Sums of Distances on Graphs and Embeddings into Euclidean Space

TL;DR

This paper studies a greedy vertex selection process on graphs based on maximizing distance sums, revealing that the resulting measures and embeddings suggest the graph's intrinsic low-dimensional structure.

Contribution

It introduces a vertex selection method based on distance sums, analyzes the limiting measures, and constructs explicit low-dimensional embeddings into  spaces.

Findings

Convergence of vertex frequency measures to low-dimensional support

Existence of explicit  embeddings into  spaces with good properties

Graph's structure can be characterized as at most 'm-dimensional'

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.