"Spatial Random Networks" David Aldous, University of California Berkeley I will give an overview of ongoing research concerning networks linking random vertices in two-dimensional space, emphasising two aspects.

(1)

If we get to choose edges, subject to a given total length, then how efficient can we make the network in the sense of providing routes between vertices whose route-length is not much larger than straight-line distance, and what precise statistic is most useful for measuring this notion of "efficient?"

(2)

There is a general class of "proximity graphs," defined for arbitrary vertices, which are always connected. Applying to random points gives a class of random networks which are connected and have bounded mean degree. This class has scarcely been studied, but seems an appealing modeling alternative to the classical geometric random graphs for which one cannot have both connectivity and bounded mean degree.