Derivation of the percolation threshold for the network model of Barabási and Albert.
The percolation threshold of the network model by Barabási and Albert [Science 286, 509 (1999)] (BA) has thus far only been "guessed" based on simulations and comparison with other models. Due to the still uncertain influence of correlations, the reference to other models cannot be justified. In this paper, we explicitly derive the well-known values for the BA model. To underline the importance of a null model like that of Barabási and Albert, we close with two basic remarks. First, we establish a connection between the abundance of scale-free networks in nature and the fact that power-law tails in the degree distribution result only from (at least asymptotically) linear preferential attachment: Only in the case of linear preferential attachment does a minimum of topological knowledge about the network suffice for the attachment process. Second, we propose a very simple and realistic extension of the BA model that accounts for clustering. We discuss the influence of clustering on the percolation properties.
Pietsch
Wolfgang
W
Department for Philosophy of Science, University of Augsburg, Universitätsstrasse 10, D-86135 Augsburg, Germany. wpietsch@gmx.de