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 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× enriched for the 13-gene KEGG circadian set (7 of the 8 set genes in the 454-gene backbone; corrected Benjamini-Hochberg (BH) pBH=4×10-6) and 4.54× enriched for the 111-gene Reactome set (8 of 16 genes; pBH=1.6×10-4), 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.