Condensation in random geometric graphs


Description Phase 2

The project we propose is a continuation of the project of the same name in the first funding period. In both parts of the project we explore the condensation phenomena arising when a geometric random graph is conditioned on certain unlikely events. The first part of the project was focused on the event of the graph having exceptionally many edges, in which case the condensation can manifest itself in the emergence of vertices with exceptionally high degree or of localized clusters of exceptionally high connectivity. In the second part our principal aim is to extend this work to more general large deviation events, for example the graph having unusually many triangles, untypically long paths or large connected components. These events are also based on a condensation effect, which we will explore in greater depth. We will also use and extend recent new methods developed in the analysis of sharp phase transitions in geometric random graphs and other percolation models.

Description Phase 1

In a random geometric graph points are randomly distributed in space, and each point is given a random radius. Two points are joined by an edge if their distance is smaller than the sum of the two radii. When such a random graph is conditioned to have more edges than it would normally have, the excess edges can form different geometries. For example, a small number of points could get a large radius and therefore be adjacent to a high number of edges, or points could accumulate in a small spatial area and form a complete graph by virtue of their proximity. The purpose of the project is to investigate the different geometric configurations arising in these condensing graphs in dependence on the distribution of the individual radii.

Preprints/Publications

Anna Brandenberger, Serte Donderwinkel, Céline Kerriou, Gábor Lugosi, Rivka Mitchell: Temporal connectivity of Random Geometric Graphs (02/2025)

Peter Gracar, Marilyn Korfhage, Peter Mörters: Robustness in the Poisson Boolean model with convex grains (10/2024)

Emmanuel Jacob, Amitai Linker, Peter Mörters: Metastability of the contact process on slowly evolving scale-free networks (07/2024)

Remco van der Hofstad, Pim van der Hoorn, Céline Kerriou, Neeladri Maitra, Peter Mörters: Condensation in scale-free geometric graphs with excess edges (05/2024)

Jakob E. Björnberg, Cécile Mailler, Peter Mörters, Daniel Ueltschi: A two-table theorem for a disordered Chinese restaurant process (03/2023)

Peter Gracar, Arne Grauer: The contact process on scale-free geometric random graphs (08/2022)

Céline Kerriou, Peter Mörters: The fewest-big-jumps principle and an application to random graphs (06/2022)

Emmanuel Jacob, Amitai Linker, Peter Mörters: The contact process on dynamical scale-free networks (06/2022)

Peter Gracar, Arne Grauer, Peter Mörters: Chemical distance in geometric random graphs with long edges and scale-free degree distribution. (08/2021) published

Peter Gracar, Markus Heydenreich, Christian Mönch, Peter Mörters: Transience Versus Recurrence for Scale-Free Spatial Networks (09/2020) published

Peter Gracar, Lukas Lüchtrath, Peter Mörters: Percolation phase transition in weight-dependent random connection models. (03/2020) published

Peter Gracar, Markus Heydenreich, Christian Mönch, Peter Mörters: Recurrence vs transience for weight-dependent random connection models. (11/2019) published

Members

  • member's portrait

    Prof. Dr. Peter Mörters

    Universität zu Köln
    Principal Investigator
  • member's portrait

    M. Sc. Céline Kerriou

    Universität zu Köln
    Associated Scientist
  • member's portrait

    M. Sc. Marilyn Korfhage

    Universität zu Köln
    Associated Scientist
  • member's portrait

    M. Sc. Lukas Luechtrath

    Weierstrass Institute
    Associated Scientist
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    M. Sc. Lucas Schätze

    Universität zu Köln
    Associated Scientist
  • member's portrait

    M. Sc. Nick Schleicher

    Universität zu Köln
    Associated Scientist

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