From Persistence to Resilience: New Betti Numbers for Analyzing Robustness in Simplicial Complex Networks
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913839960817664 |
|---|---|
| author | Hernández-García, Pablo Serrano, Daniel Hernández Gómez, Darío Sánchez |
| author_facet | Hernández-García, Pablo Serrano, Daniel Hernández Gómez, Darío Sánchez |
| contents | Persistent homology is a fundamental tool in topological data analysis; however, it lacks methods to quantify the fragility or fineness of cycles, anticipate their formation or disappearance, or evaluate their stability beyond persistence. Furthermore, classical Betti numbers fail to capture key structural properties such as simplicial dimensions and higher-order adjacencies. In this work, we investigate the robustness of simplicial networks by analyzing cycle thickness and their resilience to failures or attacks. To address these limitations, we draw inspiration from persistent homology to introduce filtrations that model distinct simplicial elimination rules, leading to the definition of two novel Betti number families: thick and cohesive Betti numbers. These improved invariants capture richer structural information, enabling the measurement of the thickness of the links in the homology cycle and the assessment of the strength of their connections. This enhances and refines classical topological descriptors and our approach provides deeper insights into the structural dynamics of simplicial complexes and establishes a theoretical framework for assessing robustness in higher-order networks. Finally, we establish that the resilience of topological features to simplicial attacks can be systematically examined through biparameter persistence modules, wherein one parameter encodes the progression of the attack, and the other captures structural refinements informed by thickness or cohesiveness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_10467 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From Persistence to Resilience: New Betti Numbers for Analyzing Robustness in Simplicial Complex Networks Hernández-García, Pablo Serrano, Daniel Hernández Gómez, Darío Sánchez Algebraic Topology Primary 55N31, Secondary 55U10, 05E45, 05C82, 92C42 Persistent homology is a fundamental tool in topological data analysis; however, it lacks methods to quantify the fragility or fineness of cycles, anticipate their formation or disappearance, or evaluate their stability beyond persistence. Furthermore, classical Betti numbers fail to capture key structural properties such as simplicial dimensions and higher-order adjacencies. In this work, we investigate the robustness of simplicial networks by analyzing cycle thickness and their resilience to failures or attacks. To address these limitations, we draw inspiration from persistent homology to introduce filtrations that model distinct simplicial elimination rules, leading to the definition of two novel Betti number families: thick and cohesive Betti numbers. These improved invariants capture richer structural information, enabling the measurement of the thickness of the links in the homology cycle and the assessment of the strength of their connections. This enhances and refines classical topological descriptors and our approach provides deeper insights into the structural dynamics of simplicial complexes and establishes a theoretical framework for assessing robustness in higher-order networks. Finally, we establish that the resilience of topological features to simplicial attacks can be systematically examined through biparameter persistence modules, wherein one parameter encodes the progression of the attack, and the other captures structural refinements informed by thickness or cohesiveness. |
| title | From Persistence to Resilience: New Betti Numbers for Analyzing Robustness in Simplicial Complex Networks |
| topic | Algebraic Topology Primary 55N31, Secondary 55U10, 05E45, 05C82, 92C42 |
| url | https://arxiv.org/abs/2505.10467 |