This paper studies the average distances in random graphs with given expected degrees. It shows that for certain families of such graphs, the average distance is almost surely of order log n / log d, where d is the second-order average degree. For power law graphs with exponent β > 3, the average distance is also of order log n / log d. However, for power law graphs with exponents 2 < β < 3, the average distance is almost surely of order log n, but the diameter is of order log n. These graphs contain a dense subgraph called the core, with diameter O(log log n), and almost all vertices are within distance O(log log n) from the core. The paper also discusses the small-world phenomenon, where any two strangers are connected through a short chain of acquaintances. It examines random graph models of large complex graphs and their implications for random graph theory. The study of realistic large graphs provides insights for random graph theory. The paper considers a general model G(w) for random graphs with given expected degree sequences. It proves that for admissible expected degree sequences, the average distance is almost surely (1 + o(1)) log n / log d. For specially admissible sequences, the diameter is almost surely Θ(log n / log d). For power law graphs with β > 3, the average distance is (1 + o(1)) log n / log d, while for β between 2 and 3, the average distance is O(log log n) and the diameter is O(log n). The paper also discusses the phase transition at β = 3, where the average distance becomes Θ(log n / log log n). It provides proofs for these results and discusses the implications for real-world networks such as the Internet, social, and citation networks. The paper concludes that power law graphs with β > 3 have average distance of order log n, while those with 2 < β < 3 have average distance of order log log n. The core of these graphs plays a crucial role in determining the average distance and diameter.This paper studies the average distances in random graphs with given expected degrees. It shows that for certain families of such graphs, the average distance is almost surely of order log n / log d, where d is the second-order average degree. For power law graphs with exponent β > 3, the average distance is also of order log n / log d. However, for power law graphs with exponents 2 < β < 3, the average distance is almost surely of order log n, but the diameter is of order log n. These graphs contain a dense subgraph called the core, with diameter O(log log n), and almost all vertices are within distance O(log log n) from the core. The paper also discusses the small-world phenomenon, where any two strangers are connected through a short chain of acquaintances. It examines random graph models of large complex graphs and their implications for random graph theory. The study of realistic large graphs provides insights for random graph theory. The paper considers a general model G(w) for random graphs with given expected degree sequences. It proves that for admissible expected degree sequences, the average distance is almost surely (1 + o(1)) log n / log d. For specially admissible sequences, the diameter is almost surely Θ(log n / log d). For power law graphs with β > 3, the average distance is (1 + o(1)) log n / log d, while for β between 2 and 3, the average distance is O(log log n) and the diameter is O(log n). The paper also discusses the phase transition at β = 3, where the average distance becomes Θ(log n / log log n). It provides proofs for these results and discusses the implications for real-world networks such as the Internet, social, and citation networks. The paper concludes that power law graphs with β > 3 have average distance of order log n, while those with 2 < β < 3 have average distance of order log log n. The core of these graphs plays a crucial role in determining the average distance and diameter.