Owada, T and Samorodnitsky, G and Thoppe, G (2021) Limit theorems for topological invariants of the dynamic multi-parameter simplicial complex. In: Stochastic Processes and their Applications, 138 . pp. 56-95.
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Abstract
The topological study of existing random simplicial complexes is non-trivial and has led to several seminal works. However, the applicability of such studies is limited since a single parameter usually governs the randomness in these models. With this in mind, we focus here on the topology of the recently proposed multi-parameter random simplicial complex. In particular, we introduce a dynamic variant of this model and look at how its topology evolves. In this dynamic setup, the temporal evolution of simplices is determined by stationary and possibly non-Markovian processes with a renewal structure. Special cases of this setup include the dynamic versions of the clique complex and the Linial�Meshulam complex. Our key result concerns the regime where the face-count of a particular dimension dominates. We show that the Betti number corresponding to this dimension and the Euler characteristic satisfy a functional strong law of large numbers and a functional central limit theorem. Surprisingly, in the latter result, the limiting process depends only upon the dynamics in the smallest non-trivial dimension. © 2021 Elsevier B.V.
| Item Type: | Journal Article |
|---|---|
| Publication: | Stochastic Processes and their Applications |
| Publisher: | Elsevier B.V. |
| Additional Information: | The copyright for this article belongs to Authors |
| Keywords: | Euler equations, Betti numbers; Euler characteristic; Functional central limit theorem; Functional strong law of large number; Limit theorem; Multi-parameter simplicial complex; Multiparameters; Non-trivial; Simplicial complex; Topological invariants, Topology |
| Department/Centre: | Division of Electrical Sciences > Computer Science & Automation |
| Date Deposited: | 23 Jul 2021 06:52 |
| Last Modified: | 23 Jul 2021 06:52 |
| URI: | http://eprints.iisc.ac.in/id/eprint/68863 |
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