Quantum Graphity models have been proposed as one approach towards Quantum Gravity in which a fully quantum mechanical, background independent system in the high energy gives rise to a low energy state that looks like ordinary spacetime. The high energy states of the Quantum Graphity system are random graphs with microscopic degrees of freedom that live on the edges and vertices. In the low emergy limit, a discretized regular lattice emerges. While this program is promising, much work remains to be done to show that such a low dimensional structure and dynamics results through statistical mechanics. In this talk, I will describe how a uniform dimension emerges in the low energy starting with an assumption of a high energy Hamiltonian that is quadratic in the vertex degrees of the nodes of the graph.