Poisson convergence can yield very sharp transitions in geometric random graphs
نویسندگان
چکیده
We investigate how quickly phase transitions can occur in some geometric random graphs where n points are distributed uniformly and independently in the unit cube [0, 1] for some positive integer d. In the case of graph connectivity for the one-dimensional case, we show that the transition width behaves like n (when the number n of users is large), a significant improvement over general asymptotic bounds given recently by Goel et al. for monotone graph properties. We outline how the approach used here could be applied to higher dimensional graphs and to other graph properties. The key ingredient is the availability of a Poisson paradigm complementing the “zero-one” law usually occurring for many graph properties.
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