We use methods from computational algebraic topology to study functional brain networks in which nodes represent brain regions and weighted edges encode the similarity of functional magnetic resonance imaging (fMRI) time series from each region. With these tools, which allow one to characterize topological invariants such as loops in high-dimensional data, we are able to gain understanding of low-dimensional structures in networks in a way that complements traditional approaches that are based on pairwise interactions. In the present paper, we use persistent homology to analyze networks that we construct from task-based fMRI data from schizophrenia patients, healthy controls, and healthy siblings of schizophrenia patients. We thereby explore the persistence of topological structures such as loops at different scales in these networks. We use persistence landscapes and persistence images to represent the output of our persistent-homology calculations, and we study the persistence landscapes and persistence images using k-means clustering and community detection. Based on our analysis of persistence landscapes, we find that the members of the sibling cohort have topological features (specifically, their one-dimensional loops) that are distinct from the other two cohorts. From the persistence images, we are able to distinguish all three subject groups and to determine the brain regions in the loops (with four or more edges) that allow us to make these distinctions.

我们使用计算代数拓扑的方法来研究功能性大脑网络，其中节点代表大脑区域，加权边编码每个区域的功能磁共振成像 (fMRI) 时间序列的相似性。借助这些工具，我们可以表征高维数据中的拓扑不变量（例如循环），我们能够以一种补充基于成对交互的传统方法的方式来理解网络中的低维结构。在本文中，我们使用持久同源性来分析我们从精神分裂症患者、健康对照者和精神分裂症患者的健康兄弟姐妹的基于任务的 fMRI 数据构建的网络。因此，我们探索了拓扑结构的持久性，例如这些网络中不同尺度的循环。k-均值聚类和社区检测。根据我们对持久性景观的分析，我们发现同级队列的成员具有不同于其他两个队列的拓扑特征（特别是它们的一维循环）。从持久性图像中，我们能够区分所有三个主题组并确定允许我们进行这些区分的循环中的大脑区域（具有四个或更多边缘）。