Isolation and Connectivity in Random Geometric Graphs with Self-similar Intensity Measures
- University of Bristol (United Kingdom)
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.
- OSTI ID:
- 22783790
- Journal Information:
- Journal of Statistical Physics, Vol. 172, Issue 3; Other Information: Copyright (c) 2018 Springer Science+Business Media, LLC, part of Springer Nature; Article Copyright (c) 2018 The Author(s); http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-4715
- Country of Publication:
- United States
- Language:
- English
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