Nonperturbative quantum gravity has to be a theory independent of any underlying background spacetime geometry. In general, a background independent quantum theory will have to be defined fully relationally without any reference to external auxiliary or background fields. Understanding the mechanism by which geometric notions such as dimension and curvature could emerge as low energy effective degrees of freedom from a background-independent quantum gravity will shed light on nonperturbative quantum gravity. In Quantum Graphity, a fully relational, background-independent quantum system is defined on graphs and low energy Einstein geometry is expected to emerge through statistical mechanics. However, the mechanism by which low energy spacetime geometry emerges is unclear due to two main challenges. First, the graph-theoretic notions that correspond to geometric concepts of dimension and curvature are not clearly understood. Second, the Hamiltonian producing the statistical mechanical dynamics that can lead to low energy Einstein geometry is unknown. In this talk, we demonstrate that an Ising-model-like Hamiltonian dynamics with recursively defined graph dimension (Knill dimension), can produce a geometric dimension and uniform Knill curvature in the low energy limit.