FIGURE 2. Changes in the diameter d of the network as a function of the fraction f of the removed nodes.

a, Comparison between the exponential (E) and scale-free (SF) network models, each containing N = 10,000 nodes and 20,000 links (that is, k = 4). The blue symbols correspond to the diameter of the exponential (triangles) and the scale-free (squares) networks when a fraction f of the nodes are removed randomly (error tolerance). Red symbols show the response of the exponential (diamonds) and the scale-free (circles) networks to attacks, when the most connected nodes are removed. We determined the f dependence of the diameter for different system sizes (N = 1,000; 5,000; 20,000) and found that the obtained curves, apart from a logarithmic size correction, overlap with those shown in a, indicating that the results are independent of the size of the system. We note that the diameter of the unperturbed ( f = 0) scale-free network is smaller than that of the exponential network, indicating that scale-free networks use the links available to them more efficiently, generating a more interconnected web. b, The changes in the diameter of the Internet under random failures (squares) or attacks (circles). We used the topological map of the Internet, containing 6,209 nodes and 12,200 links (k = 3.4), collected by the National Laboratory for Applied Network Research http://moat.nlanr.net/Routing/rawdata/. c, Error (squares) and attack (circles) survivability of the World-Wide Web, measured on a sample containing 325,729 nodes and 1,498,353 links³, such that k = 4.59.