There is a nifty paper out in the Proceedings of the National Academy of Sciences looking at scale-free networks. While such things — networks where the majority of nodes have few connections, but some nodes have many connections — are showing up in more and more places, from protein networks to the world-wide web, people are over-enthusiastic in their analyses: They assume that chunks of scale-free networks are themselves scale-free, and such is not the case.
… the sampled network can deviate significantly from a power law with nearly all of the nodes having low connectivity in extreme cases. Interestingly (and rather worryingly) this pattern is seen (and often dismissed) in some putative examples of power laws.
…there seems to have been a recent trend to apply the name “scale-free” to any kind of network with a fat-tailed degree distribution, without a detailed statistical assessment. To understand the role of networks in biology or elsewhere, it is, however, important to focus on the entire network and not just the tail; as we have seen, it is the nodes with low to medium connectivities that are most severely affected by sampling.