Abstract
We address the question of locality in random graphs. In particular, we study a geometric random tree model _T_<sub>α,<em>n</em> </sub> which is a variant of the FKP model proposed in [Fabrikant et al. 02]. We choose vertices υ<sub>1</sub>, . . . , υ<sub>n</sub> 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 υ<sub> <em>t</em> </sub> to the vertex in υ<sub>1</sub>, . . . , υ<sub>t-1</sub> that minimizes α‖υ<sub> <em>t</em> </sub> − υ<sub> <em>i</em> </sub>‖ + ‖υ<sub> <em>i</em> </sub> − o‖ for _i_ < _t_, where α > 0. We categorize an edge υ<sub> <em>i</em> </sub> → υ<sub> <em>j</em> </sub> in this graph as local or global depending on the edge length relative to the distance from υ<sub> <em>i</em> </sub> to o. We study the extent to which the tree is composed of either global or local edges and, in particular, show that it undergoes a transition at α = 1.
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