Euclidean vs Graph Metric

The theory of sparse graph limits concerns itself with versions of local convergence and global convergence, see e.g. [41]. Informally, in local convergence we look at a large neighborhood around a random uniformly chosen vertex in a graph and in global convergence we observe the whole graph from afar. In this note rather than surveying the general theory we will consider some concrete examples and problems of global and local convergence, with a geometric viewpoint. We will discuss how well large graph approximate continuous spaces such as the Euclidean space. Or how properties of Euclidean space such as scale invariance and rotational invariance can appear in large graphs. For the theory of unimodular random graphs and stationary graphs see [2] [8]. The first sections consider approximating the Euclidean and Finsler metrics by graphs. We study the emergence of rotational, scale and conformal invariance in large graph metrics. We then move on to comment on random graph metrics. Starting with graphs obtained by perturbing the Euclidean metric, and then moving on to random graphs that are restricted to have a planar topology. In particular, we will study graphs generated by random subdivisions. Local and global graph limits will be woven into the whole discussion.