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.

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