Communications in Mathematical Sciences

On the degree properties of generalized random graphs

Yi Y. Shi and Hong Qian

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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.

Article information

Commun. Math. Sci., Volume 7, Number 1 (2009), 175-187.

First available in Project Euclid: 27 March 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 05C40: Connectivity

Random graph degree distribution connectivity giant component


Shi, Yi Y.; Qian, Hong. On the degree properties of generalized random graphs. Commun. Math. Sci. 7 (2009), no. 1, 175--187.

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