## Communications in Mathematical Sciences

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

### On the degree properties of generalized random graphs

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 27 March 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.cms/1238158611

**Mathematical Reviews number (MathSciNet)**

MR2512839

**Zentralblatt MATH identifier**

1173.05042

**Subjects**

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

**Keywords**

Random graph degree distribution connectivity giant component

#### Citation

Shi, Yi Y.; Qian, Hong. On the degree properties of generalized random graphs. Commun. Math. Sci. 7 (2009), no. 1, 175--187. https://projecteuclid.org/euclid.cms/1238158611