Limits of Embedded Graphs, and Universality Conjectures for the Network Flow

Abstract

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.