The intrinsic dimensionality of graphs

We resolve the following conjecture raised by Levin together with Linial, London, and Rabinovich [16]. Let Z ∞ d be the infinite graph whose vertex set is Z d and which has an edge (u,v) whenever ||u-v|| ∞ = 1. Let dim(G) be the smallest d such that G occurs as a (not necessarily induced) subgraph of Z ∞ d . The growth rate of G, denoted ρ G , is the minimum ρ such that every ball of radius r > 1 in G contains at most r ρ vertices. By simple volume arguments, dim(G) = Ω(ρ G ). Levin conjectured that this lower bound is tight, i.e., that dim(G) = O(ρ G ) for every graph G.Previously, it was not known whether dim(G) could be upper bounded by any function of ρ G , even in the special case of trees. We show that a weaker form of Levin's conjecture holds by proving that, for every graph G, dim(G) = O(ρ G log ρ G ). We disprove, however, the specific bound of the conjecture and show that our upper bound is tight by exhibiting graphs for which dim(G) =Ω(ρ G log ρ G ). For families of graphs which exclude a fixed minor, we salvage the strong form, showing that dim(G) = O(ρ G ). This holds also for graphs without long induced simple cycles. Our results extend to a variant of the conjecture for finite-dimensional Euclidean spaces due to Linial[15].

Authors

• Robert Krauthgamer
• James R. Lee

Keywords

• embeddings
• finite metric spaces
• graph decomposition
• growth restricted graphs
• lovasz local lemma
• low--dimensional