non-zero (i.e., allowed vary, rather than being identically zero):
Sp (m) Cr.i 0).
(15)
Crucially, the spectral graph invariant under parity transformation'. which corresponds to the transformation 2 2 in the polynomial. Let P' E) = P(= E). The support of this parity-
Lee., spatial inversion about the origin (r + -r), flipping the ID lattice from left to right. In terms of H this corresponds to - T-j. which is equivalent to transposing the matrix (H HT The transpose does not change the eigenvalues.
22

---

transformed polynomial is:
Sp. ((-n, (n.m) E Sp}.
(16)
The characteristic polynomial class Cp is defined as the equivalence class represented by this pair of supports:
Cp = (Sp. }pr).
(17)
This classification ensures that polynomials related by spatial inversion, which necessarily yield the same spectral graph, belong to the same class. If the polynomial structure is palindromic in 2 (i.e., Sp = S,r), the class is simply identified by the single support set Sp.
We find that the characteristic polynomial class Cp is key criterion for classifying spectral graph topologies and is thus the target for inverse classification tasks. Varying the specific values of the coefficients Ca.a within fixed class Cp may deform the geometry of the spectral graph but typically preserves its fundamental topology.
Examples:
1. Single-band example (s = 1) from Figure A3. P(a.E) = -2- + 2² E. The monomials present are =- Eu E and E The support is Sp (), (1,0). (4, 0), (0, 1)}- The parity-transformed polynomial P(= F.) = -
2 z- +
E. The transformed support Sp {(2,0), (2.00),(111))) (), 0), (0, 1)}. The
class is Cp = S
2. Two-band example (s = 2). Consider class defined by the structure (similar to the dataset construction example): P(z, E) E2 +2 3- 2
+c ++E=+EEz The support is Sp = {(0, (-3, 0), (3, 0), (-1 0), (1, 1), (2, 1)}. This support explicitly captures the interplay between hopping range and energy bands, defining the class.
(-3,0),
B.3.5 RECAP OF KEY CONCEPTS (HIERARCHY OF ABSTRACTIONS)
Objects and their roles.
Real-space Hamiltonian Hraal: an infinite banded (block) Toeplitz operator acting on (2) C formed from hopping blocks (T))-
Bloch Hamiltonian Hz)): the matrix Laurent polynomial in equation 9, the Fourier
transform of Hreal-
Characteristic polynomial P(z, E): the scalar Laurent polynomial equation 10; bivari- ate polynomial in and E (after clearing denominators in 2).
ChP class Cp: an equivalence class of characteristic polynomials defined by fixing the monomial support in (z, E) and accounting for parity symmetry (see equation 17).
Spectral graph G: the subset of C traced by eigen-energies E' in the thermodynamic limit under OBC,
-= (E E C: E GBZ, P(z, E) 0).
A specific Hamiltonian (or specific P) maps to a specific spectral graph; ChP class maps a family of spectral graphs parameterized by its coefficients.
How the abstractions relate.
Hreal
Fourier H(z)
det[--BI]]
P)z,E)
Support Symmetry
Cp
P(2, solve in (GB7)
C.
Each arrow forgets inessential microscopic details while preserving spectral information relevant at the next level. This hierarchy clarifies terminology used in this work.
23

---

Object
Symbol
Mathematical type
Typical size
Real-space Hamiltomian
Head
banded Toeplitz operator
(finite) or infinite
Bloch Hamiltomian
(())
Laurent-poly, matrix
x (s bands/orbitals)
Characteristic polynomial
P(z, E)
Laurent poly. in 2 & Poly. in E set of P's with fixed support up to parity symmetry
(pp, (P) in 2; degree s in E
ChP class
Cp
Spectral graph
g
spatial planar multigraph on C (energy plane)
B.4
THE SHORTCUT TO SPECTRAL GRAPH VIA ELECTROSTATIC ANALOGY.
B.4.1
DENSITY OF STATES pEE) AND THE SPECTRAL POTENTIAL ((E)
The density of states (DOS) describes the continuous spectrum. It is defined the density of eigenstates on the complex energy plane. An example of DOS is shown in Figure A3c.
Recent developments in non-Bloch theory map the problem of finding the spectral graph and DOS to classic 2D electrostatic problem (Xiong & Hu, 2023; Yang et al., 2022; Wang et al., 2024). If we treat the eigenvalues en (for a system of size N) as electric charges of strength 1/N, we can define the Coulomb potential o (E), also called the spectral potential, point E $ G:
(EE) = lim
log E en Σ Ex
N +00
= =- ((E") log IE E'1 de
(18)
The DOS is related to the potential by the Poisson equation:
p 2π
(19)
where 722 PRoE + P is the Laplacian operator on the complex energy plane. The Laplacian extracts curvature. Geometrically, this implies that the loci of the spectral graph G, where the DOS concentrated, reside on the ridges the Coulomb potential landscape (EE), as suggested in figure A5 and figure 2.
B.4.2 EFFICIENT CALCULATION OF ((F)
Leveraging Szegö's strong limit theorem, the spectral potential ((E) in equation 18 can be reduced computationally efficient form based directly the characteristic polynomial P(z, E):
PP PP-
(E) log layp (E)l +
Σ (x(E)
(20)
impp+1
Here, pp and qp are the lowest and highest degrees of 2 in P(z, E), respectively (see equation 10). agp (E) the coefficient of zar (E) - log [E)] are the inverse decay lengths associated with the largest roots of P(z, E) = 0 (these are 2pr+t....., 2pr-trr in the sorted list).
Although equation 18 is strictly defined for E' & G. the expression in equation 20 can be analyti- cally continued to the entire complex plane (Xiong & Hu, 2023). This allows the construction of the potential landscape & (E) merely by knowing the characteristic polynomial P(2, E). thereby obviat- ing the need for direct diagonalization of large real-space Hamiltonians and avoiding the numerical errors associated with such diagonalizations.
C POLY2GRAPH PIPELINE DETAILS
Armed with the above theoretical guidance, we implement the transformations numerically, and then integrate few computer vision techniques (Lee et al., 1994; Wang et al., 2018; Nunez-Iglesias et al., 2018) to construct the spectral graph given its characteristic polynomial (or Bloch Hamil- tonian). This appendix complements section 2. The core procedure of Poly2Graph algorithm ("Characteristic Polynomial to Spectral Graph") is summarized in algorithm 1.
24

---

Algorithm 1: P:ooyyryrapph Characteristic Polynomial to Spectral Graph
Input: (1) (z) or P(z, F.) == det (=)
t Hami an or its characteri stie polynomia a
Input: (2) Energy Domain: Ω C C such that 2 9 (spectral graph)
Output: Spectral Graph:
begin
# Buil ld the characteristic polynomial if only H()) was given
input (z) then
P(2, E) det H E1 -
# Stage 1: Coarse computation over initial energy grid Ω (Parallel) for do
# Solve roots
[z(E)} = Sort[Roots(P(2; E)] s.t. |ai S ... lzptol
# Compute spectral potential
(F)) = log (F) E log|
# Compute Density of States (DOS)
( = 2-
(E)
# Identify ff regions of interest
coarse_mask di (etiatttttariia(((t)))ee))
# Define refined energy grid
Q' oeg.eeaaeeeeesssiicccooooo..
# Stage (Parallel) for E E Ω2' do
Refined computation within masked regions O
# Re solve roots at higher resolution
(2(E)) Sort|Roots(P(2;
Recompute spectral potential
(EE) - log la, Di log (B)
Recompute DOS
(((E) - (L)
Combine coarse and refined DOS for full high-resol ution image
Mhiaa combin(((n())) (
Binarize high-resclution DOS
= iiri BEa)
Extract one-pixel-wide skeleton
graph skeleton skeleton = skeletonize (fina (inaaiiaiieiiallannnnaeeeeee image)
Convert skeleton to graph object
g = skeletton2graphgrapp..oooo skeleton)
# Post-processing
G merge tolerance)
G remove.isollteeedddee(((
C.1 INITIALIZATION AND INPUT
Poly2Graph accepts diverse input formats the 1-D tight-binding crystal. can initialize from Bloch Hamiltonian H (2) directly from its characteristic polynomial P(z, E). Supported formats for (zz, E3) include sympy Matrix (for (z), (k)), sympy. Polyy, or a raw string expression of the polynomial. During initialization, Poly2Graph automatically computes full set of different representations and properties. See the tutorial section appendix or visit our repository https: //github. com/ sarinsteiin yan htt//ooly2ee /tlpp/olapl.l....//p
The energy domain C C, which must fully contain the spectral graph G, is also required. While users specify custom Ω and its discretization, by default Poly2Graph can automatically esti-
25

---

mate suitable region by diagonalizing small real-space Hamiltonian (typically L 40 unit cells) and applying a small padding.
Sidenotes: notions of "size" appeared so far.
1. System size - I. (real-space lattice length): The number of unit cells. The operator real is L x L for finite L and becomes an infinite operator in the thermodynamic limit L + oo.
2. Internal size (band/orbital count): The matrix dimension of (z). This equals the maximal degree in of P(z, E).
3. Laurent degree in of P(z, E) - the pair (pp, (P) (or total d2 - pp + qp): Governing the number of roots at fixed E. It controlled by the degree range of H (z)'s elements via equation 12.
4. Rasterization resolution of spectral images N (pixels):
The size of the spectral images that serve as intermediate construct between input charac- teristic and output spectral graph. Not property of the crystal.
C.2 ACCELERATED RoOT FINDING
As detailed in section 2, solving for the roots {E) of E) discretized domain Q is the primary computational bottleneck. To achieve the reported performance gains (five orders of magnitude speedup and higher memory efficiency over previous methods (Tai & Lee, 2023) on default settings), we employ specialized root-finding strategy.
0 for each energy E in the
For characteristic polynomial P(2, E) Ej=-p - aj (E)²², its roots are equivalent to the eigenval- of its Frobenius companion matrix F(E). For polynomial of degree d p q. the companion matrix is d x d matrix constructed from the coefficients:
a- - (E)/a(())))
1
0
a p+1 (E)//a/)
F(E)-
0
a p22 (F)
(21)
-ag (E)/ (E)/aa (E)
This formulation holds for complex coefficients (E) C.
Poly2Graph constructs a batch of these companion matrices for each E E Ω, where f2 the dis- cretized grid of energy values. This batch is then processed by parallelized eigensolver. The implementation automatically detects the availability of TensorFlow or PyTorch backends, leveraging them for hardware acceleration, including optional GPU support via CUDA. To opti- mize memory and computation, calculations are performed using single precision (float32), which has been found sufficient for high-fidelity spectral graph extraction.
C.3 ADAPTIVE RESOLUTION AND IMAGE PROCESSING
The adaptive resolution strategy, outlined in section 2, is crucial for computational tractability.
The spectral potential & (EE) (equation 3) and DOS p(E) (equation 4) computed on an initial, moderately resolved grid (c.g., 256 x 256). The DOS image is binarized and morphologically dilated to create a conservative mask !2' covering potential graph regions.
Coarse Identification:
Refined Calculation: Within this mask Ω' each pixel subdivided (e.g., into a 4 x 4 subgrid), and (EE) and p( E) are recomputed at this higher resolution.
This two-stage process achieves high effective resolution (e.g., 1024 x 1024) while minimizing redundant calculations empty regions of the complex energy plane.
The resulting high-resolution DOS image is again binarized. We currently employ global mean threshold (p(E) > (p(E")) r'en) for binarization, as has empirically outperformed pool of other common global and adaptive thresholding heuristics, including Otsu, Li, Yen, Triangle, Isodata,
26

---

local adaptive, and hysteresis variants for datasets. Subsequently, iterative morphological thin- ning operations (Lee et al., 1994) are applied to reduce the binarized features to a one- aie-pixel-wide skeleton, revealing the graph topology.
C.4
GRAPH EXTRACTION AND POST-PROCESSING
The skeleton2graph submodule converts the binary skeleton into graph representation. It identifies pixels as junction nodes (three or more neighbors), leaf nodes (one neighbor), or edge points (two neighbors). The output is NetworkX Mult Graph object, where each edge in particular stores its geometric path as an ordered sequence of (Re(E), Em))) coordinates.
To handle numerical artifacts, two post- post-procsssing steps implemented as shown in algorithm 1:
1. merrge.eearb.nnddes Nodes within predefined Euclidean distance tolerance are merged. This helps consolidate fragmented junctions.
2. semove.isolatedd.nde. Nodes with no connecting edges after the merging step are re- moved.
C.5 BENCHMARK POLY2GRAPH SPEED-UP
The primary innovation is the end-to-end automation of the spectral graph extraction process, which was previously reliant on manual inspection. This automation makes systematic research on spectral graphs feasible.
The speed and memory efficiency are secondary, but critical, feature that make large-scale research on spectral graphs feasible for the first time.
Since Poly2Graph is the first tool of its kind that fully automates the entire pipeline, a true pre- decessor for an end-to-end comparison does not exist. Here, in table A3, only benchmark the computation bottleneck (spectral potential calculation) against the best available code from Ref. (Tai & Lee, 2023), which does not automate the graph extraction.
Table A3: Speed comparison between Poly2Graph and the best available code from Ref. (Tai & Lee, 2023). The time used per sample (i.c. for an input (z)/P(z,, E)) for cach method, and the speed-up that the Poly2Graph obtained reported. The comparison conducted the computation bottleneck (spectral potential calculation), as Ref. (Tai & Lee, 2023) does automate the graph extraction. Poly2Graph's time complexity is polynomial in degree range (d; = p q).
degree range
Poly2Graph
Ref. (Tai & Lee, 2023)
Speed-up
p+q=2
13.1 = 0.3 ms
3025 s
2.3e5 x
p+q=3
20.8 = 0.1 ms
3423
1.6e5 x
p+q=4
28.6 + 0.3 ms
3921 s
1.4e5 x
p+q=5
38.8 0.2 ms
5177 s
1.3e5 x
p+q=6
50.8 # 0.3 ms
6199
1.2e5 x
C.6 CAVEATS: COMPONENT FRAGMENTATION
A notable challenge, particularly for systems with large degree ranges many bands, is phe- nomenon we termed component fragmentation. As illustrated in the bottom row of figure A4, this refers to the spurious disconnection of spectral branches that should ideally form single connected component. Fragmentation typically arises at junction nodes where the surrounding DOS is ex- ceptionally low. In such cases, the spectral potential landscape (4 E)) around these junctions is virtually flat, making the corresponding ridges (which correspond to edges) fall below the detection threshold of the binarization and thinning processes, due finite floating-poit point precision.
While the current global mean thresholding for binarization is robust general choice, it may strug- gle with complicated spectra. Ultra-low-DOS edges be missed, leading to missing pixels in the skeleton and thus fragmentation. While more sophisticated ridge-following or adaptive local thresh- olding algorithms might offer improvements, they often come significant cost Poly2Graph's
27

---

speed and memory efficiency. Addressing this intrinsic limitation robustly remains area for future development.
D DATASET DETAILS
D.1 COMPARISON WITH 45 OTHER DATASETS
We present a comprehensive comparison of our dataset terms both structural properties and statistical metrics. Table A4 compiles all prominent graph datasets to the best of our knowledge. Each column is described the caption. As illustrated, while some spatial graph datasets do ex- ist, they generally lack rich connection geometry (RCG. Nontrivial edge patterns beyond simple straight-line link) or support for multiple parallel edges between nodes. The dataset most similar to HSG-12M 1.G in these respects is OpenStreetMap: however, it is not designed with any ML downstream tasks, contains far fewer graphs, and is medium-scall judged by OGB criteria [39]. Moreover, al- though it supports non-linear edge shapes streets connecting pair of destinations straight-line--the complexity of its connectivity is limited. In contrast, the edge geometries in our setting exhibit much richer geometric variation. Consequently, prior this work, the absence of large-scale multigraph learning challenge remain unaddressed.
usually
Moreover, shown in the table, our large-scale T-HSG-5M dataset is the only temporal dataset that includes class labels. This is particularly valuable in our setting, as different classes may exhibit distinct temporal evolution patterns. We leave the investigation of these dynamics future research.
D.1.1 COMPARISON WITH OTHER DATASETS IN PHYSICAL SCIENCES
Additionally, as illustrated in figure 1, HSM-12M stands out as the largest classification dataset in terms of both graph count and class diversity. Although our dataset designed for classification, it is still informative to compare it with others based on the total number of graphs and nodes. By these metrics, certain large-scale computer science datasets- such as TpuGraphs, Tenset, and Mal- Net-contain larger overall volumes. However, fairer comparison emerges when we evaluate our dataset alongside those from the natural sciences, as they are constructed using similar methodolo- gies.
To facilitate this comparison, we selected the largest datasets from the table and visualized them in figure A6. The results show that even among natural science datasets not constrained classification tasks, ours stands out as not only competitive but also the largest in scale. These findings highlight the scope and impact of this work.
D.2 CLARIFICATION ON HSG-ICPOLOG DATASET VARIANT
The HSG-topolocy variant contains only pairwise non-isomorphic graphs. Within each ChP class retain exactly representative for every unique connectivity cnnteme-quiialenttly one per isomorphism class that have the same adjacency matrix (i.c., up to a relabeling of nodes) and discard the rest isomorphic samples. This construction makes the dataset purely combinatorial.
This filtering induces a pronounced class imbalance: different Hamiltonian families generate markedly different numbers of distinct topologies (i.e. isomorphisms).
This design is physically motivated: we are interested how opectral-graph topology varies with the underlying Hamiltonian parameters.
28

---

dataset: #Graphs gives total number of graphs; #Classes the number of target labels; #Nodes and #Edges report the average number of nodes and edges per graph: Scale column follows the OGB Large-Scale Challenge definitions [39]: Small (S) datasets fall below the large-scale thresholds; Medium (M) datasets contain > million nodes or > 10 million edges: Large (L) datasets exceed 100 million nodes billion edges.; Attributed indicates whether hoth node and edge features present; Spatial denotes whether the graphs carry geometric coordinate in- formation (e.g. 2D, 3D, geographic coordinate system-GCS); Temporal flags static (S) or time-serics graph data (T); Multi marks support for multiple edges between node pairs; and RCG (Rich Connection Geometry) indicates datasets whose edge geometry exhibits nontrivial patterns that go beyond simple straight- line connec- tions. "?" entries indicate information stated in the original paper. In addition to values extracted from the literature, some benchmark statisties were sourced from Refs. [39; 119; 21; 78].
Table A4: Overview of graph-level benchmark datasets. Each row corresponds
Dataset
#Graphs #Classes
#Nodes
#Edges
Scale Attributed Spatial Temporal Multi RCG
Biology
ENZYMES [5: 75]
6000
32.6
62.1
8
S
PROTEINS [5; 75]
I.IK
2
39.1
72.8
S
[75:20] 20]
1.2K
284.3
715.7
S
ProFold [32]
76K
8.0
?
3D
T
NeuroGraph [87]
Skeleton (NTU-RGB+D) [88]
56K
60
25.0
24
M
3D
T
ppa [126; 38]
158K
37
243.4
266.1
S
Skeleton (Kinetics) [51]
260K
400
18.0
17
M
2D
T
Chemistry
MUTAG [56; 75]
19.8
SIDER [105: 1]
1.4K
2
33.6
35.4
BACE [105; 90]
1.5K
2
34.1
36,9
ClinTox [105: 29]
1.5K
26.2
27.9
S
BBBP [105; 71]
2.0K
2
24.1
25.9
Tox21 [105:
7.8K
19,3
M
ToxCast [105: 8:3
8.6K
18.8
19.3
M
Peptides func [24]]
Peptides-struct1 [24]
15.5
150.9
307.3
M
3D
MolHIV [105: 38]
41.1K
25.5
27.5
MUV [105; 85]
93.1K
24.2
26.3
QM9 [81]
129K
12
18.0
18.6
M
3D
MOSES
MolOpi [48]
229K
24
$3
M
3D
ZINC250K [45]
MolPCBA [105; 38]
437.9K
26.0
28.1
M
PCOM-Contact [24]
529.4K
30.1
61.1
M
CbEMBL [72]
1.8M
27.0
58
M
3D
PCQM4M<2 [39]
3.7
14.1
14.6
I.
Social Networks
IMDB-BINAR [75:
1K
2
19.8
96,5
AMDB-MULTI [75; 116]
1.5K
13.0
65.9
REDDIT BINARY [75: 116]
2K
429.6
497.8
REDDIT-MULTI-SS [75; 116]
5.0
508.5
594.9
M
REDDIT-MULTI-I2K 116]
11.9
391.4
CollabNet [92]
2.3K
303K
207.6K
L
GCS
Computer Science
CIFARI0 [23;
60K
117.6
MNIST [23; 60]
70K
10
70.6
$64.5
M
2D
Database [36]
300.0
MalNet [27]
1.3
696
15.4K
35.2K
SILI
I.
TpuGraphs (Tile) [78]
TpuGraphs (Layout)
3L1M
7.7K
TenSet [124]
51.6
5.0-10.0
Gengraphy
METR-LA [46]
34K
327.0
2.4
M
GCS GCS
T
PeMS-BAY [15]
50K
207
1.5
T
OpenStreetMMa
110K
500
1.2K
M
S
Physies
N-body-spring [54]
10
T
N-body-charged [54]
3.4M
25.0
M
2D
T
THIG--M SM
5.IM
1401
13.8
28.9
C-plane C -laan
T
HSG- 12M
11.6M
1401
13.8
28.9
I.
S
29

---

BSGG-1MM
10'
11.6 Graphs
Biology
T-ISG-5M
Chemistry
N-body spring
Geogruphy
Physics
POQMAMV2
N-body charged
Simple Graphs
# of Graphs
10
x
Spatial Multigraphs
CHEMOL
PCQM-Contact
161
(Kinetics)
Nodes
ZINC250K
MOSES
10
X
ProFold
Mppp
PeMS-BAY
METR-LA
Peptides-fume fune
10
10
10
Total Number of Nodes
Figure A6: Number of graphs total number of nodes in HSG-12M compared other natural science datasets. HSG-12M stands out with the highest data volume across all natural science datasets, including those not designed for classification.
E BENCHMARK DETAILS
E.1
DATA PREPROCESSING (INCLUDING SPATIAL MULTIEDGE FEATURIZATION)
Feature type
Dim.
Components/ / Description
Complex position coordinates (2D)
Node
4
Spectral potential at the node (ID)
DOS at the node (1D)
(Curved) edge length (ID: also used the edge weight) Straight-line distance between endpoints (ID)
Edge
Midpoint coordinates (2D)
Average spectral potential along the edge (ID)
Average DOS along the edge (ID)
Graph
None.
E.2
TRAINING CONFIGURATION
Baseline Models.
Graph Convolutional Network (GCN) (Kipf & Welling, 2017): first-order spectral graph con- volution that aggregates normalized neighbor features and applies shared linear transform.
Graph Attention Network (GAT) (Veličković al,, 2018): masked self-attention over neighbor- hoods with multi-head weighting to adaptively combine messages.
Modified Graph Attention Network (GATv2) (Brody et al., 2022): reformulated attention with dynamic queyy dependence for greater expressiveness and more stable training.
Graph Isomorphism Network (GIN) (Xu al., 2019): sum-aggregating message passing with an MLP update, designed for expressivity comparable to the Seisfeiler-Lehman test.
Edge-conditioned Graph Isomorphism Network (GINE) (Hu et al., 2019): GIN variant that injects edge features into the message MI.P, improving performance edge-featured graphs.
30

---

Molecular Fingerprints (MF) (Duvenaud al., 2015): uses learnable neighborhood transforms and softmax pooling, yielding a fixed-length real-valued fingerprint aggregated over nodes and layers.
Crystal Graph Convolutional Neural Network (CGCNN) (Xie & Grossman, 2018): crystal- graph convolutions over periodic structures (periodic crystal graph) that aggregate atom-bond interactions within a distance cutoff for materials prediction.
GraphSAGE (Hamilton al., 2017): inductive nighhor-ssmpliing with learnalle aggregators (e.g., mean) enabling generalization to unseen graphs.
All architectures are pooled via global add pooling, followed by a multi-layer perceptron (MLP) to produce the final class logits.
Global training setup. Data splits, training hyperparameters, optimizer, learning rate scheduler, hardwares, trainer settings, and common model hyperparameters are:
Group
Hyperparameter
Value / Notes
Data
Splits ratio
8:1:1 Stratified random splits (preserve class/target proportions across train/val/test)
Split strategy
Training
Batch size
6000
Max epochs
100
Max steps
1000
Seeds
2025, 666)
Model (common)
#Conv layers
4
#MLP layers
2
Activation
ReLU
Dropout
0.0
#Heads (GAT/GATT2)
Optimizer
Algorithm
Adam (AMSGrad) [69]
Init LR To
1 x 10-3
Min LR Tmin
1 1 10-5
Weight decay
0.0
Scheduler
Policy
Cosine annealing [68]
Period To
100
Trainer
Devices
Two RTX A5000, 24GB each Distributed Data Parallel (DDP)
Strategy
Dataset-specific hyperparameter: hidden dimension post-convolution MLP. The post- convolution MLP hidden dimension is tuned to be larger than the number of classes per dataset, to ensure sufficient capacity for final classification.
dim
MLP one-band
two-band three-band topology
HSG-12M
Value
128
256
1500
1500
1500
Model-specific hyperparameters: hidden dimension of graph convolution layers. To ensure fair comparison, for each dataset, tune the hidden dimension of graph convolution layers for each model such that the total number trainable parameters is within 4% relative difference across all models.
31

---

dim
Conv
MF
GCN
SAGE
GAT
GIN
GINE
CGCNN
GATv2
band
100
467
330
452
312
312
202
330
1SG-two-bandd
200
933
661
933
621
621
410
661
HSG-three-band
300
1279
963
1279
852
852
601
963
HSG-topology
300
1279
963
1279
852
852
601
963
HSG-12M
300
1279
963
1279
852
852
601
963
E.3 EVALUATION METRICS
We evaluate single-label multiclass prediction over dataset D {(G,, 3>) with y {I....,C) C}. Let Si,e be the model score (logit or probability) for class с on example i, and 0: arg max. Si.c.
Accuracy (micro-Fi). Overall fraction of correct predictions:
Acc Σ T(ū y
(22)
In single-label multiclass settings, Accuracy equals the micro-averaged Fj. is intuitive and stable when class frequencies are roughly balanced. The chance baseline is 1/C.
Macro Fj. Compute one-vs-rest counts for each class
TP Σ
I( = 0 = c), FPc Σ I& + Os = cc FN: = Σ Ilys = c. 0s + ct.
(23)
Per-class precision/recall and F1, with 0/0 = 0:
TP.
TP.
Fl. = 2P.Re
Pe TP. FP.
Rs = TP. + FN.
(24)
P. + R
Macro-average classes:
Macro-F, 1 Σ Fla
(25)
Macro-F, weights all classes equally and is therefore sensitive to minority-class perfor- mance-crucia. when C is large or labels are imbalanced.
Top-k Accuracy. Let
denote the indices of the l largest scores. The Top-k
metric
a Top-k Σ Iue TopK( k)
(26)
We report & E {5, 10}. Top-k captures ranking quality and is directly aligned with inverse-design workflows that accept shortlitt for subsequent physics-based re-ranking. Random- handom-guess guess base- lines scale as k/C (e.g., 5/24 Rs 20.83%, 10/24 RS 41.67%; for C' 1401: 5/1401 R 0.357%, 10/1401 R 0.714%).
Reporting and interpretation. All metrics are reported as mean averaged over seeds on the held-out test split. For smaller or moderately balanced label spaces, Accuracy is informative and easy to compare; for highly imbalanced or very large C, Macro-F, is emphasized to surface minority-clas recall, while Top-5/10 quantify the usefulness of the model as candidate sandidate-generator for downstream, physics- constric-o refinement.
E.4 ADDITIONAL BENCHMARK RESULTS AND ANALYSIS
Tables A5-A8 reports additional benchmark results including test loss, test top-5, and training statis- tics, including throughput and device utilization.
Overall accuracy and stability. Across all five static variants, seed variance is consistently small, indicating stable training.
32

---

Edge attributes matter. Methods that explicitly consume edge features (e.g. GINE) consis- tently superior to their edge-agm agge-agostic counterparts (e.g. GIN), For example on two-band, GINE (.518).049) outperforms GIN (.343 On HSG-12M, plain GIN essentially collapses (.063 1.031. while GINE remains competitive (.460...22). This aligns with the dataset design: multi-edge geom- etry (length, straight-line distance, midpoint, average spectral potential, average DOS) carries irre- ducible spatial information; architectures which propagate and transform edge states are expected to succeed.
Performance degrades with task difficulty. Averaging over models, test metrics degrade from seyy-thheee-aad--tteee
This monotonic decay is expected: eigher-band Hamiltonians induce larger graphs with richer multi-edge geometry and more challenging class diversity (up to 1,401), stressing both representation and optimization. Moreover, as expected, per-graph memory usage scales with dataset complexity for every model (e.g., SAGE: 0.066 on one-band -> 0.544 on three-band).
one wwt-ndd-ttw--bad----e
Top-k is high promising for inverse design. Despite moderate Top-1 I accuracy on the largest settings, Top-10 accuracy very high (e.g., on HSG-12M SAGE 95.2% CGCNN 94.8%). For easier subsets, Top-10 essentially saturates (99%+ on one one-band for all models). This pattern implies that models almost always retrieve a small candidate set of plausible Hamiltonian families. This is encouraging for inverse design workflows (retrieve top-k families, then re-rank/verify physically in experiments, e.g. design few meta-matetial candidates and observe if targeted spectral properties are obtained).
Attention is not a free lunch here. GAT / GATv2 lag SAGE / CGCNN on all splits, while also incurring the highest peak GPU memory per graph (e.g., on HSG- 2M ~1.93-2.11 SAGE's 0.51). In spatial multigraphs with high effective degrees (many parallel edges), attention softmaxes can become dominated by edge multiplicity/noise and impose additional compute/memnory over- head without commensurate accuracy gains under our budgets. In other words, dense multi-edge neighborhoods amplify attention's 's quadratic costs and may dilute useful geometrie cues when our direction-ignorant summary features are the only edge signal.
GraphSAGE excels with limited budget and comparable parameters constraint. GraphSAGE is consistently the strongest baseline. It attains the best Top-1 I accuracy and macro-Fı on every subset. Under matched trainable parameter constraints (54% difference) and fixed training budget (max_epochs = 100, max_steps 1000), these consistent gains suggest that:
1. Either, other more expressive architectures (e.g., attentive or edge-MLP-based require larger training budgets to fully realize their potential;
Or, other expressive baselines are more parameter-hungry. and thus under trainable parameter constraints, they are not as efficient as lightweight baselines for large-scale benchmarks;
3. Or surprisingly, GraphSAGE's neighborhood aggregation is better inductive bias for our spatial multigraphs than attention or vanilla spectral convolutions.
One could explore relaxing the fixed budget and hyperparameter optimization to unlock each archi- tecture's 's full potential. However this is far beyond the computing resources currently available to us, and we leave this to future work.
Implications for spatial multigraph learning. With our fixed-size, direction-agnostie edge sum- maries (length, straight-line distance, midpoint, avg. potential/DOS), relatively simple, locality- biased architectures already capture much of the diseriminative signal.
However, the persistent fop-1++Topp-1 gap between edge-aware baselines and their edge adge-agnostic counterparts, and difficulty high class diversity, together indicate that fine-grained geometric information along multi-edge curves (e.g., curvature, torsion, higher-order moments, spline/Bezier edge parameterizations, sequence encodings of the edge polyline) are promising routes to push performance further, especially on large-scale challenges like three -and and HSG-122 12M.
33

---

Table A5: Additional Graph-level classification results for Test Loss on the HSG dataset variants. Cells show +sd over three random seeds; best model per dataset Bold.
Model
one-band
two-band
three-band
topology
HSG-12M
GCN
.723 .088
1.695 1.047
2.522 1.522..063
2.062.1.022
2.357
GAT
.841 008
1.776
2,465 1.068
1.8254.0s.
2.330
GATv2
9926-01 018
1.853+.018
2.809+.111
1.968
2.401
GIN
.07
2.216+.402
4.926+.422
4.179+.612
4.764
GINE
-054-027
1,4664.183
2.298
1.3164.057
1.7994.124
MF
1.056
2.392
2.852+.006
2.222++0551
2.658
CGCNN
.474-011
1.218
1.730+.066
1.191
1.485+.021
GraphSAGE
.355 2011
.932 1.022
1,561 1.073
1.019 1.029
1.434 1.34_.009
Table A6:
Additional Graph-level classification results for Top-5 Accuracy on the HSG dataset variants. Cells show sea over three random seeds; best model per dataset Bold.
Model
one-band
two-band
three-band
topology
HSG-12M
GCN
.988
.847 1.009
.693
_715_.007
..25
GAT
.981 0001
.8304.007
.702.1.016
.752
.7264.009
GATv2
.973 002
.815+.003
4.27
.723 005
.711
GIN
.95_ 000
.746..087
..84
.276
.222+.098
GINE
.996
.884
.755
.848
.8304.018
MF
-974-001
.703+.026
.6164.006
.673
.655..015
CGCNN
.995-000
.920+.012
.8364.009
.870_.00
.8764.003
GraphSAGE
.998 .000
.953 1.002
8631.012
.898 ..002
8822..02
Table A7: Additional Graph-level classification results for Throughput (graphs see ") on the HSG dataset variants. Cells show mean over three random seeds; best model per dataset in Bold.
Model
one-band
two-band
three-band topology
HSG-12M
GCN
113580+1840
839754717
50803
44065453
52945
GAT
103757+879
62717+302
31885
25862429
341934228
GATv2
104282+2065
63092++26
30074+11
23825+15
32051
GIN
111905
84581+354
61019
55879498
62134
GINE
112075 1:619
81356
52408 1186
45984
53332
MF
112493 1.3766
859801111
65223
61507 132
66829 1729
CGCNN
105543+1818
62122+280
26752+17
204 14
2867 +226
GraphSAGE
113781 11378141799
855904467
544734266
48479473
560644553
Table A8: Additional Graph-level classification results for Average Peak GPU Memory (MB/-
graph) on the HSG dataset variants. Cells show mean over three random seeds; best model per dataset in Bold.
Model
one-band
two-band
three-band
topology
HSG-12M
GCN
.0851+ + 0001
.3308 3338&+0000 0002
.7274+0004
.88244.0015
16713
GAT
.2418 1.0001
1.0276 .003
2.301
2.8788 1.0040
2.1065 1.0056
GATv2
.2100 1.002
.8809 1 0002
2.1102
2.6580 1.0036
1.9301 1.0056
GIN
.0882 0002
.2554 1.0002
.6375 1.0014
.7535 1.0014
.5918
1. 00009
GINE
.1209 +
.48 6
1.0676
1.3142 0.22
.98124
MF
.0494 1194400000 0002
-1766
.4054
.4701 +000
.3772+
CGCNN
.1876 00000
.798 I +.0004
1.9284 1001
2,4368 4.004
1.7636...050
GraphSAGE
.0669 00002
.2407 1.001
55112
.6542
.5108. 1.
34

---

E.5 PHYSICAL MOTIVATION OF THE BENCHMARK TASK
In crystalline, mesoscopic solids and meta-materials, observable spectral signatures- disper- sions, density of states pl a generated by an underlying Hamiltonian (=) whose structure is dictated by crystal geometry, orbital content, and the pattern of allowed hoppings. The forward map
poly2graph: - (
(27)
from tight-binding (or effective) Hamiltonian to a spectral graph G deterministic yet typically many-to-one: different microscopic parameterizations inside the same Hamiltonian family-more precisely, the same happing-pattern can induce very similar spectra.
For materials discovery and interpretation of experiments, one interesting and practically rel- evant question is the inverse problem of Hamiltonian inference from spectral data:
Given a desired spectral signature (the Hamiltonian spectralgraph), what "class" of Hamiltonians- material structure/hopping pattern-could realize it?
We cast this inverse-design query as supervised categorical retrieval from spectral graphs to a dis- crete characteristic polynomial (ChP) class (a peopiing-patemm)... Learning predictor
fo G G++C Cp
(28)
coarse- consse-rrins the ill- -oo-posed inverse map into a small set of plausible Hamiltonian families. High Top- k accuracy means we can enumerate TopK(Jo (G)) k)
compact candidate list, thus reducing searching over a huge number of hopping patterns to a manageable shortlist.
This framing is physically meaningful for reasons. First, the spectral features that guide human intuition (e.g. gaps and gap sizes) are controlled primarily by lattice symmetries and connectivity rather than precise parameter values; predicting the family therefore the right first step. Second, our spatial multigraph featurization of spectra (node- and edge-level geometric and spectral statistics) is engineered to expose invariants that tie back local real-space structure, allowing GNNs learn robust surrogates of poly2graph-?
Beyond enabling inverse design, the HSG provides a large-scale resource to study how Hamilto- nian parameters control spectral-graph morphology and to pre-train scientific foundation models on physically grounded graph signals, contributing to A14Science community (Yan, 2025; Yan & Pan, 2025). short, the mapping Spectral Graph - ChP Class directly operationalizes task that materials physicists and chemists already perform by hand, but at scale and with principled uncertainty via Top-k retrieval.
F UNIVERSALITY OF SPECTRAL GRAPHS THROUGH TOEPLITZ DECOMPOSITION
At the heart of our framework is the Poly2Graph algorithm, function that establishes a direct mapping from the algebraic domain of polynomials and matrices to the structural domain of spectral graphs. This connection is most naturally illustrated with Toeplitz matrices. As demonstrated in appendix B, a generic Toeplitz matrix can be interpreted as single-band bight-binding Hamiltonian (Eq.8), which corresponds to unique signature spectral graph.
The significance this specific result is vastly amplified by foundational theorem in linear algebra:
Any matrix can be expressed as a product of Toeplitz matrices (Ye & Lim, 2015).
Specifically, for any matrix M C"xn, decomposition into product of r Toeplitz matrices exists, where [n/2] + < r < 2n + 5. While this decomposition is not unique without further constraints, its existence is guaranteed.
Consequently, by associating a spectral graph with each Toeplitz component, we can represent any arbitrary matrix multiset of these graphs. Although the decomposition is not unique, can define the smallest multiset as the canonical topology signature of the input system. Henceforth. this provides universal procedure for translating complex matrices into graph-based representation:
35

---

1. Start with generic matrix M E Cax=
2. Decompose M into product of Toeplitz matrices, M T.T2....
3. Apply the Poly Poly2Graph algorithm each Toeplitz factor T; to extract corresponding spectral graph G,. The resulting multiset (91...., Gr is the spectral graph representation M.
This procedure highlights the remarkable generality of our framework. Since the characteristics of most scientific systems be expressed as matrix, polynomial, more generically vector (which can be standardized and treated as the coefficients a univariate polynomial), our approach offers novel analytical lens. This opens up avenues for research across numerous fields, direction we are actively exploring and invite the broader community to join.
G TUTORIAL OF POLY2GRAPE PACKAGE
poly2graph a Python package for automatic Hamiltonian spectral graph construction. It takes in characteristic polynomial or Bloch Hamiltonian and returns the spectral graph.
G.1 FEATURES
gigh-performance
Fast construction of spectral graph from any one-dimensional models
Adaptive resolution to reduce floating operation cost and memory usage
Automatic backend for computation bottleneck. If tensorflow / torch is avail- able, any device (e.g. /GPU: / /TPU:0, 0, ouda: etc.) that they support can be used for acceleration.
Cover generic topological lattices
Support generic one-band and multi-band models
Flexxible multiple input choices, they characteristic polynomials or Bloch Hamilto- nians; formats include strings, sympy Polly, y, and sympy Matrii
Automatic and Robust
By default, no parameerss are needed. Just input the characteristic of your model and pol p2laggr handles the rest
Automatic spectral boundary inference
Relatively robust on multiband models that are prone to "component fragmentation"
Helper functionalities generally useful
skeleton2graph module: Convert skeleton image its graph representation
hami ielooiiinnn an module: Conversion among different Hamiltonian representations and efficient computation of range of properties
G.2 INSTALLATION
You can install the package via pip:
pip install poly2graph
clone the repository and install it manually:
git clone httpp://tthaaaaccceerrraaaaeeees
cd poly2graph
pip install
36

---

Optionally, if TensorFlow PyTorch is available, poly2graph will make of them automati- cally to accelerate the computation bottleneck. Priority: tensorflow torch > numpy.
Check the installation:
import poly2graph p2g print (p2g. version
G.3 USAGE
See the Poly2Graph Tutorial JupyterNotebook for quick interactive start.
p2g. SpectralGraph and p2g. CharPolyClass are the two main classes in the package.
p2g SpectralGraph investigates the spectral graph topology of specific given characteristic polynomial or Bloch Hamiltonian. p2g. CharPolyClas alyllasssss investigates a class parametrized characteristic polynomials Bloch Hamiltonians, and is optimized for generating spectral proper- ties in parallel.
import numpy as np import networkx as nx import sympy as sp
4 import matplotib.pyplot as plt
from matplotiib import colors a always start by initializing the symbols for k, 2, sp.symbols ('k', real-True)
2, E sp. symbols ('z E', complex=True)
G.3.1 A GENERIC ONE-BAND EXAMPLE (225.SPECTRAAGRAPHH))
Characteristic polynomial:
P(E,)) =- h(2) E =
E
Its Bloch Hamiltonian (Fourier transformed Hamiltonian in momentum space) is scalar function:
h(2) = 22-=22-
where the phase factor is defined as == ett Expressed in terms of crystal momentum k:
h(k) 4ik et E -2kk
The valid input formats to initialize p2g. www.Sppearllraeeec object are:
1. Characteristic polynomial in terms of z and E:
string of the Poly in terms of and E
as sympy. Poly with {z, 1 1/2, E} as generators
2. Bloch Hamiltonian in terms of k or sympy Matrix in terms of k
as a sympy Matrix in terms of z
All the following characteristi are valid and will initialize to the same characteristic poly- nomial and therefore produce the same spectral graph:
37

---

char_poly_stz 1-2es-2 2*04
char_poly_Pon -sp.Poly(
-2AA-2 E 22+ 2**4, 2, 1/2, generators are 2, 1/2,
phase_K sp. exp (sp. I+k)
char Mhar_hamil_ =
phase_k + phx4e_kex4[)
II
char haar_hamil: sp.Matrix (1-2AA22
2++4])
Let us just use the string to initialize and see a set of properties that are computed automatically:
sg
Specta (char poly str, k=k, z=2, E=E)
Characteristic polynomial:
sg. ChP
>>> Poly( (2 - - 3 E, 2, 1 E, domain = Z)
Bloch Hamiltonian:
For one- one-band model, it is a unique, rank-0 0 matrix (scalar)
sg.h. k
>>>
[e eth -2ik
hg h
>>>
9 +1
The Frobenius companion matrix of (E) (z):
treating E as parameter and z as variable
Its eigenvalues are the roots the characteristic polynomial at a fixed complex energy E. Thus is useful calculate the GBZ (generalized Brillouin zone), the spectral potential (Ronkin function), etc.
sg. companion E
38

---

>>>
0
1
0
0
E
0
0
0
0
1
Number of bands & hopping range:
print (' ('Number of bands:', , sg. num cam_aands) print 'Max hopping length the right:', sg. sg.poly p) print ('Max hopping length to the left:', sg.poly q)
>>>
Number bands:
Max hopping length to the right: Max hopping length to the left: 4
A real-space Hamiltonian of finite chain and its energy spectrum:
sg.real_space___ N-40,
number of unit cells
pbc-False, max_dim=500 maximum dimension of the Hamiltonian matrix (for numerical accuracy)
open boundary conditions
energy np..iinalg.eigvals (H)
fig, sut.loss (figsize=:1, 31)
ax. ax.plet venergy.real. energy.imag. 'k.', markersize=5) u ax. (xlabel= Re (E) ylabel='Im(B)
xlim=sg.spectnn poneetrallsquar([1220 h spcrrra_aurrrarrrr1222 (2:])
plt tlt.tight_layout (); showshow()
Im(E)
THE SET OF SPECTRAL FUNCTIONS
phi, dos, binaried dos sg. spwctral...ccm.. images ()
39

---

fig, plt. suoplots figsize= (8, sharex-True,
sharey=Irue)
axes [0] inshow extent=sg. spectralpppeterreeeeee
cmap='terrain')
axes [0] set (E) ylabel='Im (E) title= title='Spectral Potential')
p2, p98 np.percentile (dcs, (2, 98))
Clip extreme DOS to increase visibility.
norm colors. Normalize (vmin=p2, vmax=p98)
axes [xessl]]inshow (dos, extent=sg. spectral cmap='viridis', nerm=norm)
axes 11 set (xlabel=
title= of States')
12 axes [2] imshow (binaried des, Detent=sq.spectzza squae, cmap= 'gzay)
13
axes [2] set (xlabel= (2) , title= title-'Graph skeleton')
plt.tight_layou ()
plt.show()
Spectral Potential
Density of
Graph Skeleton
2
Im(E)
-2
Re(E)
Re(E)
Re(
THE SPECTRAL GRAPH G
graph sg. spectrall
fig, ax plt..auoplotssffigiieeee pos www.et_ooee_attrrbuuesssgggggg nx. draw wwrrw_nnetwrkx_noeeee nodes (goaes (graph, pos, alpha=0.8, 8, ax=ax,
3)}
'pos')
node_color= mode_size=td nede color= volor='4460622 #A60628")
Bettwwnnetworxxeeeeeeee (graph, pos, alpha=0. 8, ax=ax, nx. draw www.nnewwwweeeeee
width=5, edge color=1#348AB)
plt. tight_layouttt ()) plt.show
40

---

If "tensorflow or torch is available, poly2graph will automatically use them and run CPU by default. If other device, e.g. GPU / TPU is available, one can pass device (device string) to method apectraal.mmgges and spectral.grape
(device=//cpu:3)
apeccraaldtapt..pccccaaa___p spevice=//ppul11
Spectraal@raph..eectaal. iigaeeteeaiimamseeeeeeee (device= ccpu)) specrralstrph..secttaaa. (de (device= cuda:0
However, some functions may not have gpu kernel in t E//oorch in which case the computation will fallback to CPU.
G.3.2 A GENERIC MULTI-BAND EXAMPLE (f2G. SEE.TTATTLAAGRPE)
Characteristic polynomial (four bands):
2(E,)) det(h(2) E E)
2²
1/2² 22 + E E
One of its possible Bloch Hamiltonians in terms of 2:
0
0
0
x²
1/22
+
h(
0
2
0
1 0
0
0
1
mu Spect ralGraph(( 22
1/2**2 + Baz
E**4", k, 2, B)
Characteristic polynomial:
multi. ChP
Poly (2² +2E+ 22 E 2, 2, E, domain Z.)
Bloch Hamiltonian:
For multi-band model, the p2g. Spect is not initialized with a sympy Matrix then poly2graph will use the companion matrix of the characteristic polyno- mial P (z) (E) (treating parameter and E as variable) as the Bloch Hamiltonian this possible band Hamiltonians that possesses the same energy spectrum
is one of the and thus the same spectral graph.
sg multi.b.k
0
0
0
2 cos (2k)
0
tk
0
0
0
1
0
 nnTtT." Da
2
2
41

---

1
0
The Frobenius companion matrix of (E) (z):
sg. multi. companion_F
>>>
0 0
0 0
0
E4
0
0 1
E
Number of bands & hopping range:
print ('Number of bands:', sg_multi, num_bands)
print ('Max hopping length to the right:', sg_multi. momulti.poly.... p)
print ('Max hopping length to the left:', sg_multi.poll
>>>
Number of bands: 4
Max hopping length to the right: 2 hopping length the left: 2
real-space Hamiltonian of finite chain and its energy spectrum:
mmulti sg_multi. www.mmitittelaeeeemm.m seaae_ppaee___e H
N-40, pbc-ralse,
number of unit cells
open boundary conditions
max dim=500 maximum dimension of the Hamiltonian matrix (for
numerical accuracy)
energy_multi = [g.
fig, ax aut.ssubplotsssfigiiieee 3))
ax. ax.plot (teey energy_mull imag, 'k.', markersize=5)
u ax. (xlabel= Re (E) ylabel=' (m)
www.iiisppcttra.c...m plt.tight_layout (); plo.show()
ylim=sg_mult multi. spectral_squaee (2:1)
42

---

10
0.5
Im((]
-1.0
THE SET OF SPECTRAL FUNCTIONS
@ni_multi, dos_multi, Linaried_dos_milte
(device=1/cpp)
fig, axes plpp (1, 3, figsize=(8, 3), sharex-True, sharey-True)
axes|0] inw..iiimwwwwhhhh........ ipsiioh axtent=sg.mutt..spctt.....
- cmap='terrain'')
ylabel-' Im (8) title- Spectrall Potential
[eea[1]]insshoow@@s__mltttt extent=sq_mule http:/sss__ttt.ppppaa____spppr axes
cmap= cmap='viridi'' nom=noorm)
axcs[1].set (xael)))_cc(llaaee===+((11)) title- Density of states')
axes [tst::/_mmmooooocaaiiddddddd__ad
extent-sg_multd
spectral_square,, oop
cmap='gray')
[xes[2] (xlabe1='Re (E) title= Skeleton')
blt.tight_layout (); plt.show
Spectral Potential
Density of States
Graph Skeleton
0.5
im(E)
0.0
0.5
-1
Re(E)
RetE)
Re(E)
THE SPECTRAL GRAPH g
graph multi sg multi appectral_grappe
node pairs or edges with distance < threshold pixels are merged
fig, ax plt. subplots (figsize= 3)) pos pos_multi = nx. _tttiideettttruutee (graph 'pos') nx, draw hwwp:/ggraaaaaaaaaaaa pos mos_multi, ax=ax,
me-1_size=10, 2 " '348AHD', width=2, alpha-0.8)
43

---

pit.tight layout () plt. show()
G.3.3 NODE AND EDGE ATTRIBUTES OF THE SPECTRAL GRAPH OBJECT
The spectral graph is networkx. Mu 1tultn Graph object.
Node Attributes
1. pos: (2.)-numpy array
the position of the node (Re(E), Im(E)
2. dos float
the density of states the node potential. float
the spectral potential at the node
Edge Attributes
1. weight float
the weight of the edge, which the length of the edge in the complex energy plane
2. pts: : (w, 2)-numpy array
the positions of the points constituting the edge, where is the number of points along the edge, i.e., the length of the edge, equals weight
avg tvg_dos: float
the average density of states along the edge
4. avg-potent.i i all: float
the average spectral potential along the edge
node node_attr= = dict (graph. nodes (data=True) ]
edge_attr = list (graph.eddoog (data=True)] print ('The ('ehe attributes of the first node\n', [0], 'An') print ('The attributes of the first edgein', edge_attr [0] [-1], (An")
>>>
The attributes the first node I'pos': array (1-0 (1-0.20403848, -2.11668106)), 'dos': 0. 0.0111655922889855,, 'potential'; -0. 6086558770388881136
The attributes the first edge
('weight': 1.4176547247884077
'pts': array 0raay(1112200033445e00,,, 021162220000.,,22211111111100000 [-1. 1-1.99792882c-01, -2.11243496-+001,
II
[ 942283966-01, -1..00775--0000000----0-0-00000
44

---

"avg_dos" dos': 0.10761458, avg_potentiall -0.5068641)
G.3.4 A GENERIC MULTI-BAND CLASS (P2g. CHARPOLYCIASS):
Let us add two parameters {a, to the aforementioned multi-band example and construct p2g CharPolyClass object:
a, b sp. symbols ('a b', real-True)
cp p2g. CharPoLyClass
"z++2 + a/2++2 b+E+2 E**4", k-k, 2=2, K-
parans- b), pass parameters set
>>>
Derived Bloch Hami itonian
'h_z" with bands.
View few auto-computed properties
Characteristic polynomial:
cp.ChP
>>> Poly(2" a + b=E EA, 1 E, domain Z (a,
Bloch Hamiltonian:
ck
>>>
0 0
(a + e dikiee 20k
0
0
betk
0
1
0
cp.
>>>
0 0
+
22
0
0
The Frobenius companion matrix of (E) (z)
cp. companion E
45

---

0
0
E
0
0 1
Fb
AN ARRAY SPECTRAL FUNCTIONS
To get an array spectral images or spectral graphs, we first prepare the values of the parameters b}
a_array np. linspace (-2,, 1,
arrray np. wwnspaccccceeeee (-1, 1, a grid, b_ p_grid np. ag.meshgridle_arge array, b p_array) param_dict {a: a_grid, b: b_grid)
('a_grid shape:', a_grid.shape, 'Anb_grid shape:', b_grid. shape)
>>>
a_grid shape: (6, 6)
b_grid shape: (6,
Note that the value array of the parameters should have the same shape, which is also the shape of the output array of spectral images
phi_arr, dos_arr, binaridd_dos__ spectraa images (param_ ditp:aaaaattppamddddeea dict) ('phi_aar shape:', phi_arr.shar shape, "Andos_arr shape:', des_arr. shape, Nabinaried__oss_rr shape: binarie__ds_aarr.hhhee
spectral_squavt
>>>
phi_arr shape: (6, 6, 1024, 1024)
dos_arr shape: (6, 6, 1024, 1024)
binaried _or_aardddos_rrr arr shape: (6, 6, 1024, 1024)
from mpl toolkits gridl import ImageGrid
fig plt. fittrei (figsize= (13, 13))
grid (fgg, 111, nrows ncols= 6), axes pad=0, label_mode='1> sharo_all-True)
for ax, (i, j) in (irid [(i for i in range (6) for in range [): ax. imshow (phi i1, extent=specttal [i, i), cmap='terrain")
ax. (x1abe1== (K) yl textext
0.03, 0.97, f'a [a_ (a_arrayld): b (b bbararayllla .2E)', ha='left', va='top', transformaxx, traassxxssstsssess...... fontsize=10, color='tab:red', bbox=dict (alphaa0 8, facecolor= (ancol))="white')
12
13
14
46

---

plt. pit.tight. layout ()
plt. savefig /ttps//ssssssssaaeeeeeeeeeeeeeee /atp://edo...eeccaaooooootaaaaaa dpi-72)
19
plt. show)how()
10..6--1.00
00,00 2.
0.5
Ind3)
-0.5
ImtE]
-0.5
1.0
knte
-0.5
1.0
0.0
0-0.0,0
A.40440.
3.00
1.0
0.0
-1.0
1.00 P- 1.00
D- 3.00 0.60
3.00 0.20
1.00, 0- 0.20
1.00 0.60
0..0 3.00
1.0
-1.0
-1
RelE)
AN ARRAY OF SPECTRAL GRAPHIS
graph_ flat, param_dict flat cp. sp.spteraal. graph paar dict) print (graph_flat, '\n') print (param_diat_flatt
[<networkx. wwwwsse..mlltiiieee..el <networkx classes. www.sss.m.liagara......... object at 0x000001966DFC3CF00
object at 0x00000019600<c011390>
thetwerkx.clasee wwwp:/caasss..sliiioooo www.imraphh...mmmm MultiGraph object at 0x0000016600CC77000
(a:
47

---

array -2 - -, -0.8, 0. 0.4, 1. -2
-1 -0.8, -0.2,
0.4,
2
-. -0.8 8, -0.2, 2, 0.4,
-2 -1.4, -0.8,
+ -0.2, 0. 4, 1. -2. -1.4, 4, 0.88
0.
0.4, 1. -2. -1.4,
-0.8, - 0.2, 0.4, 1. 1),
b:
array (1-1. -1.
-1. -1. -1. -0.6, 6, -0.6 -0.6, 6, -0.6, 6,
10.6, -0.6, 6, -0.2, -0.2, -0,2, -0.2, -3.2, -0.2, 0.2, 0.2, 0.2,
- 0,2, 0.2, 0.2, 0.6, 0.6, 0.6, 0.6, 0.6, 0.6, 1. 1.
1. 1. 1. 1. 11}
The spectral graph networkx. www..Muliiiraaarr object, which cannot be directly returned as multi-dimensional numpy array of MultiGraph, except for the case of 1D array. In- stead, we return flattened list of networkx MultiGraph objects, and the accompanying param.dict dicttfcccfll is the dictionary that contains the corresponding flattened parameter val- ues.
It's recommended to pass the values of the parameters vectors (ID arrays) instead of higher dimensional ND arrays to avoid the overhead of reshaping the output and the difficulty to retrieve postprocess the spectral graphs.
H LIMITATIONS AND FUTURE WORK
Component fragmentation. Our extraction pipeline struggles when the hopping range or band number becomes large (e.g. for pure theoretical interests that fall outside realistic domain), because extremely low densitics states make the graph skeleton fragile, occasionally fragmenting a con- nected component (as shown in the bottom in figure A4). We term this phenomenon component fragmentation and note that it an intrinsic limitation of the spectral graph per se (see appendix C.6).
comprehensive benchmark. Our contribution centers the dataset and its generator; the benchmark is intended to catalyze follow-up work. We invite the community to perform comprehensive, large-scale, and carefully designed evaluations.
Better designed,
Representation gap. Our reference PyG conversion uses fixed-size, direction-agnostic edge sum- maries, which can discard full continuous details of multi-edge geometry. Future encoders could operate directly on edge coordinate sequences, e.g. explore spline/Bezier bases, curvature/shape descriptors.
Multi-edge modeling. Vanilla attention and pooling are not tailored for heavy edge multiplicity. Multi-edge calti-edge-aware aware mechanisms btpee//uuuydddddddddddddeedd edges, odge-gated updates, sparsified geometric tention, or dual-graph pooling over edges may better exploit information carried by parallel curves.
Temporal modeling (T-HSG-5M), Our static benchmark does cover dynamic tasks. T-HSG-5M enables carly-sequence classification, temporal extrapolation, and Shange-point detec- tion that leverage continuous geometric evolution along Hamiltonian parameters.
48


Reviews:
[{"review_id": "99zalDNDMc", "reviewer": "Reviewer_K5Rg", "summary": "This paper introduces Poly2Graph, an open-source pipeline that converts one-dimensional non-Hermitian crystal Hamiltonians into Hamiltonian spectral graphs (HSGs): spatial multigraphs representing the geometry of complex energy spectra. The authors build HSG-12M, a dataset of 11.6 M static and 5.1 M dynamic multigraphs spanning 1,401 characteristic-polynomial classes. Each graph encodes spectral topology and geometry via node coordinates and multi-edge trajectories on the complex-energy plane. The dataset is positioned as the first large-scale benchmark for spatial multigraph learning, and as a bridge between algebraic physics data and graph machine learning. Benchmarks with popular GNNs (GCN, GAT, GINE, GraphSAGE, etc.) show that edge-aware models outperform edge-agnostic ones, while overall Top-1 accuracies remain moderate ( 30–60 %) and Top-10 accuracies high (95 %).", "questions": "1. Why is a graph representation preferable to treating the spectra as 2-D arrays or polynomial coefficients?\n2. Can you show any downstream task where learning on graphs leads to qualitatively different or improved results than learning on images?\n3. How robust are the extracted graphs to numerical perturbations or thresholding choices?\n4. Would CNNs or transformers on spectral images achieve comparable or better accuracy?\n5. Are there physical or real-data applications planned where HSG-12M would provide measurable benefit?", "limitations": "", "rating": 2, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. Impressive scale and engineering: Poly2Graph automates a previously manual physics workflow, generating > 10 M graphs and compressing 177 TB of raw spectra into 256 GB.\n2. First large-scale multigraph dataset\n3. Novel cross-disciplinary framing: Establishes a link between non-Hermitian band theory and graph representation learning, potentially inspiring new geometry-aware GNNs.\n4. Solid baseline benchmarking: Eight GNNs are compared with consistent training budgets. Results reveal real performance gaps between edge-aware and edge-agnostic models.", "weaknesses": "1. Limited justification of scientific or ML value: The paper convincingly shows that the data can be generated, but not why learning from these graphs is necessary or insightful. The benchmark task, namely classifying Hamiltonian families, appears artificial, with no demonstrated physical or methodological payoff.\n2. Unclear advantage of the graph representation: The authors do not compare to simpler baselines such as CNNs on spectral images, MLPs on polynomial coefficients, or models trained directly on spectral arrays. It remains unproven that representing spectra as graphs yields better or different information.\n3. Representation loss and stability: The extraction from spectra to graph skeletons may discard quantitative details and can fragment edges (acknowledged in Appendix H). No analysis quantifies how much information or robustness is lost.\n4. Synthetic and self-contained: All data are algorithmically generated. No connection is made to experimental measurements or to existing real-world multigraph domains.\n5. Benchmark insight is shallow: Standard GNNs achieve modest accuracies, but this mainly reflects task complexity and training budget, not necessarily new modeling challenges.\n6. Incremental as a benchmark contribution: The work is an engineering milestone rather than a conceptual one. Its usefulness for advancing ML is not clear.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "dviOwTxj7x", "reviewer": "Reviewer_7GxK", "summary": "This submission introduces Poly2Graph, a high-throughput package that converts 1D crystal Hamiltonians into Hamiltonian Spectral Graphs (HSGs). Using Poly2Graph, the authors construct and release HSG-12M which is a large static spatial multigraph dataset with 11.6M static and 5.1M dynamic Hamiltonian spectral graphs across 1,401 classes; For the HSGs, each graph class corresponds to a Hamiltonian family (hopping pattern), the multi-edge spatial geometry is essential and cannot be simplified without loss, and  a GNN surrogate may enable inverse design from desired spectral graphs to candidate material structures (e.g., acoustic metamaterials, circuits, photonic crystals).", "questions": "N/A", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Clear problem framing & significance. Casting Hamiltonian systems as spatial multigraphs is compelling and bridges non-Hermitian quantum physics with graph representation learning.\n\n2. Scale & engineering contribution. The scale (11.6M static; 5.1M dynamic; 1,401 classes) of the graph data is notable.\n\n3. Breadth of tasks. Static graph classification plus temporal, graph-level tasks opens avenues beyond typical node/edge prediction settings in temporal graphs.\n\n4. Inverse-design angle. The connection between hopping patterns ↔ spectral graphs ↔ structure motivates differentiable surrogates and downstream inverse design—a high-impact direction.\n\n5. Writing & organization. The paper reads smoothly; motivation, pipeline, and applications are explained with good intuition.", "weaknesses": "1. What’s fundamentally new on the modeling side?\nBeyond scale and the presence of multi-edges + spatial coordinates, it’s not yet clear which modeling challenges are truly novel relative to existing large-scale domains (molecules, traffic, social). For example: Do standard GNNs (e.g., message passing with geometric encoders) already handle these graphs well? Which failure modes emerge uniquely from spatial multigraphs that are not captured in simple GNNs?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "MCePrSHuXF", "reviewer": "Reviewer_JuyR", "summary": "The paper presents a computational framework (Poly2Graph) and large-scale dataset that map non-Hermitian Hamiltonians of one-dimensional crystals into spectral graphs, i.e., geometric multigraphs embedded in the complex energy plane. The framework provides efficient construction of spectral graphs from Hamiltonians, enabling systematic dataset generation across varying scales. The largest constructed dataset, HSG-12M, contains 11.6 million graphs spanning 1,401 distinct Hamiltonian (characteristic polynomial) classes. The authors also benchmark multiple graph neural network (GNN) architectures on a graph-level classification task: predicting the underlying Hamiltonian class from the corresponding spectral graph.", "questions": "See Weaknesses.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- Non-Hermitian physics has emerged as an important area in condensed matter and atomic physics. This work establishes a physics-grounded benchmark for evaluating and developing machine learning models on spatial multigraphs derived from physical systems.\n- The dataset is impressive in scale ($\\approx$12 million graphs), and the automated spectral graph extraction pipeline is technically solid, with potential benefits for the broader physics community.", "weaknesses": "The paper does not clearly explain whether the proposed task, predicting the Hamiltonian class from its spectral graph, corresponds to a physically meaningful scenario. Specially, my question is two-fold:\n- This formulation implicitly assumes situations where the Hamiltonian is unknown but the spectral graph itself can somehow be measured, possibly through experiments. A key question is whether the spectral graph, as defined in this work, represents a physically measurable quantity that can be constructed from experimental data.\n- Also, the authors may elaborate on why predicting the Hamiltonian class represents a meaningful task in physics. \n\nI would be glad to raise my score if the authors can adequately address these concerns.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "elwv4ljZex", "reviewer": "Reviewer_aSbR", "summary": "The paper introduces Poly2Graph, an automated pipeline converting 1-D crystal Hamiltonians into spectral graphs. This enables the creation of HSG-12M, a large-scale dataset of spatial multigraphs, which is a new benchmark for geometry-aware graph learning.", "questions": "Please see the weaknesses.", "limitations": "", "rating": 6, "confidence": 1, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The work introduces HSG-12M, a large-scale dataset. It addresses a clear gap by providing the first large-scale benchmark focused on spatial multigraphs , which are underrepresented in graph ML.\n\nThe authors release the Poly2Graph pipeline. This open-source tool enables researchers to generate new spectral graph datasets , promoting reproducibility and future study.", "weaknesses": "As the reviewer is not an expert in this area, the following points are offered from a more general perspective:\n\n1. The paper’s heavy reliance on domain-specific concepts from non-Hermitian physics and algebraic geometry (e.g., Ronkin functions)  might make the data generation process inaccessible to the broader graph ML community. The physical motivation for the benchmark task (inverse design) feels highly specialized.\n\n2. The benchmark evaluation focuses on standard GNNs (GCN, GAT, GIN)  not explicitly designed for spatial multigraphs. The paper identifies this as a new challenge but stops short of benchmarking against more suitable, geometry-aware architectures, weakening the analysis of the dataset's specific challenges.\n\n3. A key feature is the rich, geometric nature of the multi-edges. However, the reference benchmark discards this by collapsing edge geometry into fixed-size summary features (e.g., length, midpoint). This featurization, while practical, may underutilize the dataset's novel geometric complexity.\n\n4. The authors acknowledge a limitation in the Poly2Graph pipeline called \"component fragmentation\", where extremely low densities of states can cause graphs to be spuriously disconnected. This raises potential concerns about the topological fidelity of the generated graphs in complex cases.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)