Sharp detection and triangle likelihood limits for quadratic spherical kernel graphs

We study whether quadratic spherical geometry can be detected from a single observed graph. Latent positions are independent and uniform on $\mathbb S^{d-1}$; conditional on them, edges are independent with connection probability $K_{d,β}(t)=[1+β(dt^2-1)/(d-1)]/2$, where $0<|β|\le1$. We test this mo