  The closed loops or cycles in a brain network embeds higher order signal
transmission paths, which provide fundamental insights into the functioning of
the brain. In this work, we propose an efficient algorithm for systematic
identification and modeling of cycles using persistent homology and the Hodge
Laplacian. Various statistical inference procedures on cycles are developed. We
validate the our methods on simulations and apply to brain networks obtained
through the resting state functional magnetic resonance imaging. The computer
codes for the Hodge Laplacian are given in
https://github.com/laplcebeltrami/hodge.
