Recovering Candidate Circadian Regulators of Arrhythmic Pituitary Hormone Genes Using Reliability-Weighted Magnetic Laplacian with rwMagLap

We study how to recover candidate circadian-clock regulators of pituitary hormone genes that are important for women's health but do not show a clear 24-hour rhythm in bulk tissue, aiming to nominate clock-linked regulatory targets that could inform future chronopharmacologic and chronotherapeutic strategies. We propose \textbf{rwMagLap}, which builds a graph on rhythmic backbone genes. For each edge, we combine 24-hour fit quality with peak-time phase, represented as a complex unit-circle value, yielding a Hermitian adjacency matrix and a magnetic Laplacian. We insert arrhythmic hormone genes, treated as anchors, by a reliability-weighted nearest-neighbor projection. The projected anchor-neighbor weights are pooled into a soft teleport distribution, and complex personalized PageRank then ranks rhythmic backbone genes by the magnitude of their PageRank scores. In pituitary data, we find that all 11 women's-health anchors are arrhythmic. Even so, we find that the top-50 list is $7.95\times$ enriched for the 13-gene KEGG circadian set (7 of the 8 set genes in the 454-gene backbone; corrected Benjamini-Hochberg (BH) $p_{\mathrm{BH}}=4\times10^{-6}$) and $4.54\times$ enriched for the 111-gene Reactome set (8 of 16 genes; $p_{\mathrm{BH}}=1.6\times10^{-4}$), while a phase-blind real-valued baseline recovers none. We recover candidates through reliability weighting and phase-aware seeding rather than through magnetic propagation. The magnetic phase adds a different capability: it represents temporal order. On pituitary backbone, the magnetic embedding recovers measured peak-time order of connected pituitary genes with accuracy $0.971$, while $q{=}0$, i.e., no magnetic charge, is at chance.