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Human whole-brain functional connectivity networks have been shown to exhibit both local/quasilocal (e.g., a set of functional sub-circuits induced by node or edge attributes) and non-local (e.g., higher-order functional coordination patterns) properties. Nonetheless, the non-local properties of topological strata induced by local/quasilocal functional sub-circuits have yet to be addressed. To that end, we proposed a homological formalism that enables the quantification of higher-order characteristics of human brain functional sub-circuits. Our results indicate that each homological order uniquely unravels diverse, complementary properties of human brain functional sub-circuits. Noticeably, the H1 homological distance between rest and motor task was observed at both the whole-brain and sub-circuit consolidated levels, which suggested the self-similarity property of human brain functional connectivity unraveled by a homological kernel. Furthermore, at the whole-brain level, the rest–task differentiation was found to be most prominent between rest and different tasks at different homological orders: (i) Emotion task (H0), (ii) Motor task (H1), and (iii) Working memory task (H2). At the functional sub-circuit level, the rest–task functional dichotomy of the default mode network is found to be mostly prominent at the first and second homological scaffolds. Also at such scale, we found that the limbic network plays a significant role in homological reconfiguration across both the task and subject domains, which paves the way for subsequent investigations on the complex neuro-physiological role of such network. From a wider perspective, our formalism can be applied, beyond brain connectomics, to study the non-localized coordination patterns of localized structures stretching across complex network fibers. 

RNA sequencing data have been abundantly generated in biomedical research for biomarker discovery and other studies. Such data at the exon level are usually heavily tailed and correlated. Conventional statistical tests based on the mean or median difference for differential expression likely suffer from low power when the between-group difference occurs mostly in the upper or lower tail of the distribution of gene expression. We propose a tail-based test to make comparisons between groups in terms of a specific distribution area rather than a single location. The proposed test, which is derived from quantile regression, adjusts for covariates and accounts for within-sample dependence among the exons through a specified correlation structure. Through Monte Carlo simulation studies, we show that the proposed test is generally more powerful and robust in detecting differential expression than commonly used tests based on the mean or a single quantile. An application to TCGA lung adenocarcinoma data demonstrates the promise of the proposed method in terms of biomarker discovery.