Sums of Distances on Graphs and Embeddings into Euclidean Space

Abstract

Let $G=(V,E)$ be a finite, connected graph. We consider a greedy selection of vertices: given a list of vertices $x_1,\dots ,x_k$, take $x_{k+1}$ to be any vertex maximizing the sum of distances to the existing vertices and iterate: we keep adding the `most remote' vertex. The frequency with which the vertices of the graph appear in this sequence converges to a set of probability measures with nice properties. The support of these measures is, generically, given by a rather small number of vertices $m\ll |V|$. We prove that this suggests that the graph $G$ is at most '$m$-dimensional' by exhibiting an explicit $1-$Lipschitz embedding $\varphi :G\to {\ell }^1({\mathbb{R}}^m)$ with good properties.