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arXiv · 2204.13278

Sums of Distances on Graphs and Embeddings into Euclidean Space

Stefan Steinerberger
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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 $\phi: G \rightarrow \ell^1(\mathbb{R}^m)$ with good properties.

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Stefan Steinerberger. 2022-04-28. Sums of Distances on Graphs and Embeddings into Euclidean Space. https://arxiv.org/abs/2204.13278

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