Graphity models are characterized by configuration spaces in which states correspond to graphs and Hamiltonians that depend on local properties of graphs such as degrees of vertices and numbers of shortcycles. It has been argued that such models can be useful in studying how an extended geometry might emerge from a background independent dynamical system. As statistical systems, graphity models can be studied analytically by estimating their partition functions or numerically by Monte Carlo simulations. In this talk I will present recent results obtained using both of these approaches. In particular, I will describe the transition between the high and low temperature regimes and arguethat matter degrees of freedom must play an important role in order for the graph states dominating in the low temperature regime to resemble interesting extended geometries.