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.