A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction P(k) of nodes in the network having k connections to other nodes goes for large values of k as

where is a parameter whose value is typically in the range 2 < < 3, although occasionally it may lie outside these bounds.

Many networks have been reported to be scale-free, although statistical analysis has refuted many of these claims and seriously questioned others.Preferential attachment and the fitness model have been proposed as mechanisms to explain conjectured power law degree distributions in real networks.

History

In studies of the networks of citations between scientific papers, Derek de Solla Price showed in 1965 that the number of links to papers—i.e., the number of citations they receive—had a heavy-tailed distribution following a Pareto distribution or power law, and thus that the citation network is scale-free. He did not however use the term "scale-free network", which was not coined until some decades later. In a later paper in 1976, Price also proposed a mechanism to explain the occurrence of power laws in citation networks, which he called "cumulative advantage" but which is today more commonly known under the name preferential attachment.