Higher-order statistics for constructing centered edge functional connectivity

Functional connectivity is often constructed to understand functional organizations in neural systems, where the brain regions and their pairwise interactions are viewed as nodes and edges, respectively. In practice, functional connectivity is commonly estimated via the correlation of pairs of brain regions. One limitation is that the correlation coefficient captures only the pairwise linear dependence relationship between pairs of nodes and may fail to capture complex higher-order relationships. Recently, a novel concept known as edge-centric functional connectivity (eFC) has been introduced to measure interactions between pairs of edges based on the cofluctuation of two nodal time series, offering a new perspective for understanding brain networks. Nevertheless, eFC considers the absolute levels of edge time series, that is, their mean values, in estimation. If the parameter of interest is the covariation between a pair of edges, incorporating mean values of edge time series can introduce bias or deviation, resulting in a skewed estimation. In this manuscript, we propose an alternative approach to estimate the unbiased covariation between pairs of edges, termed centered edge functional connectivity (ceFC), with theoretical foundations. We demonstrate that the proposed estimator is consistent with a sufficient sample size or number of time frames. Additionally, we develop a multiple hypothesis testing framework with a controlled false discovery rate to evaluate the strength of the unbiased covariation among edges. Furthermore, we employ thresholding to obtain a thresholded estimator that has been shown to converge to the true ceFC matrix in high-dimensional settings in which the number of nodes is much larger than the number of samples or time frames. We validate the finite sample performance of the proposed methods via numerical studies and a data application using the Midnight Scan Club dataset.