How to Realize a Graph on Random Points

We are given an integer $d$, a graph $G=(V,E)$, and a uniformly random embedding $f : V rightarrow {0,1}^d$ of the vertices. We are interested in the probability that $G$ can be realized by a scaled Euclidean norm on $mathbb{R}^d$, in the sense that there exists a non-negative scaling $w in mathbb{R}^d$ and a real threshold $theta > 0$ so that [ (u,v) in E qquad text{if and only if} qquad Vert f(u) - f(v) Vert_w^2 < theta,, ] where $| x |_w^2 = sum_i w_i x_i^2$. These constraints are similar to those found in the Euclidean minimum spanning tree (EMST) realization problem. A crucial difference is that the realization map is (partially) determined by the random variable $f$. In this paper, we consider embeddings $f : V rightarrow { x, y}^d$ for arbitrary $x, y in mathbb{R}$. We prove that arbitrary trees can be realized with high probability when $d = Omega(n log n)$. We prove an analogous result for graphs parametrized by the arboricity: specifically, we show that an arbitrary graph $G$ with arboricity $a$ can be realized with high probability when $d = Omega(n a^2 log n)$. Additionally, if $r$ is the minimum effective resistance of the edges, $G$ can be realized with high probability when $d=Omegaleft((n/r^2)log nright)$. Next, we show that it is necessary to have $d geq binom{n}{2}/6$ to realize random graphs, or $d geq n/2$ to realize random spanning trees of the complete graph. This is true even if we permit an arbitrary embedding $f : V rightarrow { x, y}^d$ for any $x, y in mathbb{R}$ or negative weights. Along the way, we prove a probabilistic analog of Radons theorem for convex sets in ${0,1}^d$. Our tree-realization result can complement existing results on statistical inference for gene expression data which involves realizing a tree, such as [GJP15].