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We use topological data analysis to study "functional networks" that weconstruct from time-series data from both experimental and synthetic sources.We use persistent homology with a weight rank clique filtration to gaininsights into these functional networks, and we use persistence landscapes tointerpret our results. Our first example uses time-series output from networksof coupled Kuramoto oscillators. Our second example consists of biological datain the form of functional magnetic resonance imaging (fMRI) data that wasacquired from human subjects during a simple motor-learning task in whichsubjects were monitored on three days in a five-day period. With theseexamples, we demonstrate that (1) using persistent homology to study functionalnetworks provides fascinating insights into their properties and (2) theposition of the features in a filtration can sometimes play a more vital rolethan persistence in the interpretation of topological features, even thoughconventionally the latter is used to distinguish between signal and noise. Wefind that persistent homology can detect differences in synchronizationpatterns in our data sets over time, giving insight both on changes incommunity structure in the networks and on increased synchronization betweenbrain regions that form loops in a functional network during motor learning.For the motor-learning data, persistence landscapes also reveal that on averagethe majority of changes in the network loops take place on the second of thethree days of the learning process.

Persistent homology of time-dependent functional networks constructed from coupled time series