Infinite random graphs and properties of metrics [chapter]

Anthony Bonato, Jeannette Janssen
2016 IMA Volumes in Mathematics and its Applications  
We give a survey of recent developments in the theory of countably infinite random geometric graphs. Classical results of Erdős and Rényi establish that countably infinite random graphs are isomorphic with probability 1. Infinite random graphs have vertices identified with points in a metric space, and edges are added with a given probability dependent on the relative location of their endpoints. The probability that infinite random geometric graphs are isomorphic is considered. The metric
more » ... s where such a unique isotype emerges are indeed fairly rare, and specifically arise in the context of finite dimensional normed spaces equipped with the ℓ ∞ -metric. We survey negative results for random geometric graphs in the cases of the Euclidean and hexagonal metric. Recent work which considers infinite random geometric graphs in the general setting of normed linear spaces is described. Open problems in the area are provided in the final section. Geometric random graph models play an emerging role in the modelling of realworld networks such as on-line social networks [8, 9, 13] , wireless networks [21] , and the web graph [1, 20] . In such stochastic models, vertices of the network are represented by points in a suitably chosen metric space, and edges are chosen by a mixture of relative proximity of the vertices and probabilistic rules. In real-world networks, the underlying metric space is a representation of the hidden reality that leads to the formation of edges. Such networks can be viewed as embedded in a
doi:10.1007/978-3-319-24298-9_11 fatcat:44rsfbbm55hp3oawfhdxbhbd3u