Sums of Distances on Graphs and Embeddings into Euclidean Space

04/28/2022

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by   Stefan Steinerberger, et al.

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Let G=(V,E) be a finite, connected graph. We consider a greedy selection of vertices: given a list of vertices x_1, …, 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 ≪ |V|. We prove that this suggests that the graph G is at most 'm-dimensional' by exhibiting an explicit 1-Lipschitz embedding ϕ: G →ℓ^1(ℝ^m) with good properties.