论文标题
图形上的距离和嵌入到欧几里得空间中的总和
Sums of Distances on Graphs and Embeddings into Euclidean Space
论文作者
论文摘要
令$ g =（v，e）$为有限的连接图。我们考虑了贪婪的顶点选择：给定顶点$ x_1，\ dots，x_k $，以$ x_ {k+1} $为$ x_1，x_k $，以使其最大化现有顶点的距离之和最大化的任何顶点并迭代：我们不断添加“最远程”顶点。该序列中图的顶点出现的频率收敛到具有良好属性的一组概率度量。通常，这些度量的支持是由少数顶点$ m \ ll | v | $给出的。我们证明，这表明图形$ g $最多是“ $ m $ dimensional”，即通过展示$ 1- $ LIPSCHITZ嵌入$ ϕ：G \ rightarrow \ ell^1（\ Mathbb {r}^m）$具有良好的财产。
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 $ϕ: G \rightarrow \ell^1(\mathbb{R}^m)$ with good properties.
