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rather difficult to obtain good data about real-world network degree distributions. For example, the data used by Barabási and colleagues for analyzing metabolic networks came from a Web-based database to which biologists from all over the world contributed information. Such biological databases, while invaluable to research, are invariably incomplete and error-ridden. Barabási and colleagues had to rely on statistics and curve-fitting to determine the degree distributions in various metabolic networks—an imperfect method, yet the one that is most often used in analyzing real-world data. A number of networks previously identified to be “scale-free” using such techniques have later been shown to in fact have non-scale-free distributions.

As noted by philosopher and historian of biology Evelyn Fox Keller, “Current assessments of the commonality of power laws are probably overestimates.” Physicist and network scientist Cosma Shalizi had a less polite phrasing of the same sentiments: “Our tendency to hallucinate power laws is a disgrace.” As I write this, there are still considerable controversies over which real-world networks are indeed scale-free.

Even for networks that are actually scale-free, there are many possible causes for power law degree distributions in networks; preferential attachment is not necessarily the one that actually occurs in nature. As Cosma Shalizi succinctly said: “there turn out to be nine and sixty ways of constructing power laws, and every single one of them is right.” When I was at the Santa Fe Institute, it seemed that there was a lecture every other day on a new hypothesized mechanism that resulted in power law distributions. Some are similar to preferential attachment, some work quite differently. It’s not obvious how to decide which ones are the mechanisms that are actually causing the power laws observed in the real world.

The claimed significance of network science relies on models that are overly simplified and based on unrealistic assumptions. The small-world and scale-free network models are just that—models—which means that they make simplifying assumptions that might not be true of real-world networks. The hope in creating such simplified models is that they will capture at least some aspects of the phenomenon they are designed to represent. As we have seen, these two network models, in particular the scale-free model, indeed seem to capture something about degree-distributions, clustering, and resilience in a large number of real-world systems (though point 1 above suggests that the number might not be as large as some think).

However, simplified models of networks, in and of themselves, cannot explain everything about their real-world counterparts. In both the small-world and scale-free models, all nodes are assumed to be identical except for their degree; and all links are the same type and have the same strength. This is not the case in real-world networks. For example, in the real version of my social network (whose simplified model was shown in figure 14.2), some friendship links are stronger than others. Kim and Gar are both friends of mine but I know Kim much better, so I might be more likely to tell her about important personal events in my life. Furthermore, Kim is a woman and Gar is a man, which might increase my likelihood of confiding in her but not in Gar. Similarly, my friend Greg knows and cares a lot more about math than Kim, so if I wanted to share some neat mathematical fact I learned, I’d be much more likely to tell Greg about it than Kim. Such differences in link and node types as well as link strength can have very significant effects on how information spreads in a network, effects that are not captured by the simplified network models.

Information Spreading and Cascading Failure in Networks

In fact, understanding the ways in which information spreads in networks is one of the most important open problems in network science. The results I have described in this and the previous chapter are all about the structure of networks—e.g., their static degree