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^-α. 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, pr, that needs to be removed before the network disintegrates. We show analytically and numerically that for α ≤ 3 the transition never takes place, unless the network is finite. In the special case of the physical structure of the Internet (α = 2.5), we find that it is impressively robust, with pc > 0.99.