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Scale-Free Networks

The article by Watts and Strogatz inspired many follow-up articles but one of the most important once was the paper by Albert-László Barabási and Réka Albert on the emergence of scaling in random networks [Barabasi1999]. In this article they reported that many real-world networks exhibit a so-called scale-free degree distribution. A degree distribution describes how many nodes in a network have how many neighbors. A common assumption for any distribution is that it is normal, i.e., that most items display a value around the mean and that nobody is very far from the mean. In a random graph in which every pair of edges has a probability of p to be connected, the degree distribution will follow a normal distribution quite closely.
A scale-free degree distribution is very distinct from that: in it, most of the items have a very low value while others have a very high value, far from the mean.

The authors showed on the actor-collaboration graph, some portion of the WWW and the power grid data that the degree distributions of these networks were far from normal. We will discuss later whether the degree distributions are really scale-free (which requires a certain mathematical form), but in any case they were far from the distribution of a random graph.

Barabási and Albert also introduced a model, the so-called preferential attachment model that gave a possible explanation for the evolution of networks with a scale-free degree distribution.
In a second article, Albert, Jeong and Barabási showed that the robustness of networks depends highly on their degree distribution [Albert2000].