The degree distribution of random planar graphs

Abstract

The degree distribution is a very important characteristic of a network and has been well studied in various kinds of real-word networks (like the internet or social networks) and also from a theoretical point of view in several probabilistic network models. In this paper we present an overview over recent results on the degree distribution of planar graphs and maps, where one assumes that every planar graph with n vertices is equally likely. The main result says that for every fixed k ≥ 1 the average number of vertices of degree k in a planar graph of size n is (asymptotically) proportional to dk n, where dk > 0.