TawaDiff: Interferometric Crowd-Epidemic Co-Diffusion on Cayley-Structured Pilgrimage Networks

The Hajj pilgrimage draws over 2.5 million pilgrims annually to Makkah, creating the world's largest mass gathering with two coupled public health risks: crowd crush and infectious disease transmission. Standard graph models treat pilgrimage contact networks as random or scale-free, missing the rigid group-theoretic structure that Hajj ritual imposes on crowd movement. We introduce TawaDiff, a Cayley-structured Interferometric Graph Neural Network for joint crowd-density and epidemic co-diffusion modeling over pilgrimage networks. Each Hajj ritual phase -- Tawaf, Sa'i, and Mina encampment -- induces a distinct algebraic group structure on the contact network, expressible as a Cayley graph over a finite group. TawaDiff propagates a dual complex-valued signal encoding both crowd pressure and infection probability via phase-coupled interferometric message passing. We prove the Crush-Epidemic Amplification Theorem, bounding the multiplicative increase in expected transmission when crowd density exceeds a critical threshold, expressed via the Kazhdan constant of the ritual phase. On HajjRitualBench, a new synthetic benchmark calibrated to published Hajj demographic and spatial data, TawaDiff achieves 26.1% lower crowd-density prediction error and 34.8% lower outbreak containment error versus the strongest GNN baseline.