Bachelor Thesis - Graphs as Adjacency Processes

Abstract

We develop a point process framework for local weak limits of labelled random graphs. The framework represents rooted graphs by rooted adjacency processes, point processes on product spaces that encode both vertices and edges while retaining the label-space structure from which the graph is induced. Our main theorem shows that, under weak convergence of rooted adjacency processes, weak convergence of the induced rooted graphs in the Benjamini-Schramm topology is equivalent to radial tightness, a compact-containment condition for rooted neighborhoods in the underlying label space. This reduces weak convergence of graphs to adjacency process convergence together with a sharp non-escape criterion. We then combine this construction with Palm theory to treat uniform rooting of finite graphs and obtain local weak convergence results for a broad class of spatial inhomogeneous random graphs under general conditions, with concrete results for mixed-binomial and Cox vertex processes with labels in and .

Notes

This thesis, conducted at the Technical University of Eindhoven, was the start of my ongoing research programme into the connection between random graph theory and the theory of point processes (adjacency processes specifically).

Acknowledgements

I would like to give a warm acknowledgement to my supervisor, Dr. Pim van der Hoorn. Pim was exceptionally supportive throughout the project, helped me develop my mathematical maturity and research skills, and was always willing to set aside time to offer advice on academic and career matters. He was deeply engaged with and invested in the project, and I feel very grateful to have worked under his supervision.