Scale-free network

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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.^[1][2] 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".

Many networks have been reported to be scale-free, although statistical analysis has refuted many of these claims and seriously questioned others.^[3][4] Additionally, some have argued that simply knowing that a degree-distribution is fat-tailed is more important than knowing whether a network is scale-free according to statistically rigorous definitions.^[5][6] Preferential attachment and the fitness model have been proposed as mechanisms to explain conjectured 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.^[7][8]