  We define notions of local topological convergence and local geometric
convergence for embedded graphs in $\mathbb{R}^n,$ and study their properties.
The former is related to Benjamini-Schramm convergence, and the latter to weak
convergence of probability measures with respect to a certain topology on the
space of embedded graphs. These are used to state universality conjectures for
the long-term behavior of the network flow, or curvature flow on embedded
graphs. To provide evidence these conjectures, we develop and apply
computational methods to test for local topological and local geometric
convergence.
