The diameter of a scale-free random graph
We consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number m of earlier vertices, where each earlier vertex is chosen with probability proportional to its degree. This process was introduced by Barabási and Albert, as a simple model of the growth of real-world graphs such as the world-wide web. Computer experiments presented by Barabási, Albert and Jeong and heuristic arguments given by Newman, Strogatz and Watts suggest that after n steps the resulting graph should have diameter approximately log n. We show that while this holds for m = 1, for m≥2 the diameter is asymptotically log n/log log n.
Bollobás, B., & Riordan, O. (2004). The diameter of a scale-free random graph. Combinatorica, 24 (1), 5-34. https://doi.org/10.1007/s00493-004-0002-2