Multiparameter Persistent Homology of Hypergraphs

Student Number

25306

Degree

Master of Science in Mathematics

Department

Department of Mathematical Sciences

Faculty/School

School of Mathematics and Computer Science (SMCS)

Date of Award

Summer 2025

Advisor

Dr. Danish Ali, Assistant Professor, Department of Mathematical Sciences, School of Mathematics and Computer Science (SMCS)

Committee Member 1

Dr. Shaheen Nazeer, Examiner, LUMS

Committee Member 2

Dr. Amir Bashir, Assistant Professor and Chairperson of Mathematical Sciences, Department of Mathematical Sciences

Project Type

Thesis

Access Type

Restricted Access

Document Version

Final

Pages

97

Keywords

Homology, Topological Data Analysis, Persistent Homology, Multiparamter Persistent Homology, Hypergraphs, Category Theory, Abelian Category

Abstract

To model the relational type data, Hypergraphs capture the multiway relationship. Multiple parameters often affect this data. The Multiparameter Persistent Homology of Hypergraphs builds upon the work of Jie Wu and Shiquan Ren on “The Stability of Persistent Homology of Hypergraphs” [77], which introduced embedded homology to capture the homological features of hypergraphs. The embedded homology positions the homology of a hypergraph H between the homology of its associated and lower-associated simplicial complexes. This concept inspired the development of multiparameter persistent homology for hypergraphs, where the embedded structure evolves “together” through the poset P to define the embedded homology of the hypergraph at each point. This structure is modeled as an I-shaped diagram in the category of hypergraphs HypGph, forming the category of such functors HypGphI . A filtration functor F : P → HypGphI is defined, along with a persistent module M : P → VectI , where VectI denotes the category of I-shaped diagrams in Vect. The interleaving distance between these persistent modules is defined by inspiration from Lesnick’s work [47]. It is also observed that the category of Persistent Modules is abelian. The main results include the stability of multiparameter sublevel filtrations and the multiparameter stability of morphisms between hypergraphs. The stability of a morphism means that the persistent maps induced by a morphism between two hypergraphs remain stable under perturbations of the filtrations applied to both the source and target hypergraphs. This result is proven using the properties of abelian categories.

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