Section: New Results

Lecture notes on random geometric models — random graphs, point processes and stochastic geometry

[32] The goal of this sequence of lessons is to provide quick access to some popular models of random geometric structures used in many applications: from communication networks, including social, transportation, wireless networks, to geology, material sciences and astronomy. The course is composed of the following 15 lessons: (1) Bond percolation on the square lattice, (2) Galton-Watson tree, (3) Erdős-Rényi graph — emergence of the giant component, (4) Graphs with a given node degree distribution, (5) Typical nodes and random unimodular graphs, (6) Erdős-Rényi graph — emergence of the full connectivity, (7) Poisson point process, (8) Point conditioning and Palm theory for point processes, (9) Hard-core point processes, (10) Stationary point processes and mass transport principle, (11) Stationary Voronoi tessellation, (12) Ergodicity and point-shift invariance, (13) Random closed sets, (14) Boolean model and coverage processes, (15) Connectedness of random sets and continuum percolation. Usually, these subjects are presented in different monographs: random graphs (lessons 2–6), point processes (7-12), stochastic geometry (13-14), with percolation models presented in lesson 1 and 15 often addressed separately. Having them in one course gives us an opportunity to observe some similarities and even fundamental relations between different models. Examples of such connections are:

• Similar phase transitions regarding the emergence of big components observed in different discrete, lattice and continuous euclidean models (lessons 1–4, 15).

• Single isolated nodes being the last obstacle in the emergence of the full connectivity in some discrete and euclidean graphs exhibiting enough independence (lessons 6, 15).

• A mass transport principle as a fundamental property for unimodular random graphs and Palm theory for stationary point processes; with both theories seeking to define the typical node/point of a homogeneous structure (lessons 5, 10–12).

• Poisson-Galton-Watson tree and Poisson process playing a similar role in the theory of random graphs and point processes, respectively: for both models independence and Poisson distribution are the key assumptions, both appear as natural limits, and both rooted/conditioned to a typical node/point preserve the distribution of the remaining part of the structure (lessons 2,5, 7–8).

• Size biased sampling appearing in several, apparently different, conditioning scenarios, as unimodular trees (lesson 5), Palm distributions for point process (lesson 8), zero cell of the stationary tessellations (lessons 11).

The goal of this series of lectures is to present some spectrum of models and ideas. When doing this, we sometimes skip more technical proof details, sending the reader for them to more specialised monographs. Some theoretical and computer exercises are provided after each lesson to let the reader practice his/her skills. Regarding the prerequisites, the reader will benefit from having had some prior exposure to probability and measure theory, but this is not absolutely necessary.

The content of the course has been evolving while the author teaches it within the master programme Probabilité et modelés aléatoires at the University Pierre and Marie Curie in Paris. The present notes were thoroughly revised when the author was presenting them as a specially appointed professor at the School of Computing, Tokyo Institute of Technology, in the autumn term 2017.