## Mathew D. Penrose

Connectivity and singletons in soft geometric graphs

### Abstract:

Consider a random graph on n vertices scattered
uniformly at random in the unit d-cube (d >1
fixed), in which any two vertices distant
at most r apart are connected with probability p.
This generalizes some well-known random graph models.
We describe how for n large under *any*
choice of the parameter sequence ((r(n),p(n)), n ≥ 1), the probability that
the resulting graph is connected is governed
by the probability that it is free of isolated
vertices, and the number of isolated vertices
is approximately Poisson. We describe
some aspects of the proof.