We shall consider a geometric graph model on the "hyperbolic" space, which is characterized by a negative Gaussian curvature. Among several equivalent models representing the hyperbolic space, we treat the most commonly used d-dimensional Poincare ball. One of the main characteristics of geometric graphs on the hyperbolic space is tree-like hierarchical structure. From this viewpoint, we discuss the asymptotic behavior of subtree counts. It then turns out that the spatial distribution of subtrees is crucially determined by an underlying curvature of the space. For example, if the space gets flatter and closer to the Euclidean space, subtrees are asymptotically scattered near the center of the Poincare ball. On the contrary, if the space becomes "more hyperbolic" (i.e., more negatively curved), the distribution of trees is asymptotically determined by those concentrated near the boundary of the Poincare ball.

This is joint work with Yogeshwaran D. at Indian Statistical Institute.