Persistent Entropy in Topological Analysis

• Persistent entropy is a topological statistic that uses Shannon entropy to summarize normalized feature lifetimes, revealing structural complexity and redundancy.
• It is stable under perturbations, scale-invariant, and efficiently computable, making it applicable to various data types like time series, point clouds, networks, and images.
• Persistent entropy aids in detecting phase transitions, differentiating signal classes, and isolating genuine topological features amid noise for actionable insights.

Persistent entropy is a topological statistic that summarizes the distribution of lifetimes in a persistence diagram using the Shannon entropy. Serving as a robust, low-dimensional descriptor of topological complexity, persistent entropy is stable under perturbations, scale-invariant, and broadly applicable to time series, point clouds, networks, images, and dynamical systems. It is frequently employed to detect structural transitions, differentiate signal classes, quantify regularity, and identify genuine topological features amid noise.

1. Mathematical Definition and Basic Properties

Let $B = \{[b_i, d_i)\}_{i=1}^m$ be a persistence barcode for a given filtration (derived from, e.g., a time series, point cloud, graph, or function). Each bar $[b_i,d_i)$ represents a feature born at $b_i$ and dying at $d_i$, with corresponding lifetime $\ell_i = d_i - b_i$. Form the vector of normalized lifetimes:

$$p_i = \frac{\ell_i}{L}, \quad L = \sum_{j=1}^m \ell_j$$

The persistent entropy of $B$ is then the Shannon entropy of this normalized length vector:

$PE(B) = -\sum_{i=1}^{m} p_i \log p_i$

By convention, $0\log 0 = 0$ and only finite bars are considered ($d_i < \infty$). This yields $[b_i,d_i)$0, with $[b_i,d_i)$1 (minimum) for a diagram dominated by one bar, and $[b_i,d_i)$2 (maximum) when all bars have equal length. The measure is permutation invariant, sensitive to the distribution of lifetimes, and encodes redundancy or fragility in topological structure (Atienza et al., 2018, Atienza et al., 2016, Rucco, 8 Feb 2026).

2. Stability, Invariance, and Analytical Theory

Persistent entropy inherits key stability properties from the theory of persistent homology:

• Stability under perturbations: If two filtrations $[b_i,d_i)$3 and $[b_i,d_i)$4 satisfy $[b_i,d_i)$5, and their barcodes differ by at most $[b_i,d_i)$6 in the bottleneck distance, then $[b_i,d_i)$7 is $[b_i,d_i)$8 (Rucco et al., 2015, Atienza et al., 2018). Explicit bounds are available in terms of the diagram distance and total bar length.
• Scale invariance: $[b_i,d_i)$9 is invariant under uniform scaling of $b_i$0, since the normalized proportions $b_i$1 remain unchanged (Atienza et al., 2018, Toscano-Duran et al., 8 Sep 2025).
• Continuity and convergence: Under mild assumptions (finite total persistence, bounded near-diagonal mass), persistent entropy is continuous with respect to convergence of diagrams in bottleneck or $b_i$2-Wasserstein distance. Under large $b_i$3, $b_i$4 converges in probability (and $b_i$5) to $b_i$6 if $b_i$7 (Rucco, 8 Feb 2026).
• Information-theoretic interpretation: As a true Shannon entropy, $b_i$8 quantifies “uncertainty” or “disorder” in the distribution of topological lifetimes—the degree of redundancy (many similar bars) or concentration (few dominant bars) (Atienza et al., 2016).

3. Algorithms and Computational Workflows

The generic pipeline for computing persistent entropy is as follows (Rucco et al., 2015, Alvarado et al., 26 May 2026, Atienza et al., 2016):

1. Construct a filtration from the data (e.g., Vietoris–Rips, Čech, Alpha, lower-star) in the chosen domain.
2. Compute persistent homology to extract birth–death pairs $b_i$9 for a chosen homological degree.
3. Calculate lifetimes: $d_i$0.
4. Normalize: $d_i$1, with $d_i$2.
5. Evaluate PE: $d_i$3.
6. For time-resolved or dynamic systems, repeat over sequential snapshots; for spatial data, apply over relevant subsets or scales.

For application-specific contexts, such as dynamic networks (Alvarado et al., 26 May 2026), image analysis (Atienza et al., 2018), or continual learning (Basterrech, 2024), preprocessing and the construction of the underlying metric or filtration must be adapted accordingly. Efficient computation is $d_i$4 in the number of bars, dominated by the complexity of homology computation (e.g., union-find for $d_i$5 is nearly linear).

4. Applications and Use Cases

Persistent entropy has been successfully deployed in a spectrum of data-driven and theoretical settings:

• Phase transition detection in complex systems: PE provides a provably robust order parameter for phase transitions in stochastic, deterministic, and data-driven contexts. Theoretical results guarantee an asymptotic entropy gap across phases with critical control parameter $d_i$6, provided there is a macroscopic change in diagram structure (Rucco, 8 Feb 2026). Topological stabilization of PE accurately identifies critical parameters in models such as Kuramoto synchronization, Vicsek flocking, and neural network training.
• Time-varying networks: Applied, for example, to the Eastern Mediterranean trade network (0–400 CE), PE of $d_i$7 barcodes on adaptive Rips filtrations quantifies structural regimes, cycle redundancy, and fragility under historical perturbations (Alvarado et al., 26 May 2026).
• Signal classification and time-series analysis: Persistent entropy of the $d_i$8 barcode effectively discriminates between classes (e.g., healthy/faulty motor signals, emotion in speech), showing high classification accuracy and robustness to noise (Rucco et al., 2015, Gonzalez-Diaz et al., 2018).
• Image and texture analysis: PE provides a scalar descriptor of topological/geometric organization in images (e.g., tissue microscopy), distinguishing states along morphogenetic axes and degrees of disorder (Atienza et al., 2018, Atienza et al., 2018).
• Point cloud and shape analysis: As a stable, parameter-free measure for separating topological features from noise, PE supports adaptive denoising and feature selection in high-dimensional geometric data (Atienza et al., 2017, Atienza et al., 2016).
• Cosmology and dynamical fields: Persistent entropy of topological features in excursion sets quantifies the impact of nonlinear and linear redshift space distortions in cosmic matter fields, robustly isolating truly large-scale structure (Abedi et al., 2024).
• Machine learning and continual learning: PE serves as a compact summary statistic for monitoring topological shifts (concept drift) in data streams via topology-preserving projections, outperforming linear dimension reduction methods in detecting regime changes (Basterrech, 2024).
• Biomolecular structure and protein classification: Multiscale persistent entropy, parametrized by a resolution scale, yields natural descriptors of structural regularity and disorder, e.g., in dihedral angle distributions, and supports robust protein classification and the definition of protein structure indices (Xia et al., 2016).

5. Extensions, Variants, and Summary Functions

Several extensions of classical persistent entropy have been developed:

• Length-weighted persistent entropy (LWPE): Instead of normalizing bar lengths, LWPE weights the Shannon entropy directly by the raw lifetime, i.e., $d_i$9, enhancing sensitivity to long-lived features and absolute scale (Toscano-Duran et al., 8 Sep 2025).
• Entropy summary and normalized entropy summary functions: $\ell_i = d_i - b_i$0 refines the Betti curve by incorporating the global persistence significance of features alive at time $\ell_i = d_i - b_i$1 (Atienza et al., 2018).
• Multiscale persistent entropy: Incorporates a resolution parameter $\ell_i = d_i - b_i$2 in rigidity/density filtrations, yielding a continuous family of entropy values that encapsulate topological organization across scales. In biomolecular contexts, this enables scale-dependent discrimination of structural motifs (Xia et al., 2016).

These variants address limitations of the classical scale-invariant entropy and enable richer representational power for function approximation, learning, and interpretability.

6. Theoretical Implications and Limitations

Persistent entropy directly encodes the diversity and concentration of topological feature lifetimes, providing a bridge between algebraic topology and information theory. Its strengths include:

• Broad applicability across domains and data modalities.
• Compatibility with theoretical stability guarantees.
• Parameter-free integrative summary of persistence diagrams.

However, persistent entropy alone may fail to discriminate datasets with identical normalized distributions but distinct absolute feature scales (addressed by LWPE) (Toscano-Duran et al., 8 Sep 2025). Its ability to separate signal from noise can be limited in settings with moderate numbers of medium-length bars or if true features are not well-separated from noise in bar length distribution (Atienza et al., 2016, Atienza et al., 2017).

7. Summary Table: Core Properties of persistent entropy

Property	Description	Source(s)
Formula	$\ell_i = d_i - b_i$3	(Rucco et al., 2015, Rucco, 8 Feb 2026)
Stability	Lipschitz in bottleneck/Wasserstein distance	(Rucco et al., 2015, Atienza et al., 2018)
Scale invariance	Yes (classical PE), Not for LWPE	(Atienza et al., 2018, Toscano-Duran et al., 8 Sep 2025)
Sensitivity	High to bar-length diversity; low when one bar dominates	(Atienza et al., 2016, Rucco et al., 2015)
Computational complexity	$\ell_i = d_i - b_i$4 (entropy); $\ell_i = d_i - b_i$5 for $\ell_i = d_i - b_i$6 diagram	(Rucco et al., 2015)

Persistent entropy is now a standard, rigorously analyzed tool for transforming topological summaries into actionable quantitative features broadly applicable throughout topological data analysis, dynamical systems, statistical learning, and network theory (Rucco, 8 Feb 2026, Alvarado et al., 26 May 2026, Atienza et al., 2018, Atienza et al., 2016, Basterrech, 2024, Atienza et al., 2017, Toscano-Duran et al., 8 Sep 2025, Xia et al., 2016).