Resilience of the Internet to random breakdowns

A common property of many large networks, including the Internet, is that the connectivity of the various nodes follows a scale-free power-law distribution, P(k)=ck^-a. We study the stability of such networks with respect to crashes, such as random removal of sites. Our approach, based on percolation theory, leads to a general condition for the critical fraction of nodes, p_c, that need to be removed before the network disintegrates. We show that for a<=3 the transition never takes place, unless the network is finite. In the special case of the Internet (a=2.5), we find that it is impressively robust, where p_c is approximately 0.99.