A Geometric Preferential Attachment Model of Networks II

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Abstract

A detailed understanding of expansion in complex networks can greatly aid in the design and analysis of algorithms for a variety of important network tasks, including routing messages, ranking nodes, and compressing graphs. This has motivated several recent investigations of expansion properties in real-world graphs and also in random models of real-world graphs, like the preferential attachment graph. The results point to a gap between real-world observations and theoretical models. Some real-world graphs are expanders and others are not, but a graph generated by the preferential attachment model is an expander whp .

We study a random graph G n that combines certain aspects of geometric random graphs and preferential attachment graphs. This model yields a graph with power-law degree distribution where the expansion property depends on a tunable parameter of the model.

The vertices of G n are n sequentially generated points x 1,x 2,...,x n chosen uniformly at random from the unit sphere in . After generating x t , we randomly connect it to m points from those points in x 1,x 2,...,x t − 1 ....