Scaling limits of evolving spanning trees and of random walks on evolving spanning trees


Description Phase 2

In this project we study scaling limits of evolving uniform spanning trees (UST) on further classes of networks (including Erdös-Renyi graphs, sequences of densely connected expander graphs and low-dimensional tori). The main motivation comes from modeling large and sparsely connected networks. Trees are the extreme cases of sparsely connected networks. In real world networks, the structure of the network might change over time. One emphasis of the project concerns a particular network dynamics. This is the Aldous-Broder algorithm which is a tree-valued stochastic process that generates the UST. A random walk is a simple stochastic process on a network which allows to explore the structure of the network. In the context of communication networks (e.g.\ internet, wifi) it can be understood as a message sent from device to device. In the current research random walks on dynamic network models are compared with random walks on static networks. In this project we determine the space and time scales on which the random walk has a diffusive scaling limit.

Description Phase 1

In this project we want to study scaling limits of evolving spanning trees on different graphs (in particular, on d-dimensional tori, where d ≥ 2) and work towards tools for the construction and analytical characterization of diffusions on evolving continuum trees. The main motivation comes from modeling large and sparsely connected networks. Trees are the extreme cases of sparsely connected networks.

In real world networks, the structure of the network might change over time. One emphasis of the project concerns the study of the scaling limit of a particular network dynamics. This is the Aldous-Broder chain which is a tree-valued Markov chain generating uniform spanning trees.

A random walk is a simple stochastic process on a network which allows to explore the structure of the network. In the context of communication networks (e.g. internet, wifi) it can be understood as a message sent from device to device. Recently, random walks were studied on dynamic network models and compared with random walks on static networks. The new feature of this project is the construction of scaling limits of random walks on evolving sparsely connected graphs.

Preprints/Publications

Osvaldo Angtuncio Hernández, Gabriel Berzunza Ojeda, Anita Winter: Scaling limit of the Aldous-Broder chain on regular graphs: the transient regime (02/2025)

Andreas Greven, Frank den Hollander, Anton Klimovsky, Anita Winter: Continuum graph dynamics via population dynamics: well-posedness, duality and equilibria (10/2023) published

Andreas Greven, Frank den Hollander, Anton Klimovsky, Anita Winter: The grapheme-valued Wright-Fisher diffusion with mutation (08/2023) published

Gabriel Berzunza Ojeda, Anita Winter: Convergence of the pruning processes of stable Galton-Watson trees and other ℝ-trees (04/2023)

Josué Nussbaumer, Viet Chi Tran, Anita Winter: Algebraic two-level measure trees (07/2022) published

Josué Nussbaumer, Anita Winter: The algebraic α-Ford tree under evolution (06/2020)

Members

  • member's portrait

    Prof. Dr. Anita Winter

    Universität Duisburg-Essen
    Principal Investigator
  • member's portrait

    PhD Osvaldo Angtuncio-Hernández

    Universität Duisburg-Essen
    Associated Scientist
  • member's portrait

    M. Sc. Roman Gambelin

    Universität Duisburg-Essen
    Associated Scientist
  • member's portrait

    M. Sc. Vinita Mulay

    Universität Duisburg-Essen
    Associated Scientist
  • member's portrait

    M. Sc. Max Siemen

    Universität Duisburg-Essen
    Associated Scientist

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