Spatial Generalizations of Random Graphs

Abstract

The most important model of a random graph where nodes are connected at random was proposed by Erdos and Renyi [242] and constitutes an archetype—or at least a benchmark—for constructing more complex random graphs. It is then natural to ask if we can extend this model to the case where nodes are located in space. In this chapter, we will discuss some of the possible extensions that were proposed in the literature. In particular, we will discuss the Waxman graph which was proposed as a model for intra-domain Internet network. Another important model is the Watts–Strogatz graph [7] which interpolates between a lattice and the Erdos–Renyi random graph and is able to produce graphs with simultaneously a large clustering coefficient and a small diameter (varying as $\log N$). In this model, there is an underlying lattice and it can thus be considered as a spatial network. We will discuss some of the properties of this network and end this chapter with a presentation of navigability problems and the demonstration of Kleinberg’s result [243].