Structure and dynamics of directed scale-free networks
We propose to investigate directed scale-free network models and stochastic processes on such networks. Both spatial and non-spatial settings are considered, the former displaying realistic clustering effects while also posing far greater mathematical challenges.
Network models displaying scale-free behaviour are of great significance in both natural and social sciences. Undirected scale-free network models have been widely studied and the rigorous understanding of their behaviour is well advanced. However, particularly in biology, computer science and finance, many phenomena crucially depend on the directedness of the underlying graph structure. Considering directed networks adds additional layers of complexity to the models. The description of networks becomes considerably more involved, even locally, due to the appearance of arbitrary indegree-outdegree correlations. More importantly, the dynamics on directed networks are inherently irreversible, which renders many technical tools commonly used for the analysis of processes on networks ineffective. Therefore, mathematical results for directed networks are scarce and the effects emerging from introducing directed edges are, in general, poorly understood.
The aim of this project is to significantly contribute to the mathematical theory of directed networks, both from a structural perspective, i.e. in terms of network topology and percolation, and from the complementary process perspective, i.e. regarding both dynamics on networks and dynamical network formation.
Peter Gracar, Lukas Lüchtrath, Christian Mönch: Finiteness of the percolation threshold for inhomogeneous long-range models in one dimension (03/2022)
Christian Mönch, Amr Rizk: DAG-type Distributed Ledgers via Young-age Preferential Attachment (09/2021)
Peter Gracar, Markus Heydenreich, Christian Mönch, Peter Mörters: Transience Versus Recurrence for Scale-Free Spatial Networks (09/2020) published
Stein Andreas Bethuelsen, Christian Hirsch, Christian Mönch: Quenched invariance principle for random walks on dynamically averaging random conductances (09/2020) published
Peter Gracar, Markus Heydenreich, Christian Mönch, Peter Mörters: Recurrence vs transience for weight-dependent random connection models. (11/2019) published
Christian Mönch: Universality for Persistence Exponents of Local Times of Self-Similar Processes with Stationary Increments (11/2018) published