Euclidean vs Graph Metric: The Fixed-Source Problem

The theory of sparse graph limits concerns itself with versions of local convergence and global convergence, see e.g. [44]. 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 graphs 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.