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2007 On Locality in a Geometric Random Tree Model
Ross M. Richardson
Internet Math. 4(2-3): 149-174 (2007).

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 $\alpha=1$.

Citation

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Ross M. Richardson. "On Locality in a Geometric Random Tree Model." Internet Math. 4 (2-3) 149 - 174, 2007.

Information

Published: 2007
First available in Project Euclid: 27 May 2009

zbMATH: 1206.68241
MathSciNet: MR2522874

Rights: Copyright © 2007 A K Peters, Ltd.

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Vol.4 • No. 2-3 • 2007
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