Infinite geometric graphs and properties of metrics


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

Similar works

Full text

oaioai:CiteSeerX.psu:10.1...Last time updated on 10/30/2017

This paper was published in CiteSeerX.

Having an issue?

Is data on this page outdated, violates copyrights or anything else? Report the problem now and we will take corresponding actions after reviewing your request.