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 [3], 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 [1,5] and heuristic arguments given by Newman, Strogatz and Watts [23] suggest that after n steps the resulting graph should have diameter approximately logn. We show that while this holds for m=1, for m≥2 the diameter is asymptotically log n/log logn.

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Correspondence to Béla Bollobás*.

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* Research supported in part by NSF grant no. DSM9971788

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Bollobás*, B., Riordan, O. The Diameter of a Scale-Free Random Graph. Combinatorica 24, 5–34 (2004).

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Mathematics Subject Classification (2000):

  • 05C80