Limit Theorems for Persistent Homology of Multiparameter Complex Processes
Abstract
Multiparameter complex processes, based on the Costa-Farber model, provide a common framework for growing random simplicial complexes, including the Linial-Meshulam and random flag complex processes. We establish laws of large numbers for persistent Betti numbers of these processes and prove that, after a deterministic time change and normalization, their persistence diagrams converge weakly in probability to a deterministic finite measure. Its quadrant masses are determined by universal limits of normalized persistent Betti numbers, characterized by an explicit finite-dimensional optimization problem. We extend the diagram convergence to integrals of continuous functions with polynomial growth in the death coordinate and controlled singularities at the diagonal. These results yield convergence of additive and multiplicative lifetime measures and total persistence. We further deduce convergence in optimal transport distance. The proofs combine local weak convergence of simplicial pairs, spectral analysis of compressed Laplacians accounting for dependence through shared faces, and quantitative estimates for large death coordinates and short lifetimes.
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BibTeX
@misc{hino2026limit,
title = {{Limit Theorems for Persistent Homology of Multiparameter Complex Processes}},
author = {Masanori Hino and Shu Kanazawa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12197},
eprint = {2610.12197},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12197},
}
RIS
TY - UNPB
TI - Limit Theorems for Persistent Homology of Multiparameter Complex Processes
AU - Masanori Hino
AU - Shu Kanazawa
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12197
UR - https://arxiv.org/abs/2610.12197
AB - Multiparameter complex processes, based on the Costa-Farber model, provide a common framework for growing random simplicial complexes, including the Linial-Meshulam and random flag complex processes. We establish laws of large numbers for persistent Betti numbers of these processes and prove that, after a deterministic time change and normalization, their persistence diagrams converge weakly in probability to a deterministic finite measure. Its quadrant masses are determined by universal limits of normalized persistent Betti numbers, characterized by an explicit finite-dimensional optimization problem. We extend the diagram convergence to integrals of continuous functions with polynomial growth in the death coordinate and controlled singularities at the diagonal. These results yield convergence of additive and multiplicative lifetime measures and total persistence. We further deduce convergence in optimal transport distance. The proofs combine local weak convergence of simplicial pairs, spectral analysis of compressed Laplacians accounting for dependence through shared faces, and quantitative estimates for large death coordinates and short lifetimes.
ER -