Title: Isolation and connectivity in random geometric graphs with self-similar intensity measures,
Abstract: Random geometric graphs consist of randomly distributed nodes (points), with
pairs of nodes within a given mutual distance linked. In the usual model the
distribution of nodes is uniform on a square, and in the limit of infinitely
many nodes and shrinking linking range, the number of isolated nodes is Poisson
distributed, and the probability of no isolated nodes is equal to the
probability the whole graph is connected. Here we examine these properties for
several self-similar node distributions, including smooth and fractal, uniform
and nonuniform, and finitely ramified or otherwise. We show that nonuniformity
can break the Poisson distribution property, but it strengthens the link
between isolation and connectivity. It also stretches out the connectivity
transition. Finite ramification is another mechanism for lack of connectivity.
The same considerations apply to fractal distributions as smooth, with some
technical differences in evaluation of the integrals and analytical arguments.