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Percolation thresholds on planar Euclidean relative-neighborhood graphs
1Institut für Physik, Carl von Ossietzky Universität Oldenburg, D-26111 Oldenburg, Germany. oliver.melchert@uni-oldenburg.de
Summary
This study analyzes relative neighborhood graphs (RNGs) using numerical simulations. Results confirm RNGs exhibit standard 2D percolation behavior, crucial for network connectivity.
Area of Science:
- Graph theory
- Statistical physics
- Computational geometry
Background:
- Relative Neighborhood Graphs (RNGs) are proximity graphs encoding neighborly relationships.
- RNGs are relevant for constructing robust wireless ad hoc network backbones.
- Understanding percolation properties of RNGs is key for network reliability.
Purpose of the Study:
- To statistically analyze the topology and percolation properties of Euclidean RNGs.
- To determine asymptotic degree, diameter, and percolation thresholds of RNGs.
- To compare RNGs with regular 2D graphs and verify universality classes.
Main Methods:
- Numerical simulations were employed to study RNGs.
- Asymptotic degree and diameter were determined.
- Bond and site percolation thresholds were estimated.
Main Results:
- The asymptotic degree and diameter of RNGs were calculated.
- Bond and site percolation thresholds for RNGs were estimated.
- RNGs were found to belong to the standard 2D percolation universality class.
Conclusions:
- RNGs exhibit non-trivial percolation thresholds.
- The universality class of RNGs matches standard 2D percolation.
- Findings support the use of RNGs in network design for guaranteed connectivity.
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