Scale-free network

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

${\displaystyle P(k)\ \sim \ k^{\boldsymbol {-\gamma }}}$

where ${\displaystyle \gamma }$ is a parameter whose value is typically in the range ${\textstyle 2<\gamma <3}$ (wherein the second moment (scale parameter) of ${\displaystyle k^{\boldsymbol {-\gamma }}}$ is infinite but the first moment is finite), although occasionally it may lie outside these bounds. The name "scale-free" could be explained by the fact that some moments of the degree distribution are not defined, so that the network does not have a characteristic scale or "size".

Preferential attachment and the fitness model have been proposed as mechanisms to explain the power law degree distributions in real networks. Alternative models such as super-linear preferential attachment and second-neighbour preferential attachment may appear to generate transient scale-free networks, but the degree distribution deviates from a power law as networks become very large.