On Locality in a Geometric Random Tree Model

Abstract

We address the question of locality in random graphs. In particular, we study a geometric random tree model $\tau_{\alpha,n}$ which is a variant of the FKP model proposed in [Fabrikant et al., 2002]. We choose vertices $v_1, \ldots, v_n$ in some convex body uniformly and fix a point $\mathfrak{o}$. We then build our tree inductively, where at time $t$ we add an edge from $v_t$ to the vertex in $v_1, \ldots, v_{t-1}$ that minimizes $\alpha \| v_t - v_i \| + \| v_i - \mathfrak{o}\| $ for $i \lt t$, where $\alpha \gt 0$. We categorize an edge $v_i \to v_j$ in this graph as local or global depending on the edge length relative to the distance from $v_i$ to $\mathfrak{o}$. 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 $\alpha=1$.