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# Communications in Mathematical Sciences

## Volume 7 (2009)

### Number 1

### On the degree properties of generalized random graphs

Pages: 175 – 187

DOI: http://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