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It was observed that many real-world networks such as the Internet, social networks, biological networks, and Collaboration graphs have the so-called power law degree distributions. A graph is called a power law graph if the fraction of vertices with degree k is approximately proportional to k^{-b} for some constant b. The classical Erdos and Renyi random graph model G(n; p) is not suitable for modeling these power law graphs. Many random graphs models are developed. Among these models, we directly generalize G(n; p) into "random graphs with given expected degree sequences". We determined the size and volume of the giant component, the average distance and the diameter, and the spectra. Some theoretic results will be compared to real data.

(Joint work with Fan Chung Graham.)