The article by Fan Chung and Linyuan Lu examines the average distances in random graphs with given expected degrees, particularly focusing on power law random graphs. They show that for certain families of random graphs, the average distance is almost surely of order \(\log n / \log d\), where \(d\) is the weighted average of the sum of squares of the expected degrees. For power law random graphs with an exponent \(\beta > 3\), the average distance is also of order \(\log n / \log d\). However, for power law graphs with \(2 < \beta < 3\), the average distance is almost surely of order \(\log \log n\), while the diameter is of order \(\log n\). The study highlights a phase transition at \(\beta = 3\), where the average distance changes from \(\log n / \log d\) to \(\log \log n\). The research provides insights into the small-world phenomenon and the clustering effect in large complex networks.The article by Fan Chung and Linyuan Lu examines the average distances in random graphs with given expected degrees, particularly focusing on power law random graphs. They show that for certain families of random graphs, the average distance is almost surely of order \(\log n / \log d\), where \(d\) is the weighted average of the sum of squares of the expected degrees. For power law random graphs with an exponent \(\beta > 3\), the average distance is also of order \(\log n / \log d\). However, for power law graphs with \(2 < \beta < 3\), the average distance is almost surely of order \(\log \log n\), while the diameter is of order \(\log n\). The study highlights a phase transition at \(\beta = 3\), where the average distance changes from \(\log n / \log d\) to \(\log \log n\). The research provides insights into the small-world phenomenon and the clustering effect in large complex networks.