Random Geometric Graphs Induced by Stationary Point Processes
by Volker Schmidt
Abstract: For any locally finite set M in the Euclidean space of dimension
at least 2, the minimal spanning forest MSF(M) is a generalization
of the minimal spanning tree that was introduced originally by
Aldous and Steele (1992). They conjectured that MSF(X) is almost
surely connected if X is a homogeneous Poisson point process.
This conjecture was proven in Alexander (1995) for dimension 2.
However, it remains open for higher dimensions (and for non--Poisson
point processes X).
In the present talk we introduce a family of approximations
of the minimal spanning forest, see . Using techniques of
Blaszczyszyn and Yogeshwaran (2011) and Daley and Last (2005)
we prove the a.s. connectivity of the constructed approximations
for all stationary point processes X with finite range of dependence
and absolutely continuous second factorial moment measure.
We also derive conditions for the a.s. finiteness of cells in the
two-dimensional case and discuss some applications to spatial
stochastic modeling of telecommunication networks, see  and .
 Hirsch, C., Neuhaeuser, D. and Schmidt, V. (2012)
Connectivity of random geometric graphs related to minimal spanning forests.
 Gloaguen, C., Voss, F. and Schmidt, V. (2011)
Parametric distributions of connection lengths for the efficient analysis
of fixed access networks. Annals of Telecommunications 66, 103-118.
 Neuhaeuser, D., Hirsch, C., Gloaguen, C. and Schmidt, V. (2012)
On the distribution of typical shortest-path lengths in connected
random geometric graphs. Queueing Systems (in print).