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, $P\left(k\right){=ck}^{-\alpha }$. 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 needs to be removed before the network disintegrates. We show analytically and numerically that for $\alpha \le 3$ the transition never takes place, unless the network is finite. In the special case of the physical structure of the Internet $\left(\alpha \approx 2.5\right)$, we find that it is impressively robust, with $p_c>0.99\mathrm{}$.

• Received 11 July 2000

DOI:https://doi.org/10.1103/PhysRevLett.85.4626