Complexity_ A Guided Tour - Melanie Mitchell [120]
Where Do Scale-Free Networks Come From?
No one purposely designed the Web to be scale-free. The Web’s degree distribution, like that of the other networks I’ve mentioned above, is an emergent outcome of the way in which the network was formed, and how it grows.
In 1999 physicists Albert-László Barabási and Réka Albert proposed that a particular growing process for networks, which they called preferential attachment, is the explanation for the existence of most (if not all) scale-free networks in the real world. The idea is that networks grow in such a way that nodes with higher degree receive more new links than nodes with lower degree. Intuitively this makes sense. People with many friends tend to meet more new people and thus make more new friends than people with few friends. Web pages with many incoming links are easier to find than those with few incoming links, so more new Web pages link to the high-degree ones. In other words, the rich get richer, or perhaps the linked get more linked. Barabási and Albert showed that growth by preferential attachment leads to scale-free degree distributions. (Unbeknownst to them at the time, this process and its power-law outcome had been discovered independently at least three times before.)
The growth of so-called scientific citation networks is one example of the effects of preferential attachment. Here the nodes are papers in the scientific literature; each paper receives a link from all other papers that cite it. Thus the more citations others have given to your paper, the higher its degree in the network. One might assume that a large number of citations is an indicator of good work; for example, in academia, this measure is routinely used in making decisions about tenure, pay increases, and other rewards. However, it seems that preferential attachment often plays a large role. Suppose you and Joe Scientist have independently written excellent articles about the same topic. If I happen to cite your article but not Joe’s in my latest opus, then others who read only my paper will be more likely to cite yours (usually without reading it). Other people will read their papers, and also be more likely to cite you than to cite Joe. The situation for Joe gets worse and worse as your situation gets better and better, even though your paper and Joe’s were both of the same quality. Preferential attachment is one mechanism for getting to what the writer Malcolm Gladwell called tipping points—points at which some process, such as citation, spread of fads, and so on, starts increasing dramatically in a positive-feedback cycle. Alternatively, tipping points can refer to failures in a system that induce an accelerating systemwide spread of additional failures, which I discuss below.
Power Laws and Their Skeptics
So far I have implied that scale-free networks are ubiquitous in nature due to the adaptive properties of robustness and fast communication associated with power-law degree distributions, and that the mechanism by which they form is growth by preferential attachment. These notions have given scientists new ways of thinking about many different scientific problems.
However compelling all this may seem, scientists are supposed to be skeptical by nature, especially of new, relatively untested ideas, and even more particularly of ideas that claim generality over many disciplines. Such skepticism is not only healthy, it is also essential for the progress of science. Thus, fortunately, not everyone has jumped on the network-science bandwagon, and even many who have are skeptical concerning some of the most optimistic statements about the significance of network science for complex systems research. This skepticism is founded on the following arguments.
Too many phenomena are being described as power-law or scale-free. It’s typically