Open Access
March 2009 On the degree properties of generalized random graphs
Yi Y. Shi, Hong Qian
Commun. Math. Sci. 7(1): 175-187 (March 2009).


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.


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Yi Y. Shi. Hong Qian. "On the degree properties of generalized random graphs." Commun. Math. Sci. 7 (1) 175 - 187, March 2009.


Published: March 2009
First available in Project Euclid: 27 March 2009

zbMATH: 1173.05042
MathSciNet: MR2512839

Primary: 05C40 , 05C80

Keywords: connectivity , degree distribution , Giant component , random graph

Rights: Copyright © 2009 International Press of Boston

Vol.7 • No. 1 • March 2009
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