Random Geometric Graphs [Hardcover]

This monograph provides and explains the mathematics behind geometric graph theory, which studies the properties of a graph that consists of nodes placed in Euclidean space so that edges can be added to connect points that are close to one another. For example, a collection of trees scattered in a forest and the disease that is passed between them, a set of nests of animals or birds on a region and the communication between them or communication between communications stations or nerve cells. Aimed at graduate students and researchers in probability, statistics, combinatorics and graph theory including computer scientists, it covers topics such as: technical tools, edge and component counts, vertex degrees, clique and chromatic number, and connectivity. Applications of this theory are used in the study of neural networks, spread of disease, astrophysics and spatial statistics.

1. Introduction
2. Probabilistic ingredients
3. Subgraph and component counts
4. Typical vertex degrees
5. Geometrical ingredients
6. Maximum degree, cliques and colourings
7. Minimum degree: laws of large numbers
8. Minimum degree: convergence in distribution
9. Percolative ingredients
10. Percolation and the largest component
11. The largest component for a binomial process
12. Ordering and partitioning problems
13. Connectivity and the number of components
References
Index