  Understanding the common topological characteristics of the human brain
network across a population is central to understanding brain functions. The
abstraction of human connectome as a graph has been pivotal in gaining insights
on the topological properties of the brain network. The development of
group-level statistical inference procedures in brain graphs while accounting
for the heterogeneity and randomness still remains a difficult task. In this
study, we develop a robust statistical framework based on persistent homology
using the order statistics for analyzing brain networks. The use of order
statistics greatly simplifies the computation of the persistent barcodes. We
validate the proposed methods using comprehensive simulation studies and
subsequently apply to the resting-state functional magnetic resonance images.
We found a statistically significant topological difference between the male
and female brain networks.
