# Previous projects

**Scientific network**

## Stochastic Processes on Evolving Networks

Dr. habil. Anton Klimovsky

In the last 20 years, complex networks became a key tool to model real-world complex systems in the sciences. Yet, the majority of networks evolve over time and this can have a substantial effect on the processes unfolding on them. Moreover, the influence can also go the other way around: processes happening on a network can affect the evolution of the network itself. This leads to what is called coevolution in adaptive networks or more generally complex adaptive systems. Examples include epidemiological and ecological networks, neural networks, systems biology networks, social networks, financial markets, etc. In all these contexts, there is a great deal of uncertainty/volatility in the structure and dynamics of the complex system.

What are the emerging global patterns in complex systems? How do they come about from the behavior of the elements? These are typical questions in the sciences, economy and policy making. These questions immediately lead to severe mathematical problems about the models of complex systems but also about their relationships with the real data.

It is the purpose of this scientific network to advance the rigorous mathematical theory of stochastic processes on (co)evolving networks.

The scientific network focuses on:

- The probabilistic underpinnings of stochastic processes on (co)evolving networks.
- Analysis and synthesis of key examples coming from the areas of information/opinion exchange processes, population dynamics, infection processes, and dynamics of artificial neural networks.
- Issues of statistical inference, estimation and uncertainty quantification for (evolving) complex networks and processes on them.

**Priority program (SPP 2265)**

## Random Geometric Systems

**Project: Spatial growth and information exchange in evolving environments and on evolving networks - Phase 1** - 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.

**Research training school (RTG) 2131 **

## High-dimensional Phenomena in Probability - Fluctuations and Discontinuity

The Research Training Group (RTG) * High-dimensional Phenomena in Probability - Fluctuations and Discontinuity * offers excellent national and international graduates in the mathematical sciences the opportunity to conduct internationally visible doctoral research in probability theory. The goal of the RTG is to bring together the joint expertise on aspects of high dimension in probability. In the study of random structures in high dimensions, one frequently observes universality in limit theorems (fluctuations) as well as phase transitions (discontinuities). These aspects form the common focus of a large number of currently active research projects in stochastic processes. The cooperation of several research groups will offer the Ph.D. students the unique opportunity to gain experience beyond their own research topic, thus giving a broad scientific education. The RTG is supported by top level research groups in probability theory and its applications, stochastic analysis, stochastic geometry and mathematical physics. The research groups involved in the RTG have recently successfully carried out externally funded research projects in probability and statistics. As a rule, each doctoral student in the RTG will be supervised by two PIs.

#### Further information

**Priority Program (SPP 1590)**

## Probabilistic Structures in Evolution

**Project: Evolving pathogen phylogenies: a two-level branching approach** - Anita Winter

For many RNA viruses the lack of a proofreading mechanism in the virus' RNA polymerase results in frequent mutation. The high viral mutation rates, the large virus population size, and the short replication periods produce abundance of viral variability which is responsible for immune escape or drug resistance. Understanding in detail the forces which maintain this diversity can assist in the struggle against viral infections.

Pathogen patterns - and in particular the shapes of the phylogenies - are affected by the strength of selective pressure due to various levels of cross-immunity. We focus on the temporal structure of phylogenies associated with a persistent virus. We propose a two-level (host-pathogen) branching model with mutation and competition on both levels in dierent scaling regimes, where hosts can be either the infected patients or the infected cells within a single patient. We thereby extend our recent work on a panmitic virus population.

We will further rely on techniques developed for measure-valued (neutral) multilevel branching dynamics and two-level multi-type branching dynamics with mutation and competition.

#### Further information

**SFB/ Transregio (TRR 12 - A 7)**

## Symmetries and Universality in Mesoscopic Systems

**A 7 - Fluctuations and large deviations in nonequilibrium stochastic dynamics **- A. Altland, TP Köln, E. Frey, ASC München, J. Krug, TP Köln, C. Külske, M Bochum, A. Winter, M Duisburg-Essen

The project explores fluctuation-dominated behavior in interacting many-body systems originating from a variety of physical and biological contexts. A common methodological basis is provided by the use of large deviations principles in path space, which links the project to dynamical systems theory and semi-classical quantum mechanics. Specific problems to be addressed concern the structure of none-equilibrium measures in spin systems, fluctuation theorems for mesoscopic quantum systems, and effects of demographic and spatial fluctuations in models of biological population dynamics.