Wireless ad-hoc networks, a natural application of random geometric graphs, are often located in complex environments. We consider uniformly distributed nodes in a region with fractal boundary and line of sight condition, or nodes with a non-uniform, possibly fractal, intensity measure. We give a first, mostly heuristic and numerical, investigation of connectivity, which in the classical case is known to be controlled by isolated nodes. With a fractal boundary, the connection probability decays as a stretched exponential involving the dimension of the boundary. In the non-uniform cases considered, the connection probability varies more slowly with the number of nodes, and perhaps surprisingly, appears more closely related to isolated nodes than in the classical case.

This is work with Justin Coon (Oxford) and Orestis Georgiou (Toshiba), and was supported by the EPSRC (grant EP/N002458/1).