| This work is a generalization of the classical Erdos-Renyi (ER) random graph and recently emerging theory of "power-law" random networks. Particularly, we are interested in providing the theory of power-law random graphs with a rigorous probabilistic interpretation. We propose a generalized random graph (GRG) model, where the vertices can be either "homogeneous" or "heterogeneous" depending on their surrounding edge probabilities. 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. The issue of connectivity is also discussed.;One of the key results from this rigorous study of GRG is that power-law degree distribution arises only when there are correlations among the edge-forming processes in a graph. Hence the GRG is further extended to the random graph model with correlated edge probabilities. We classify the model into three different categories depending on the correlation type, and give a thorough analysis of the degree properties of the Type I correlated random graph model. The concept of copula is introduced in the modeling process. Simulation results along with an asymptotic analytical derivation are shown.;Finally an application of the random graph theory to the protein-protein interaction (PPI) network in biochemistry is provided. |
