From Persistence to Resilience: New Betti Numbers for Analyzing Robustness in Simplicial Complex Networks

TL;DR

The paper addresses the limitation that classical Betti numbers miss hole thickness and higher‑order adjacencies in simplicial networks by introducing two refined invariants: thick Betti numbers, which quantify how holes are supported by higher‑dimensional facets via coskeletal filtrations, and cohesive Betti numbers, which assess higher‑order adjacencies through h‑face posets and finite space cohomology. It develops a persistent viewpoint by using coskeletal and dimension‑restricted refinements and extends this with a biparameter persistence framework to study resilience under simplicial attacks along two axes: attack progression and structural refinement. Key contributions include the formal definitions and invariance properties of thick and cohesive Betti numbers, their interpretation for robustness, and the barycentric subdivision perspective that links cohesive invariants to standard computational pipelines. The work thus provides a richer topological descriptor set for higher‑order networks, enabling potentially impactful applications in neuroscience, cancer dynamics, contagion processes, and topological deep learning, while also outlining practical limitations and future research directions.

Abstract

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.

Figures (33)

• Figure 1: Three simplicial complexes with identical Betti numbers but different structural properties.
• Figure 2: We can represent a $0$-simplex as a point, a $1$-simplex as an edge, a $2$-simplex as a filled triangle, a $3$-simplex as a filled tetrahedron, etc. (A), and a simplicial complex as a set of simplices glued together in such a way that the intersection of any two simplices is another simplex (B).
• Figure 3: $(0,2)$-connected components.
• Figure 4: Cofiltration of a simplicial complex and the barcodes of the persistence modules associated with the $0$-th and $1$-st simplicial cohomology with coefficients in $\mathbb{Z}_2$.
• Figure 5: We can recursively obtain the $h$-coskeleton by eliminating from the $(h-1)$-coskeleton the simplices that are not a face of an $h$-simplex.

Theorems & Definitions (75)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Example 2.5
• Definition 2.6
• Remark 2.7
• Definition 2.8
• Remark 2.9
• Definition 2.10