  In the investigation of any causal mechanisms, such as the brain's causal
networks, the assumption of causal sufficiency plays a critical role. Notably,
neglecting this assumption can result in significant errors, a fact that is
often disregarded in the causal analysis of brain networks. In this study, we
propose an algorithmic identification approach for determining essential
exogenous nodes that satisfy the critical need for causal sufficiency to adhere
to it in such inquiries. Our approach consists of three main steps: First, by
capturing the essence of the Peter-Clark (PC) algorithm, we conduct
independence tests for pairs of regions within a network, as well as for the
same pairs conditioned on nodes from other networks. Next, we distinguish
candidate confounders by analyzing the differences between the conditional and
unconditional results, using the Kolmogorov-Smirnov test. Subsequently, we
utilize Non-Factorized identifiable Variational Autoencoders (NF-iVAE) along
with the Correlation Coefficient index (CCI) metric to identify the confounding
variables within these candidate nodes. Applying our method to the Human
Connectome Projects (HCP) movie-watching task data, we demonstrate that while
interactions exist between dorsal and ventral regions, only dorsal regions
serve as confounders for the visual networks, and vice versa. These findings
align consistently with those resulting from the neuroscientific perspective.
Finally, we show the reliability of our results by testing 30 independent runs
for NF-iVAE initialization.
