Editing Scale-free network (section)

{{short description|Network whose degree distribution follows a power law}}
[[File:Degree distribution for a network with 150000 vertices and mean degree = 6 created using the Barabasi-Albert model..png|thumb|Degree distribution for a network with 150000 vertices and mean degree = 6 created using the [[Barabási–Albert model]] (blue dots). The distribution follows an analytical form given by the ratio of two [[gamma function]]s (black line) which approximates as a power-law.]]
{{Network Science}}
A '''scale-free network''' is a [[complex network|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

:<math>
P(k) \ \sim \ k^\boldsymbol{-\gamma}
</math>

where <math>\gamma</math> is a parameter whose value is typically in the range <math display=inline>2<\gamma<3</math> (wherein the second moment ([[scale parameter]]) of <math>k^\boldsymbol{-\gamma}</math> is infinite but the first moment is finite), although occasionally it may lie outside these bounds.<ref>{{Cite journal | last1 = Onnela | first1 = J.-P. | last2 = Saramaki | first2 = J. | last3 = Hyvonen | first3 = J. | last4 = Szabo | first4 = G. | last5 = Lazer | first5 = D. | last6 = Kaski | first6 = K. | last7 = Kertesz | first7 = J. | last8 = Barabasi | first8 = A. -L. | doi = 10.1073/pnas.0610245104 | title = Structure and tie strengths in mobile communication networks | journal = Proceedings of the National Academy of Sciences | volume = 104 | issue = 18 | pages = 7332–7336 | year = 2007 | pmid =  17456605| pmc = 1863470|arxiv = physics/0610104 |bibcode = 2007PNAS..104.7332O | doi-access = free }}</ref><ref>{{Cite journal | last1 = Choromański | first1 = K. | last2 = Matuszak | first2 = M. | last3 = MiȩKisz | first3 = J. | doi = 10.1007/s10955-013-0749-1 | title = Scale-Free Graph with Preferential Attachment and Evolving Internal Vertex Structure | journal = Journal of Statistical Physics | volume = 151 | issue = 6 | pages = 1175–1183 | year = 2013 |bibcode = 2013JSP...151.1175C | doi-access = free }}</ref> 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 (network theory)|fitness model]] have been proposed as mechanisms to explain the power law degree distributions in real networks. Alternative models such as [[Non-linear preferential attachment|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.<ref name="Scale-free networks as preasymptoti">{{cite journal |last1=Krapivsky |first1=Paul |last2=Krioukov |first2=Dmitri |title=Scale-free networks as preasymptotic regimes of superlinear preferential attachment |journal=Physical Review E |date=21 August 2008 |volume=78 |issue=2 |pages=026114 |doi=10.1103/PhysRevE.78.026114|pmid=18850904 |arxiv=0804.1366 |bibcode=2008PhRvE..78b6114K |s2cid=14292535 }}</ref><ref name="Transient">{{cite journal |last1=Falkenberg |first1=Max |last2=Lee |first2=Jong-Hyeok |last3=Amano |first3=Shun-ichi |last4=Ogawa |first4=Ken-ichiro |last5=Yano |first5=Kazuo |last6=Miyake |first6=Yoshihiro |last7=Evans |first7=Tim S. |last8=Christensen |first8=Kim |title=Identifying time dependence in network growth |journal=Physical Review Research |date=18 June 2020 |volume=2 |issue=2 |pages=023352 |doi=10.1103/PhysRevResearch.2.023352 | arxiv=2001.09118|bibcode=2020PhRvR...2b3352F |doi-access=free }}</ref>