Local/Global Phenomena in Geometrically Generated Graphs

Abstract. We study a geometric random tree model Tα,n which is a variant of the FKP model proposed in [1]. We choose vertices v1,..., vn in some convex body uniformly and fix a point o. We then build our tree inductively, where at time t we add an edge from vt to the vertex in v1,..., vt−1 which minimizes α ‖vt − vi ‖ + ‖vi − o ‖ for i < t, where α> 0. We categorize an edge vi → vj in this graph as local or global depending on the edge length relative to the distance from vi to o. It is shown that for α bounded away from 1 either all edges are local or all are global a.a.s. However, as α → 1 we show that in fact the number of local and global edges are asymptotically balanced.