Homology, Homotopy and Applications

Volume 15 (2013)

Number 1

Limit theorems for Betti numbers of random simplicial complexes

Pages: 343 – 374

DOI: https://dx.doi.org/10.4310/HHA.2013.v15.n1.a17


Matthew Kahle (The Ohio State University, Columbus, Ohio, U.S.A.)

Elizabeth Meckes (Case Western Reserve University, Cleveland, Ohio, U.S.A.)


There have been several recent articles studying homology of various types of random simplicial complexes. Several theorems have concerned thresholds for vanishing of homology groups, and in some cases expectations of the Betti numbers; however, little seems known so far about limiting distributions of random Betti numbers.

In this article we establish Poisson and normal approximation theorems for Betti numbers of different kinds of random simplicial complexes: Erdös-Rényi random clique complexes, random Vietoris-Rips complexes, and random Cech complexes. These results may be of practical interest in topological data analysis.


random graph, topological data analysis, central limit theorem, random simplicial complex

2010 Mathematics Subject Classification

05C80, 60D05

Published 1 May 2013