Extended Persistent Betti Numbers

• Extended Persistent Betti Numbers are advanced topological invariants that generalize classical Betti numbers to capture the persistence and structure of homological features across multidimensional filtrations.
• They incorporate extensions like multiparameter persistence, thick and cohesive invariants, and higher-order cup modules to encode robustness and refined algebraic details.
• This framework is widely applied in topological data analysis, network resilience studies, and statistical modeling, leveraging novel computational and categorical techniques.

Extended Persistent Betti Numbers generalize the classical concept of Betti numbers in algebraic topology by quantifying not only the number and persistence of homological features across filtration parameters but also encoding additional structural or algebraic information in both single and multiparameter settings. They emerge in persistent homology, multiparameter persistence, and a variety of extensions that adapt homological invariants to richer, more structurally sensitive summaries for applications in topological data analysis, complex networks, and filtered algebraic or geometric contexts.

1. Definitions and Generalizations

Classical Betti numbers $\beta_n$ count $n$-dimensional topological features (connected components, cycles, cavities) of a space. In persistent homology, given a filtration $(K_r)_{r \ge 0}$ of a simplicial complex or cell complex, persistent Betti numbers $\beta_n^{r,s}$ count the number of $n$-dimensional features that are born at or before parameter $r$ and die strictly after $s$.

Extended Persistent Betti Numbers encompass several generalizations:

• Multiparameter Persistence: Invariants where filtration is indexed by a multivariate parameter, e.g., $(r_1, r_2)$, leading to bigraded Betti numbers or rank invariants (see (Moore, 2020, Kim et al., 2021, Guidolin et al., 2023)).
• Filtrations by Additional Structure: Incorporation of natural “coskeleton” or “cohesion” cofiltrations, or augmentation by algebraic invariants such as cup-length (see (Mémoli et al., 2022, Hernández-García et al., 15 May 2025)).
• Combinatorial and Algebraic Interpretations: Extended Betti tables, barcodes, or support loci computed from minimal free resolutions or Koszul complexes for modules over polynomial rings (Moore, 2020, Guidolin et al., 2023).
• Functional and Statistical Limits: Viewing the full family $(r, s) \mapsto \beta_q^{r,s}$ as a stochastic process or functional object, leading to asymptotic normality and covariance structure in random and dependent data models (Krebs et al., 2019, Krebs, 2019, Botnan et al., 2021).

2. Multiparameter and Bigraded Betti Numbers

The extension from one-parameter to multiparameter persistence changes the algebraic and combinatorial landscape dramatically:

• Bigraded Betti Numbers: For a module $M$ over $\mathbb{N}^2$, the bigraded Betti numbers $\beta_j^M(\alpha)$ count generators and syzygies in the minimal free resolution at each bidegree. These can be computed using combinatorial data extracted from zigzag barcode decompositions and minimal free resolutions (Moore, 2020, Kim et al., 2021).
• Pictorial Formula: For a finitely generated $\mathbb{Z}^2$-module $M$, the bigraded Betti numbers are derived via combinatorial summations over the “corners” of regions associated to intervals in the module’s support:

$$\beta_j(M)(p) = \sum_{I} \operatorname{dgm}(M)(I) \cdot \tau_j(I^+)(p)$$

where $\tau_j(I^+)(p)$ identifies corners of type $j$ in $I^+\subset\mathbb{R}^2$ (Kim et al., 2021).

• Limitation to Two Parameters: The combinatorial “pictorial formula” for bigraded Betti numbers fails in dimensions $d\geq 3$, reflecting the wild representation type of multiparameter persistence (Kim et al., 2021).

3. Structural and Robustness-Enhanced Invariants

To capture more than just “counts,” recent developments augment Betti numbers to describe robustness, thickness, and higher-order adjacency:

• Thick Betti Numbers: For a simplicial complex $X$ and integer $h \geq 0$, the $h$-coskeleton $X^h$ retains only faces that are part of some simplex of dimension at least $h$. The $(n, h)$-thick Betti number is

$\beta^{(n,h)}(X;k) := \dim H^n(X^h; k)$

(Hernández-García et al., 15 May 2025). These quantify the “thickness” (minimal covering simplex dimension) of homology features.

• Cohesive Betti Numbers: These are defined via the cohomology of the subposet of faces in selected dimensions, encoding the degree of higher-order connectivity (cohesion), and may be formulated as

$$\beta^{(n,\mathfrak{h})}(X;k) := \dim H^n(P_X^{(\mathfrak{h})}; k)$$

where $P_X^{(\mathfrak{h})}$ is the poset of all simplices in dimensions $\mathfrak{h}$ (Hernández-García et al., 15 May 2025).

• Persistent Cup Module and Higher-Order Persistent Invariants: The persistent cup module tracks not only the ranks of cohomology maps but also the graded ring structure (multiplicative information) via the (iterated) cup product (Mémoli et al., 2022). For filtration parameter $t$ and cup-length parameter $\ell$,

$$\Phi(X)_t := H^+(X_t) \supseteq (H^+(X_t))^2 \supseteq (H^+(X_t))^3 \supseteq \cdots$$

yields a two-parameter persistence structure (“persistent cup module”).

4. Statistical, Combinatorial, and Computational Aspects

• Statistical Properties: The processes $(r, s) \mapsto \beta_q^{r,s}$ in random or dependent settings are shown to admit strong laws of large numbers and central limit theorems (Krebs et al., 2019, Krebs, 2019, Botnan et al., 2021). The joint process over a grid of parameter pairs is asymptotically Gaussian, and this holds even under strong dependence and in the presence of percolation effects in the underlying random geometric structures.
• Support and Localization in Parameter Space: The support (nonzero locus) of extended Betti tables for multiparameter persistence modules can be sharply localized using the entrance grades of cells in the underlying filtration or via so-called homological critical grades in the case of bifiltrations (Guidolin et al., 2023). Discrete Morse theory helps minimize this support.
• Combinatorial Interpretations: In settings such as filtrations of flag complexes, extremal Betti numbers and persistence barcodes are sharply bounded and related to graph-theoretic objects such as Turán graphs, with explicit edgewise filtrations realizing maximal bar lengths and counts (Beers et al., 28 Feb 2025).
• Computational Methods: Quantum algorithms have been developed to estimate persistent Betti numbers in high-dimensional settings (both single- and multiparameter), leveraging block-encoding, quantum singular value transformation, and Dirac operator constructions for efficient scaling (Hayakawa, 2021, Ameneyro et al., 2022).

5. Applications, Impact, and Advanced Directions

• Network Robustness and Resilience: Extended Betti numbers (specifically thick and cohesive Betti numbers) provide sensitive markers of a network’s ability to preserve topological features under degeneration, with implications for neuroscience, epidemic modeling, and distributed systems (Hernández-García et al., 15 May 2025).
• Multiparameter Goodness-of-Fit and Hypothesis Testing: Statistical tests based on multiparameter persistent Betti numbers offer discriminative power for distinguishing competing null models in spatial statistics, outperforming classical methods like Ripley’s $K$-function in certain nontrivial settings (Botnan et al., 2021).
• Algebraic and Categorical Extensions: Persistent Betti numbers now serve as specializations of more general categorical invariants (functors on categories of modules, sheaves, or filtered objects), accommodating rank, cup-length, and even Lyusternik–Schnirelmann category, each satisfying functorial and stability properties (Mémoli et al., 2022).
• Foundational Insights: The relation between Betti tables and the combinatorics of filtered complexes builds a bridge between persistent topology, commutative algebra, and higher representation theory, with multiparameter persistence motivating new exploration at the interface of these domains.

6. Beyond the Classical Picture: Filtration Structures and Functional Representations

• Biparameter and Multiparameter Modules: The paper of biparameter (and higher) persistence modules, with support and rank invariants analyzed via Möbius inversion and pictorial formulas, reveals a sharp dichotomy: full classification is unattainable beyond the single-parameter case, but locally computable invariants (e.g., bigraded Betti numbers) retain essential geometric information (Moore, 2020, Kim et al., 2021).
• Functional and Landscape Representations: Viewing the collection $(r,s)\mapsto\beta^{r,s}_q$ as a functional object (or “landscape”) enables higher-level statistical summaries and applications to machine learning (see stabilized Betti sequence constructions and robustness properties, (Johnson et al., 2021)).
• Cobordism and Combinatorial Trees: All merger and ramification events in the lifetime of homology classes (beyond simple “birth-death” accounting) can be encoded by filtered matroids, rooted forests, and even cobordisms between spheres, yielding invariants strictly finer than the barcode (Güzel et al., 2022).

7. Limitations, Open Problems, and Future Directions

• Optimality and Classification: Extended Betti numbers in two-parameter settings achieve pictorial formulas, but there is provably no analogous construction in higher parameter dimensions (Kim et al., 2021).
• Integration with Machine Learning Pipelines: Smooth, vectorized, and stable summaries of persistent homology (e.g., stabilized Betti sequences, cumulative Betti vectors) enable integration of topological features into data science workflows (Johnson et al., 2021).
• Continued Extensions: Ongoing work is developing further categorical invariants (such as persistent LS-category), more flexible statistical functionals, and quantum-accelerated algorithms for large-scale topological data analysis (Mémoli et al., 2022, Hayakawa, 2021, Ameneyro et al., 2022).

---

In summary, Extended Persistent Betti Numbers synthesize and generalize classical homological invariants by incorporating persistence across multidimensional filtrations, robustness to structural or stochastic perturbations, higher-order algebraic interactions, and refined combinatorial data from filtered objects. They have become cornerstone invariants in topological data analysis and related fields, capturing the nuances of feature evolution in high-dimensional, heterogeneous, and multiscale data.