Exact and asymptotic results for the random connection model

Mon, Oct. 12

3-4PM

400 Cory

3-4PM

400 Cory

A random geometric graph arises by connecting
the points of a point process (random point pattern)
according to some random rule.
Such graphs are popular models for spatial
communication networks. We assume here that
the points form a stationary Poisson process.
The random connection rule might depend
on independent marks associated with the points
but is otherwise independent for different pairs of points.
In the classical
Gilbert model, for instance, the points are independently
marked with random radii and two points are connected
if the balls around them intersect. In this case
the connection does not involve any further randomness.
Another special case is the random connection model, where
the connection probability depends on the distance between
the points but not on any further marks.

In this talk we first discuss the degree distribution of a typical node. Then we turn to the probabilistic properties of the clusters, that is of the connected components of the graph. We give formulae for the intensity (mean density) of the clusters of fixed size and in particular for the intensity of isolated points. Finally we use the Stein-Malliavin method to derive optimal Berry-Esseen bounds for the normal approximation of the number of clusters within an observation window in the Wasserstein and the Kolmogorov distance. This talk is based on joint work with Franz Nestmann (Karlsruhe) and Matthias Schulte (Karlsruhe).

Bio: Guenter Last received the diploma degree in mathematics from Humboldt-University in Berlin in 1984 and the PhD. also from the same university in 1987. He worked there as a scientific assistent till 1992. Then he moved to Technical University in Braunschweig, where he received Dr.Sc. degree in 1995. In 1995/96 he spent one semester as a visiting professor at the University of Kaiserslautern and 1996/97 he spent one year as visiting professor at the University of Bonn. Since 2000 he is a full professor at the Karlsruhe Insitute of Technology (formerly University of Karlsruhe). His research interests are in stochastic processes and their application. In recent years his work is mainly focused on stochastic and integral geometry and on the theory of random measures and point processes.

In this talk we first discuss the degree distribution of a typical node. Then we turn to the probabilistic properties of the clusters, that is of the connected components of the graph. We give formulae for the intensity (mean density) of the clusters of fixed size and in particular for the intensity of isolated points. Finally we use the Stein-Malliavin method to derive optimal Berry-Esseen bounds for the normal approximation of the number of clusters within an observation window in the Wasserstein and the Kolmogorov distance. This talk is based on joint work with Franz Nestmann (Karlsruhe) and Matthias Schulte (Karlsruhe).

Bio: Guenter Last received the diploma degree in mathematics from Humboldt-University in Berlin in 1984 and the PhD. also from the same university in 1987. He worked there as a scientific assistent till 1992. Then he moved to Technical University in Braunschweig, where he received Dr.Sc. degree in 1995. In 1995/96 he spent one semester as a visiting professor at the University of Kaiserslautern and 1996/97 he spent one year as visiting professor at the University of Bonn. Since 2000 he is a full professor at the Karlsruhe Insitute of Technology (formerly University of Karlsruhe). His research interests are in stochastic processes and their application. In recent years his work is mainly focused on stochastic and integral geometry and on the theory of random measures and point processes.

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Ashwin Pananjady and Orhan Ocal

Last Modification Date: Wednesday, February 10, 2016