The Wreaths of KHAN: Uniform Graph Feature Selection with False Discovery Rate Control

Graphical models find numerous applications in biology, chemistry, sociology, neuroscience, etc. While substantial progress has been made in graph estimation, it remains largely unexplored how to select significant graph signals with uncertainty assessment, especially those graph features related to topological structures including cycles (i.e., wreaths), cliques, hubs, etc. These features play a vital role in protein substructure analysis, drug molecular design, and brain network connectivity analysis. To fill the gap, we propose a novel inferential framework for general high dimensional graphical models to select graph features with false discovery rate controlled. Our method is based on the maximum of $p$-values from single edges that comprise the topological feature of interest, thus is able to detect weak signals. Moreover, we introduce the $K$-dimensional persistent Homology Adaptive selectioN (KHAN) algorithm to select all the homological features within $K$ dimensions with the uniform control of the false discovery rate over continuous filtration levels. The KHAN method applies a novel discrete Gram-Schmidt algorithm to select statistically significant generators from the homology group. We apply the structural screening method to identify the important residues of the SARS-CoV-2 spike protein during the binding process to the ACE2 receptors. We score the residues for all domains in the spike protein by the $p$-value weighted filtration level in the network persistent homology for the closed, partially open, and open states and identify the residues crucial for protein conformational changes and thus being potential targets for inhibition.