Before revision:  One advantage of the null model we propose is efficiency in generating a null model that applies to a broad set of hypothesis testing procedures. The second key idea in this paper is using the null model to address multiple hypothesis testing via control of family-wise error rates and false discovery rates .
After revision:  One advantage of the null model we propose is efficiency in generating a null model based approach to testing for acyclicity, coupled with a Family-Wise Error Rate (FWER) control method that does not suffer from these computational costs. We adapt an False Discovery Rate (FDR) control approach to the topological setting, and show it to be compatible both with our null model approach and with previous approaches to hypothesis testing in persistent homology. By extending a limit theorem for persistent homology on samples from point processes, we provide theoretical validation for our FWER and FDR control methods .