other networks.

How do networks with short path lengths come about in the real world? The simplest explanation involves preferential attachment. Barabási and Albert showed in 1999 how, if new nodes in a network prefer to attach to nodes that already have many edges, this leads to a power-law distribution of vertex order which in turn gives rise to a [*scale-free*]{} network [@BA99], which turns out to be a more common type of network than the alpha or beta types. In a social network, for example, people who already have many friends are useful to know, so their friendship is particularly sought by newcomers. In friendship terms, the rich get richer. There are many economic contexts in which such dynamics are also of interest [@Jackson03].

The key paper for our purposes was written by Albert, Jeong, and Barabási in 2000. They observed that the connectivity of scale-free networks, which depends on the highly-connected nodes, comes at a price: the destruction of these nodes will disconnect the network. If an attacker removes the best-connected nodes one after another, then past some threshold point the size of the largest component of the graph collapses [@AJB00].

Later work by Holme, Kim, Yoon and 