# SPP 2265

**Priority Program (SPP 2265)**

## Random Geometric Systems

**Project: Scaling limits of evolving spanning trees and of random walks on evolving spanning trees **- Anita Winter

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.

**Project: Spatial growth and information exchange in evolving environments and on evolving networks** - Anton Klimovsky, Lisa Hartung (Johannes-Gutenberg University Mainz)

What is the effect of the space-time varying environment on the long time behavior of spatially structured populations of interacting particles/individuals/agents? This question is of high relevance, e.g., in life and social sciences/economics, computer science, artificial intelligence. Mathematically, the project focuses on two phenomenological models of interacting particle systems: (1) branching Brownian motion, which models population growth/spatial spreading; and (2) voter model, which models information exchange in a population of agents. The novel features that this project introduces into these classical models are: (1) space-time-correlated environments and (2) evolving networks. These play the role of the geographic spaces and substantially change the underlying spatial geometry. The project aims at investigating (1) growth vs. extinction, population size, spread; and (2) clustering vs. consensus of agents, space-time scaling limits of stochastic processes on evolving networks and evolving graph limits.

#### Further information

**Upcoming events:**

## First Annual Conference

Berlin, April 11-14, 2022