## Critical behaviour of the weight-dependent random connection model

### Description Phase 1

Many real-world networks can be modelled by spatial random graphs. We investigate a large class of random graphs on the points of a Poisson process in Euclidean space, which combine scale-free degree distributions and long-range effects. Every Poisson point carries an independent random weight, we form an edge between two points independently with a probability depending on the two weights and the distance of the points. This is a powerful continuum percolation model, which contains a number of spatial random graph models as special cases.

A typical artefact of percolation models is the occurrence of a phase transition: if the edge density is very small, then all connected components are finite. On the other hand, if the edge density is large, then an infinite component exists. While the structural properties of the infinite component in the latter case have been analysed in many cases, there are only very few mathematical results about the behavior at the phase transition point. Yet the behaviour at and close to the phase transition point is highly interesting from the viewpoint of mathematics as well as from applications.

The present proposal focusses on the percolation phase transition. Our goal is to devise a lace expansion for the weight-dependent random connection model, which then allows us to derive critical exponents for this model. In contrast to the plain random connection model, where the critical behaviour has been investigated recently, the addition of weights yields a much richer class of examples, in particular the integration of scale-free degree distributions and hierarchies.

### Preprints/Publications

Alejandro Caicedo, Matthew Dickson: Critical Exponents for Marked Random Connection Models (05/2023)

Matthew Dickson, Markus Heydenreich: The Triangle Condition for the Marked Random Connection Model (10/2022)

Markus Heydenreich, Lorenzo Taggi, Niccolo Torri: Prudent walk in dimension six and higher (10/2022)

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

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