How to Realize a Graph on Random Points
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We are given an integer d, a graph G=(V,E), and a uniformly random embedding f : V →{0,1}^d of the vertices. We are interested in the probability that G can be "realized" by a scaled Euclidean norm on R^d, in the sense that there exists a non-negative scaling w ∈R^d and a real threshold θ > 0 so that (u,v) ∈ E if and only if f(u) - f(v) _w^2 < θ , where x _w^2 = ∑_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 →{ x, y}^d for arbitrary x, y ∈R. We prove that arbitrary trees can be realized with high probability when d = Ω(n 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 = Ω(n a^2 n). Additionally, if r is the minimum effective resistance of the edges, G can be realized with high probability when d=Ω((n/r^2) n). Next, we show that it is necessary to have d ≥n2/6 to realize random graphs, or d ≥ n/2 to realize random spanning trees of the complete graph. This is true even if we permit an arbitrary embedding f : V →{ x, y}^d for any x, y ∈R or negative weights. Along the way, we prove a probabilistic analog of Radon's 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].
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