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We use methods from computational algebraic topology to study functionalbrain networks, in which nodes represent brain regions and weighted edgesencode the similarity of fMRI time series from each region. With these tools,which allow one to characterize topological invariants such as loops inhigh-dimensional data, we are able to gain understanding into low-dimensionalstructures in networks in a way that complements traditional approaches thatare based on pairwise interactions. In the present paper, we use persistenthomology to analyze networks that we construct from task-based fMRI data fromschizophrenia patients, healthy controls, and healthy siblings of schizophreniapatients. We thereby explore the persistence of topological structures such asloops at different scales in these networks. We use persistence landscapes andpersistence images to create output summaries from our persistent-homologycalculations, and we study the persistence landscapes and images using$k$-means clustering and community detection. Based on our analysis ofpersistence landscapes, we find that the members of the sibling cohort havetopological features (specifically, their 1-dimensional loops) that aredistinct from the other two cohorts. From the persistence images, we are ableto distinguish all three subject groups and to determine the brain regions inthe loops (with four or more edges) that allow us to make these distinctions.

Topological Data Analysis of Task-Based fMRI Data from Experiments on Schizophrenia