  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.
