🔬 materials science

A Topological\-Geometrical Descriptor (TGD) for Scalable and Interpretable Modeling of Two\-Dimensional Carbon Networks

This paper introduces a compact, physically interpretable Topological-Geometrical Descriptor (TGD) that effectively models the relative energies of two-dimensional carbon networks and vacancy-defected graphene, outperforming larger feature sets while maintaining direct structural interpretability.

Original authors: Rômulo Uriel Liberal de Oliveira, Raphael Matozo Tromer

Published 2026-10-15

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CC BY 4.0

Original authors: Rômulo Uriel Liberal de Oliveira, Raphael Matozo Tromer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world built entirely from carbon, the same element that forms the backbone of life, but arranged in flat, two-dimensional sheets. Scientists have long been fascinated by these materials because their properties change dramatically depending on how the atoms are connected. In a perfect sheet of graphene, every carbon atom is linked to three neighbors in a flawless honeycomb pattern, creating a material that is incredibly strong and conductive. But nature and human ingenuity often introduce imperfections. Atoms can be missing, creating holes, or the rings of atoms can twist into shapes with five or seven sides instead of six. These small changes in the geometric arrangement can drastically alter the energy and stability of the material, making it harder or easier to create, and changing how it behaves. The challenge for researchers has always been to find a simple way to describe these complex, messy patterns so they can predict the material's energy without having to run massive, time-consuming computer simulations for every single new design.

To solve this, a team of researchers at the University of Brasília developed a new method for describing these carbon sheets, which they call a Topological–Geometrical Descriptor. Think of this descriptor as a compact, 19-item checklist that captures the essence of a carbon network's shape. Instead of listing the coordinates of every single atom, which can be thousands, the method looks at the sheet as a map of rings. It counts how many rings have five sides, how many have six, seven, or eight, and calculates the average size and shape of these rings. It also measures a concept called "topological charge," which is a way of quantifying how much a ring deviates from the perfect six-sided hexagon found in graphene. By combining these counts with measurements of the ring areas and how different rings are connected to one another, the method creates a short list of numbers that acts as a fingerprint for the material's structure.

The researchers tested this new fingerprinting system on two different groups of materials. First, they looked at a collection of 120 different, published carbon structures. For each one, they used their 19-item checklist to predict the material's relative energy, which is a measure of how stable it is compared to perfect graphene. They compared their results against a much larger, more complex set of 203 standard features often used in materials science. Surprisingly, their compact 19-item list performed just as well as the massive 203-item list, achieving a high level of accuracy with far less information. This suggests that the specific geometric and topological details they chose to measure are the most important factors determining the energy of these carbon sheets, and that a simpler, more interpretable model can be just as powerful as a complex one.

The team then pushed the method further by testing it on a more difficult problem: graphene sheets with missing atoms, known as vacancies. They created 360 different versions of these sheets, with missing atoms ranging from 2% to 24% of the total. In this scenario, simply counting the number of missing atoms is not enough to predict the energy, because the arrangement of the remaining atoms matters just as much. When the researchers used their new descriptor, it successfully predicted the energy of these defective sheets with greater accuracy than methods that relied only on the number of missing atoms or the fraction of non-hexagonal rings. This proved that the descriptor captures subtle structural details that simpler counts miss, such as how the rings rearrange themselves around the holes.

Throughout the study, the researchers were careful to ensure their method was robust and not just a mathematical artifact. They tested how sensitive their results were to small changes in the rules used to define which atoms are connected, finding that the method remained stable even when those rules were adjusted slightly. They also made it clear that their approach is specifically designed for flat, single-layer carbon sheets and would not work directly for curved tubes or stacked layers without significant changes. While they found a simple mathematical relationship between the topological charge and energy for their specific dataset, they emphasized that this is a useful shortcut for this particular group of materials, not a universal law for all carbon. Ultimately, the work provides a clear, understandable way to link the physical shape of a carbon network to its energy, offering a tool that is both computationally efficient and easy for scientists to interpret.

Technical Summary: Topological–Geometrical Descriptor (TGD) for 2D Carbon Networks

Problem Statement
Two-dimensional (2D) carbon materials exhibit diverse energetic and functional properties, ranging from dense graphene-like lattices to porous $sp/sp^2$ architectures and defect networks. While general-purpose materials featurizers (e.g., Matminer) offer broad descriptor families, they often lack the compactness and physical interpretability required to expose specific structural trends in domain-specific systems. Existing ring statistics provide a foundation for describing medium-range order but often fail to integrate geometric and topological information into a single, interpretable vector suitable for machine learning. The authors address the need for a descriptor that is simultaneously compact, reproducible, and physically interpretable for planar carbon networks, while explicitly avoiding the conflation of different energy targets (e.g., relative allotrope energy vs. vacancy formation energy).

Methodology
The authors introduce the Topological–Geometrical Descriptor (TGD), a 19-feature vector derived from mapping atomic coordinates to a periodic planar graph.

Graph Construction: Structures are treated as embedded planar graphs $G=(V, E)$ with periodic boundary conditions. Neighbors are identified within a radius $r_{max} = 3.5$ Å, with an adaptive bond threshold $d_{max} = \alpha d_0$ (where $\alpha=1.40$ and $d_0$ is the shortest interatomic distance). This allows for mixed $sp$ and $sp^2$ coordination without imposing fixed valency.
Ring Definition: Faces are traced using a deterministic "turn-left" traversal of the planar embedding. This distinguishes TGD from shortest-path ring definitions. Projected ring areas are calculated via the shoelace formula.
Feature Extraction: The 19 input features are categorized as follows:
- Ring Statistics: Fractions of ring sizes ( $p_5, p_6, p_7, p_{8+}$ ) and Shannon entropy ( $S_{topo}$ ).
- Topological Charge: Moments of the charge distribution defined as $q_\ell = 6 - R_\ell$ (relative to graphene), including mean $\langle q \rangle$ , mean absolute charge $\langle |q| \rangle$ , and variance.
- Ring Geometry: Statistics of projected ring areas (mean, median, std dev, coefficients of variation, percentiles, max area, and fraction of area from large rings $R \ge 10$ ).
- Dual-Graph Adjacency: Fractions of 5–7 ring adjacencies and non-hexagon–non-hexagon adjacencies.
- Bonding Scale: The reference bond length $d_0$ .
Datasets and Targets:
1. 120-Allotrope Dataset: Predicts DFT relative energy per atom ( $\Delta E$ ) with respect to graphene.
2. Vacancy-Defected Graphene: Predicts vacancy formation energy ( $E_f^{vac}$ ) for 360 configurations spanning 12 concentrations (2–24%).
Validation Protocol:
- Allotropes: Repeated 5-fold cross-validation (20 repeats) comparing Ridge, SVR, Random Forest, and Gradient Boosting.
- Vacancies: "Leave-one-concentration-out" validation (group-aware) to prevent data leakage between similar defect densities. Baselines include vacancy concentration alone, non-hexagon fraction alone, and $\langle |q| \rangle$ alone.
- Comparison: TGD is benchmarked against a 203-feature Matminer representation under identical validation conditions.

Key Results

Predictive Accuracy (Allotropes): Using Gradient Boosting, TGD achieved a Mean Absolute Error (MAE) of 0.0826 ± 0.0138 eV atom $^{-1}$ and $R^2 = 0.803 \pm 0.074$ . This performance is comparable to the 203-feature Matminer representation (MAE 0.0929 eV atom $^{-1}$ ), despite TGD using only 19 features.
Predictive Accuracy (Vacancies): When entire vacancy concentrations were held out, TGD reduced the pooled MAE to 0.269 eV vacancy $^{-1}$ ( $R^2 = 0.861$ ). This outperformed simple baselines: vacancy concentration alone (MAE 0.373) and non-hexagon fraction alone (MAE 0.343).
Interpretability: The descriptor components map directly to observable structural motifs. For instance, $\langle |q| \rangle$ and mean projected ring area were identified as the most important features for the allotrope dataset. The study demonstrates that ring diversity (entropy) and deviation from hexagonal order (charge moments) are distinct structural properties.
Robustness: Sensitivity analysis showed that the inferred edge sets are stable to moderate changes in the adaptive bond factor ( $\alpha$ ). The descriptor generation is computationally efficient, taking significantly less time than the Matminer workflow for the vacancy dataset.
Surrogate Modeling: A one-variable rational function based on $\langle |q| \rangle$ was fitted to the allotrope data, yielding a modest improvement over a linear model but explicitly noted as a dataset-specific surrogate rather than a universal law.

Significance and Claims
The paper claims that TGD successfully combines compact prediction with direct structural interpretation. Its primary contributions are:

Integration: It unifies ring statistics, projected geometry, topological charge, and dual-graph adjacency into a single, low-dimensional vector.
Interpretability: Unlike "black-box" high-dimensional descriptors, TGD features correspond to physical observables (e.g., ring size fractions, area heterogeneity), allowing researchers to trace model decisions back to specific structural motifs.
Rigorous Validation: The study explicitly separates the validation of predictive accuracy from physical interpretability and avoids conflating different energy targets. It demonstrates that while simple metrics (like defect count) capture significant variance, the combined topological-geometric representation provides additional, necessary information for accurate prediction.
Scope and Limitations: The authors explicitly state that TGD is specialized for single-layer planar carbon networks. It is not currently applicable to strongly buckled sheets, nanotubes, multilayers, or chemically heterogeneous 2D materials without modification. The adaptive bond criterion is robust for the tested datasets but may require explicit connectivity checks for structures with extreme local strain or unusually short bonds.

The work concludes that TGD offers a computationally compact framework for comparing planar carbon networks while maintaining a transparent link between graph topology, ring geometry, and learned energetic trends.

Technical Summary: Topological–Geometrical Descriptor (TGD) for 2D Carbon Networks

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