Abstract
We study a random graph _G<sub>n</sub>_ that combines certain aspects of geometric random graphs and preferential attachment graphs. The vertices of _G<sub>n</sub>_ are _n_ sequentially generated points _x_<sub>1</sub>, _x_<sub>2</sub>, . . . , _x_<sub>n</sub> chosen uniformly at random from the unit sphere in ℝ<sup>3</sup>. After generating _x<sub>t</sub>_, we randomly connect that point to _m_ points from those points in _x_<sub>1</sub>, _x_<sub>2</sub>, . . . , _x<sub>t</sub>_-1 that are within distance _r_ of _x<sub>t</sub>_. Neighbors are chosen with probability proportional to their current degree, and a parameter a biases the choice towards self loops. We show that if _m_ is sufficiently large, if _r_ ≥ ln _n/n_<sup>1/2-<em>β</em> </sup> for some constant _β_, and if α > 2, then with high probabilty (whp) at time n the number of vertices of degree _k_ follows a power law with exponent α + 1. Unlike the preferential attachment graph, this geometric preferential attachment graph has small separators, similar to experimental observations of [Blandford et al. 03]. We further show that if _m_ ≥ _K_ ln _n_, for _K_ sufficiently large, then _G<sub>n</sub>_ is connected and has diameter _O_(ln _n/r_) whp.
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