level: research
random geometric graphs connect points when their inner product is large. this work looks at sparse versions where the average degree grows slowly with the number of nodes. the latent vectors come from a sphere or a gaussian distribution. edges are not independent because they share these hidden coordinates. the main result is a sharper bound on the difference between the adjacency matrix and its expectation.
for the spherical model, the bound is of order square root of np log n plus np times the cap threshold. this holds with high probability when np is at least log n. the result improves on a previous bound by liu, mohanty, schramm, and yang by needing weaker assumptions. a similar bound works for the gaussian case after adjusting for random vector lengths. this leads to better global synchronization guarantees for the kuramoto model, a classic model of coupled oscillators.
when the graph is denser, with np much larger than log n, the leading eigenvectors of the adjacency matrix reveal the hidden geometry. the analysis shows that the latent vectors can be recovered from the top eigenspace. this connects spectral properties to the underlying structure and provides a theoretical foundation for spectral clustering and embedding methods on such graphs.
why it matters: tighter spectral bounds give more reliable guarantees for algorithms that use eigenvectors to find clusters or synchronize phases in networks, which is common in data science applications like community detection and sensor localization.