Contents Online
Communications in Mathematical Sciences
Volume 7 (2009)
Number 1
On the degree properties of generalized random graphs
Pages: 175 – 187
DOI: https://dx.doi.org/10.4310/CMS.2009.v7.n1.a9
Authors
Abstract
A generalization of the classical Erdös and Rényi (ER) random graph is introduced and investigated. A generalized random graph (GRG) admits different values of probabilities for its edges rather than a single probability uniformly for all edges as in the ER model. In probabilistic terms, the vertices of a GRG are no longer statistically identical in general, giving rise to the pos- sibility of complex network topology. Depending on their surrounding edge probabilities, vertices of a GRG can be either "homogeneous" or "heterogeneous". We study the statistical properties of the degree of a single vertex, as well as the degree distribution over the whole GRG. We distinguish the degree distribution for the entire random graph ensemble and the degree frequency for a particular graph realization, and study the mathematical relationship between them. Finally, the connectivity of a GRG, a property which is highly related to the degree distribution, is briefly discussed and some useful results are derived.
Keywords
Random graph; degree distribution; connectivity; giant component
2010 Mathematics Subject Classification
05C40, 05C80
Published 1 January 2009