Infinite geometric graphs and properties of metrics

Abstract

Abstract. We consider isomorphism properties of infinite random geometric graphs defined over a variety of metrics. In previous work, it was shown that for Rn with the L∞-metric, the infinite random geometric graph is, with probability 1, unique up to isomorphism. However, in the case n = 2 this is false with either of the L2-metric or the hexagonal metric. We generalize this result to a large family of metrics induced by norms. In particular, we show that the infinite geometric graph is unique up to isomorphism if and only if the metric space has a new property which we name truncating: each step-isometry from a dense set to itself is an isometry. As a corollary, we derive that the infinite random geometric graph defined in Lp space is unique up to isomorphism with probability 1 only in the cases when p = 1 or p =∞. 1