Topological Descriptors: Theory & Applications

• Topological descriptors are mathematical summaries that encode the connectivity, cycles, and holes in data for quantification and comparison.
• They are computed via methods like persistent homology, Betti curves, and degree-based indices using efficient computational algorithms.
• Their application spans materials science, imaging, neuroscience, and network analysis, balancing interpretability, stability, and computational efficiency.

A topological descriptor is a mathematical or algorithmic summary that encodes the topology of a dataset, dataset-derived object, or network — that is, the aspects of data related to connectivity, cycles, holes, and higher-order features — for the purpose of quantification, comparison, statistical learning, or interpretability. These descriptors are central to topological data analysis (TDA), materials informatics, chemical graph theory, image analysis, neuroscience, and network science, among other disciplines. They encompass both invariant scalars (e.g. the Euler characteristic), multivariate summaries (e.g. Betti curves, persistence diagrams, degree-based graph indices), and vector-derived representations (e.g. persistence images), and are designed for use in computational pipelines, theoretical analyses, and machine learning workflows. The field encompasses a spectrum ranging from highly expressive, high-dimensional summaries (e.g. persistent homology transforms) to compressed, interpretable fingerprints used for regression, classification, or clustering.

1. Mathematical Foundations of Topological Descriptors

At the core, topological descriptors capture features invariant under continuous deformations by considering algebraic-topological invariants or features derived from filtrations on data structures. For a topological space or a filtered combinatorial object (simplicial complex, cubical complex, graph), the homology groups $H_k$ provide a basis for many descriptors, where the $k$-th Betti number $\beta_k = \text{rank}\,H_k$ enumerates $k$-dimensional holes — components ($k=0$), tunnels ($k=1$), voids ($k=2$), etc.

A filtration is a nested sequence $X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n$ (often induced by a scalar function on data), forming the basis for persistent homology (Yan et al., 2021). Persistent homology computes, at every scale, the homology groups $H_k(X_i)$, records the birth and death parameter for each homology generator, and encodes this information in the persistence diagram $\{(b_i,d_i)\}_{i=1}^{N_k}$ for each $k$. Alternative descriptors derived from this structure include Betti curves $\beta_k(\varepsilon)$, which count the number of $k$-dimensional features alive at parameter $\varepsilon$ (Szymanski et al., 22 Feb 2025, Zeppelzauer et al., 2016), and the Euler characteristic $\chi(X) = \sum_{i=0}^{n} (-1)^i \beta_i$ (Smith et al., 2021).

Degree-based indices (Randić, Zagreb, Sombor, etc.) in chemical graph theory form another major class, constructed as algebraic sums over edges or vertex pairs weighted by degree, sometimes extended to neighborhood degree sums (Jeyaraj et al., 27 Oct 2025, Mondal et al., 2019). These indices are topological in the sense that they depend only on the graph's combinatorial topology and are invariant under isomorphisms.

2. Construction and Algorithmic Realization

2.1 Persistent Homology and Betti Curves

Given a scalar field $f(x)$ (e.g., electron density $\rho(x)$), topological features across a filtration are quantified via the sequence of superlevel (or sublevel) sets $X(\varepsilon) = \{x \mid f(x) \geq \varepsilon\}$. For each threshold $\varepsilon$, compute homology $H_k(X(\varepsilon))$ and obtain Betti numbers $\beta_k(\varepsilon)$. These Betti numbers, sampled across a grid of filtration parameters, yield the Betti curves:

$$B_k(\varepsilon_i) = \beta_k(\varepsilon_i), \quad i=1,\ldots,M, ~k=0,1,2$$

Concatenating these vectors produces a descriptor amenable to statistical learning, as highlighted in the context of electronic structure analysis of inorganic solids (Szymanski et al., 22 Feb 2025).

Persistent diagrams are more detailed, comprising multisets of $(b_i,d_i)$ birth-death pairs, and can be vectorized as persistence landscapes, silhouettes, or images for downstream tasks (Zeppelzauer et al., 2016, Zeppelzauer et al., 2017).

2.2 Degree-Based and Neighborhood Degree-Based Indices

For molecular graphs, descriptors are computed by partitioning the edge set according to degree or neighborhood-degree pairs — for instance, $E_{i,j}$ records all edges between vertices of degrees $i$ and $j$. Indices are then constructed as

$I(G) = \sum_{i \leq j} m_{i,j} f(i,j)$

where $m_{i,j}$ is the count of such edges and $f$ is a polynomial or rational function (e.g., Randić, forgotten, Balaban, ABC, GA, Sombor) (Jeyaraj et al., 27 Oct 2025, Mondal et al., 2019, Ali et al., 2019). These are computationally inexpensive, often involving only local neighborhoods and histograms.

2.3 Computational Algorithms

Persistent homology computations employ reduction algorithms on boundary matrices, typically in $\mathcal{O}(n^3)$ time for arbitrary complexes, but often much faster for low-dimensional or regular filtrations. For Betti curves and Euler characteristic functions, union-find or breadth-first searches suffice for large images, with near-linear complexity (Smith et al., 2021). Degree-based indices exploit adjacency-list traversals and degree histograms for $\mathcal{O}(n + m)$ scaling, with constant-time per-descriptor performance once edge partitions are available (Jeyaraj et al., 27 Oct 2025). Specialized topological descriptors for large data — such as Betti curves for electron densities — use cubical complexes and periodic boundary conditions as preprocessing (Szymanski et al., 22 Feb 2025).

3. Information Content and Faithfulness

Topological descriptors trade off between expressivity and data compression. Betti curve representations, for example, can compress $10^4$-dimensional electron density fields into $O(10^2)$-long vectors, while retaining information content as measured by Shannon entropy. Both full density and Betti curve representations can reach similar entropy plateaus (e.g., $\sim10.5$ nats), but Betti curves achieve this at much lower dimensionality (Szymanski et al., 22 Feb 2025).

Descriptor faithfulness — the ability to uniquely determine the underlying shape or structure — depends strongly on descriptor type. Verbose descriptors (e.g. verbose persistence diagrams, recording ephemeral and paired events) require fewer directional samples for uniqueness, often as low as $d$ in dimension $d$, compared to concise descriptors (e.g. Betti curves, Euler characteristic functions) which typically require $d+1$ or even $\Omega(n)$ in the number of vertices for complex spaces (Fasy et al., 21 Feb 2024, Fasy et al., 15 Nov 2025).

Sampling bounds are critical in practical applications. Oversampling (dense directional sets) guarantees faithfulness but is computationally expensive, while undersampling can result in loss of structural information, missing vertices, or inability to distinguish between nonisomorphic objects (Fasy et al., 15 Nov 2025).

4. Applications Across Scientific Domains

4.1 Materials Science and Chemistry

Betti curves derived from electron density superlevel filtrations capture explicit bonding characteristics in crystalline solids and drive superior performance in prototype classification, metal/non-metal discrimination, and thermodynamic stability prediction compared to raw density grids and standard structure/composition descriptors, with up to 63 percentage point accuracy improvement for structural class prediction (Szymanski et al., 22 Feb 2025). Persistent homology–based descriptors (e.g. persistence images) enhance methane uptake prediction in nanoporous materials, outperforming and complementing handcrafted porosity features (Krishnapriyan et al., 2020).

Degree-based and neighborhood-degree-based indices underpin QSAR and QSPR models, correlating with thermodynamic, spectroscopic, and biological properties. Closed-form formulas for intricate graph families, such as hex-derived networks, facilitate scalable descriptor generation for large combinatorial libraries (Jeyaraj et al., 27 Oct 2025, Ali et al., 2019).

4.2 Computer Vision and Imaging

For 3D surface analysis, topological descriptors such as persistence diagrams, persistence images, and Betti curves robustly classify fine surface texture, providing state-of-the-art performance and strong complementarity to traditional features (DSIFT, GLCM, HOG). Persistence images show noise stability and outperform convolutional neural networks under certain conditions, achieving Dice Similarity Coefficient up to 0.79 (Zeppelzauer et al., 2017, Zeppelzauer et al., 2016).

4.3 Neuroscience

Neuronal tree morphologies are represented by topological morphology descriptors (TMDs), which are recursively generated barcodes transformed to persistence diagrams and then vectorized for stable, reproducible classification. Perturbation stability is guaranteed under 1-Wasserstein distance, ensuring reliability under imaging noise and biological variability (Beers et al., 2022).

4.4 Networks and Graphs

Topological descriptors for networks — including those under uncertainty — now admit probabilistic analysis, computing expected degree, clustering, and cluster sizes under fuzzy edge-probabilities, enabling robust inference even when network connectivity is only statistically inferred (Raimondo et al., 2020). For graph products, persistent homology descriptors formed on product filtrations strictly enhance expressive power relative to base-graph filtrations, providing new discriminative power for GNNs and classification benchmarks (Ji et al., 12 Nov 2025).

4.5 Scientific Visualization and Scalar Fields

Topological descriptors support symmetry detection, shape matching, clustering, event tracking, and ensemble summarization for scientific scalar fields via persistence diagrams, merge trees, Reeb graphs, Morse–Smale complexes, and their associated distances (bottleneck, $p$-Wasserstein, edit/interleaving) (Yan et al., 2021). Efficient implementations and stability results underpin widespread application in visualization pipelines.

5. Descriptor Comparison, Ordering, and Theoretical Advancements

Recent developments formalize a strict hierarchy and partial order among topological descriptor types (Fasy et al., 21 Feb 2024). Six common descriptors — concise/verbose persistence diagrams, Betti curves, and Euler characteristic functions — are ranked according to their faithfulness and the minimal number of directional samples needed for unique identification. Verbose types (e.g. augmented PDs) are provably stronger than concise types, often by orders of magnitude in required sampling.

Sampling theory establishes that, while exponential bounds in dimension often appear in worst-case faithfulness guarantees, structures in practice (moderate dimension, convex or sparse complexes) are well characterized by much smaller directional sets. Adaptive sampling, error quantification, and practical guidelines now enable reliable descriptor use across data-driven domains (Fasy et al., 15 Nov 2025).

6. Practical Considerations and Limitations

Topological descriptors offer unique interpretability, information compression, and theoretical stability, but their usefulness is conditioned by computational cost (especially for persistent homology in higher dimension), sensitivity to the choice of filtration function, and inherent information loss for global summaries (e.g. Euler characteristic curves). Some features, such as quantum–mechanical bonding/antibonding distinctions or high-order homological torsion, are not captured. For noisy data, statistical aggregation, entropy estimation, or fuzzy descriptors mitigate information loss and instability (Smith et al., 2021, Raimondo et al., 2020).

Descriptor integration into machine learning architectures, including topological neural networks and GNNs augmented with persistent homology, continues to enhance prediction accuracy, dataset clustering, and structural discovery across applications (Verma et al., 5 Jun 2024).

7. Current Directions and Outlook

Research continues on tightening sampling bounds, designing more expressive or adaptive descriptors, integrating phase information or multi-scale features, and efficiently vectorizing persistence-based summaries for high-throughput machine learning frameworks. Open problems include quantifying error under coarse sampling, integrating uncertainty, and extending to multi-field and high-dimensional settings (Fasy et al., 15 Nov 2025, Fasy et al., 21 Feb 2024, Yan et al., 2021).

Topological descriptors now constitute a mature toolkit, grounded by reproducible algorithms, quantitative stability, clear performance and information-content benchmarks, and an established theoretical hierarchy — enabling discovery and quantification of complex structure across the physical, chemical, biological, and computational sciences.