Persistent Betti Numbers in TDA

• Persistent Betti Numbers are quantitative invariants that measure the birth and persistence of homological features in filtered topological spaces.
• They generalize classical Betti numbers by evaluating the multiscale stability of features, with applications in both single- and multiparameter persistence.
• Their use spans disciplines such as cosmic web analysis, particle physics, and genomics, providing actionable insights for interpreting complex data structures.

Persistent Betti numbers are numerical invariants that encode, for each combination of homological degree and filtration parameter(s), the number of homological features that are born and persist up to prescribed scales in filtered topological spaces. They play a central role in persistent homology and multiparameter persistence modules, providing a systematic method to track the evolution and lifetimes of topological features across one or more varying parameters. Persistent Betti numbers generalize classical Betti numbers by resolving not only the existence but also the multiscale stability of topological features, with far-reaching implications in both theoretical and applied topology, topological data analysis (TDA), commutative algebra, and random topology.

1. Classical Definitions: Single-Parameter Persistent Homology

Let $\{K_t\}_{t\ge 0}$ denote a filtration of simplicial complexes or cell complexes, i.e., an increasing family $K_s \subseteq K_t$ for $s\le t$. For a fixed homological degree $p$ and field of coefficients $\mathbb{F}$, the $p$-th persistent Betti number at scales $(s,t)$ ($s \leq t$) is defined as

$$\beta_p^{s,t} := \dim_{\mathbb{F}} \operatorname{Im} \left[ H_p(K_s) \to H_p(K_t) \right].$$

This counts the number of $p$-dimensional homology classes created (born) by time $K_s \subseteq K_t$0 and still alive (not yet killed by boundaries) at time $K_s \subseteq K_t$1 (Güzel et al., 2022, Pranav et al., 2016, Botnan et al., 2021). The collection $K_s \subseteq K_t$2 records the instantaneous Betti number at $K_s \subseteq K_t$3, while the barcode or persistence diagram records individual birth-death pairs $K_s \subseteq K_t$4 of classes, satisfying

$K_s \subseteq K_t$5

Barcodes are a complete invariant for one-parameter persistence over fields; persistent Betti numbers (or "persistence ranks") are their direct numeric summaries (Güzel et al., 2022).

2. Multiparameter Persistent Betti Numbers and Betti Tables

In multiparameter persistence, the filtration is indexed by a poset $K_s \subseteq K_t$6 (for $K_s \subseteq K_t$7 parameters). A multiparameter persistence module $K_s \subseteq K_t$8 is a finitely generated $K_s \subseteq K_t$9-graded module over $s\le t$0, with

$s\le t$1

The multigraded Betti numbers, or Betti table, are defined as

$s\le t$2

where $s\le t$3 and $s\le t$4. For each homological degree $s\le t$5, one forms a graded persistence module $s\le t$6, and the support of its Betti table, $s\le t$7, is systematically controlled by the entrance grades at which $s\le t$8- and $s\le t$9-cells first appear in the multifiltered cell complex (Guidolin et al., 2023).

In the two-parameter case, the notions specialize to bigraded Betti numbers. For a $p$0-parameter persistence module $p$1, described as a functor $p$2, the bigraded Betti numbers $p$3 enumerate the multiplicities of standard "upper-set" modules in a minimal bigraded free resolution (Kim et al., 2021, Moore, 2020).

3. Combinatorial and Algebraic Formulas: Barcodes, Ranks, and Möbius Inversion

For $p$4-parameter persistence, persistent Betti numbers are uniquely determined by the collection of barcodes: $p$5 and the barcode can be recovered from all $p$6 (Güzel et al., 2022, Pranav et al., 2016).

In the $p$7-parameter case, Betti numbers are not determined by barcodes due to wild representation type, but the bigraded Betti numbers can be combinatorially reconstructed via the generalized rank invariant $p$8 (which assigns the rank of the canonical map $p$9 to each connected subposet $\mathbb{F}$0). Möbius inversion over the poset of connected subsets yields a generalized persistence diagram $\mathbb{F}$1, and the bigraded Betti numbers are obtained by counting corners: $\mathbb{F}$2 where $\mathbb{F}$3 counts the type-$\mathbb{F}$4 corners of $\mathbb{F}$5 at $\mathbb{F}$6 (Kim et al., 2021, Moore, 2020). This formula is optimal for $\mathbb{F}$7, but cannot be extended for $\mathbb{F}$8 due to nonrecoverability (Kim et al., 2021).

For multiparameter filtrations arising from cell complexes, the support of persistent Betti numbers is sharply constrained by the entrance grades of cells and their least upper bounds. In particular, for a one-critical multifiltration, support obeys

$\mathbb{F}$9

with refinements in the $p$0-parameter (bifiltration) case via homological critical grades $p$1 (Guidolin et al., 2023).

4. Statistical Properties and Asymptotics

For point-cloud data, persistent Betti numbers $p$2 from Čech or Vietoris–Rips filtrations admit a detailed probabilistic theory:

• In the subcritical regime $p$3, if $p$4 is the homological degree, $p$5, where $p$6 is the minimal support size for a persistent $p$7-cycle (Bauer et al., 2018).
• In the critical (thermodynamic) regime, properly normalized persistent Betti numbers converge to deterministic limits, with central limit theorems and functional CLTs established for broad classes of stationary or dependent data (Krebs et al., 2019, Krebs, 2019, Botnan et al., 2021, Krebs et al., 2020).
• For finite point clouds in $p$8, any fixed persistence window $p$9 admits a linear bound: $(s,t)$0 for an explicit constant $(s,t)$1, uniformly in all $(s,t)$2, all point configurations, and across Čech, $(s,t)$3, and Vietoris–Rips complexes (Edelsbrunner et al., 2024).

Statistical applications include consistent and asymptotically normal estimation of persistent Betti numbers for complex random models, power for goodness-of-fit tests in TDA, and bootstrapped confidence intervals using the smoothed bootstrap (Roycraft et al., 2020). Multivariate Gaussianity enables rigorous inference for vector-valued persistence summaries.

5. Computation, Extensions, and Quantum Approaches

Algorithmic pipeline: Persistent Betti numbers are computed by assembling filtered chain complexes, computing boundary operators, and reducing boundary matrices to normal forms (via, e.g., matrix reduction or discrete Morse theory) (Pranav et al., 2016, Guidolin et al., 2023). For multiparameter modules, Betti tables require minimal multigraded free resolutions (by Gröbner basis or syzygy algorithms), with complexity determined by the number of parameters and support of generators.

Quantum computation: Quantum algorithms for persistent Betti numbers leverage Dirac-type or block-encoded Laplacian operators, projecting onto their kernel blocks via phase estimation or singular value transformation, and yield exponential speedup in the dimension of the complex under appropriate gap assumptions (Ameneyro et al., 2022, Hayakawa, 2021).

Advanced variants:

• Graded Betti numbers and their persistent counterparts (as in persistent commutative algebra) generalize these invariants to module and ideal filtrations, enabling their application to, e.g., edge ideals or k-mer based models in machine learning, where persistent graded Betti numbers can be paired with transformers to achieve high predictive power (Zia et al., 27 Oct 2025, Suwayyid et al., 19 Dec 2025).
• Going beyond persistence, thick and cohesive Betti numbers capture the resilience, thickness, and higher-order connectivity patterns of cycles under more general filtrations or "attack" models, with the associated bipersistence modules encoding more nuanced structural information (Hernández-García et al., 15 May 2025).

6. Extremal, Stability, and Structural Results

• Extremal combinatorics of persistent Betti numbers reveal Turán-type results: for flag complex filtrations of graphs, maximal Betti numbers and persistent barcodes are achieved at Turán graphs, with explicit filtrations maximizing bar counts and total persistence (Beers et al., 28 Feb 2025).
• Stability of persistent Betti number summaries such as Betti sequences is subtle; the naïve Betti curve is not Lipschitz-stable under $(s,t)$4-Wasserstein perturbations of the barcode, but smoothed or cumulative versions regain stability and are usable as machine learning features (Johnson et al., 2021).
• Beyond vector counts, combinatorial refinements using matroid theory, ramification trees, and cophenetic distances enrich the description of dependencies among homology generators, connecting persistent Betti numbers to richer algebraic invariants (Güzel et al., 2022).

7. Applications and Interpretation

Persistent Betti numbers function as fundamental descriptors for multiscale topology:

• In cosmic web analysis, $(s,t)$5 and their diagrams reveal the hierarchical and morphologic structure of cluster, filament, wall, and void features from galaxy catalogs (Pranav et al., 2016, Roycraft et al., 2020).
• In particle physics, persistent Betti numbers distinguish quark and gluon jets by quantifying the longevity, multiplicity, and branching persistence of topological features in event jet topologies (Li et al., 2020).
• In biomolecular sequence analysis and alignment-free genomics, persistent graded Betti invariants extracted from edge ideals and module filtrations enable accurate classification and motif discovery by leveraging the persistence and synergy of syzygies over scale (Zia et al., 27 Oct 2025, Suwayyid et al., 19 Dec 2025).

In sum, persistent Betti numbers and their generalizations provide a robust and highly flexible algebraic, combinatorial, and statistical toolkit for tracking and interpreting topological structure in both deterministic and stochastic settings, encoding the full path of homology classes across multifiltration landscapes and supporting stable, scalable, and interpretable applications in modern data science and pure mathematics.