Although random cell complexes occur throughout the physical sciences, there does not appear to be a standard way to quantify their statistical similarities and differences. I'll introduce the method of swatches, which describes the local topology of a cell complex in terms of probability distributions of local configurations. It allows a distance to be defined which measures the similarity of the local topology of cell complexes. Convergence in this distance is related to the notion of a Benjamini Schramm graph limit. In my talk, I will use this to state universality conjectures about the long-term behavior of graphs evolving under curvature flow, and to test these conjectures computationally. This system is of both mathematical and physical interest. If time permits, I will discuss other applications of computationally topology to curvature flow on graphs, and describe recent work on a new notion of geometric graph limit.