Random Geometric Graphs

1
Béla Bollobás and Amites Sarkar
Random Geometric Graphs
www.msci.memphis.edu
The Classical Model of Erdos and Rényi, and
Gilbert 1959 Non-geometric mean-field
model

• The Disc Model of Gilbert 1961
• g
• Aim model local connections, e.g., network of
  transceivers in a desert
• Lower bound isolated vertices
• Upper bound small component/large
  component arguments

The k-nearest Neighbour Model Gn,k
Xue and Kumar 2003 natural conjecture cl
cu 1 Balister, Bollobás, Sarkar and Walters
2005 0.3043 cl cu 0.5139 Main
obstruction Small components (k1 vertices) 9k
25?r2
Need k lt 0.2739 log n, so cl
0.2739. Upper bound complicated (see
figure) Balister, Bollobás, Sarkar and Walters
2005 If k c log n ensures
connectivity, then k (c e) log n ensures
s-connectivity, where s o(log n).
Random Voronoi Percolation