Editing Scale-free network (section)

==Generalized scale-free model==
There has been a burst of activity in the modeling of [[Scale-free networks|scale-free complex networks]]. The recipe of Barabási and Albert<ref name=BA>Barabási, A.-L. and R. Albert, Science '''286''', 509 (1999).</ref> has been followed by several variations and generalizations<ref name=AB>R. Albert, and A.L. Barabási, Phys. Rev. Lett. '''85''', 5234(2000).</ref><ref name=Doro>S. N. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhim, cond-mat/0011115.</ref><ref name=Krap>P.L. Krapivsky, S. Redner, and F. Leyvraz, Phys. Rev. Lett. '''85''', 4629 (2000).</ref><ref name=Tadic>B. Tadic, Physica A '''293''', 273(2001).</ref><ref name=PSY>{{cite journal|title=Scale-free behavior of networks with the copresence of preferential and uniform attachment rules|journal=Physica D: Nonlinear Phenomena|volume=371|pages=1–12|year=2018|first1=Angelica |last1=Pachon |first2=Laura |last2=Sacerdote |first3=Shuyi |last3=Yang |doi=10.1016/j.physd.2018.01.005|arxiv=1704.08597|bibcode=2018PhyD..371....1P|s2cid=119320331}}</ref> and the revamping of previous mathematical works.<ref name=Bom>S. Bomholdt and H. Ebel, cond-mat/0008465; H.A. Simon, Bimetrika '''42''', 425(1955).</ref> 

In today's terms, if a complex network has a power-law distribution of any of its metrics, it's generally considered a scale-free network. Similarly, any model with this feature is called a scale-free model.<ref name="scale-free_mz23">{{Cite journal
 | last1 = Meng
 | first1 = Xiangyi
 | last2 = Zhou
 | first2 = Bin
 | title = Scale-Free Networks beyond Power-Law Degree Distribution
 | year = 2023
 | journal = Chaos, Solitons & Fractals
 | volume = 176
 | pages = 114173
 | doi = 10.1016/j.chaos.2023.114173
| arxiv = 2310.08110
 | bibcode = 2023CSF...17614173M
 | s2cid = 263909425
 }}</ref>

===Features===
Many real networks are (approximately) scale-free and hence require scale-free models to describe them. In Price's scheme, there are two ingredients needed to build up a scale-free model:

1. Adding or removing [[Node (networking)|nodes]]. Usually we concentrate on growing the network, i.e. adding nodes.

2. [[Preferential attachment]]: The probability <math>\Pi</math> that new nodes will be connected to the "old" node.

Note that some models (see 
Dangalchev<ref name = "CAD" /> and
Fitness model below) can work also statically, without changing the number of nodes. It should also be kept in mind that the fact that "preferential attachment" models give rise to scale-free networks does not prove that this is the mechanism underlying the evolution of real-world scale-free networks, as there might exist different mechanisms at work in real-world systems that nevertheless give rise to scaling.

===Examples===
There have been several attempts to generate scale-free network properties. Here are some examples:

====The Barabási–Albert model====
The [[Barabási–Albert model]], an undirected version of [[Price's model]] has a linear preferential attachment <math>\Pi(k_i)=\frac{k_i}{\sum_j k_j}</math> and adds one new node at every time step.

(Note, another general feature of <math>\Pi(k)</math> in real networks is that <math>\Pi(0)\neq 0</math>, i.e. there is a nonzero probability that a
new node attaches to an isolated node. Thus in general <math>\Pi(k)</math> has the form <math>\Pi(k)=A +k^\alpha</math>, where <math>A</math> is the initial attractiveness of the node.)

====Two-level network model====

Dangalchev (see <ref name = "CAD" />) builds a 2-L model by considering the importance of each of the neighbours of a target node in preferential attachment. The attractiveness of a node in the 2-L model depends not only on the number of nodes linked to it but also on the number of links in each of these nodes.

: <math>\Pi(k_i)=\frac{k_i + C \sum_{(i,j)} k_j}{\sum_j k_j + C \sum_j k_j^2},</math>

where ''C'' is a coefficient between 0 and&nbsp;1.

A variant of the 2-L model, the k2 model, where first and second neighbour nodes contribute equally to a target node's attractiveness, demonstrates the emergence of transient scale-free networks.<ref name="Transient"/> In the k2 model, the degree distribution appears approximately scale-free as long as the network is relatively small, but significant deviations from the scale-free regime emerge as the network grows larger. This results in the relative attractiveness of nodes with different degrees changing over time, a feature also observed in real networks.

====Mediation-driven attachment (MDA) model====

In the [[mediation-driven attachment model|mediation-driven attachment (MDA) model]], a new node coming with <math>m</math> edges picks an existing connected node at random and then connects itself, not with that one, but with <math>m</math> of its neighbors, also chosen at random. The probability <math>\Pi(i)</math> that the node <math>i</math> of the existing node picked is

: <math> \Pi(i) = \frac{k_i} N \frac{\sum_{j=1}^{k_i} \frac 1 {k_j} }{k_i}.</math>

The factor <math>\frac{\sum_{j=1}^{k_i} \frac 1 {k_j} }{k_i}</math> is the inverse of the harmonic mean
(IHM) of degrees of the <math>k_i</math> neighbors of a node <math>i</math>. Extensive numerical investigation suggest that for approximately <math>m> 14</math> the mean IHM value in the large <math>N</math> limit becomes a constant which means <math>\Pi(i) \propto k_i</math>. It implies that the higher the
links (degree) a node has, the higher its chance of gaining more links since they can be
reached in a larger number of ways through mediators which essentially embodies the intuitive
idea of rich get richer mechanism (or the preferential attachment rule of the Barabasi–Albert model). Therefore, the MDA network can be seen to follow
the PA rule but in disguise.<ref>{{cite journal | last1 = Hassan | first1 = M. K. | last2 = Islam | first2 = Liana | last3 = Arefinul Haque | first3 = Syed | year = 2017 | title = Degree distribution, rank-size distribution, and leadership persistence in mediation-driven attachment networks | doi = 10.1016/j.physa.2016.11.001 | journal = Physica A | volume = 469 | pages = 23–30 | arxiv = 1411.3444 | bibcode = 2017PhyA..469...23H | s2cid = 51976352 }}</ref>

However, for <math>m=1</math> it describes the winner takes it all mechanism as we find that almost <math>99\%</math> of the total nodes has degree one and one is super-rich in degree. As <math>m</math> value increases the disparity between the super rich and poor decreases and as <math>m>14</math> we find a transition from rich get super richer to rich get richer mechanism.

====Non-linear preferential attachment====
{{See also|Non-linear preferential attachment}}
The Barabási–Albert model assumes that the probability <math>\Pi(k)</math> that a node attaches to node <math>i</math> is proportional to the [[degree (graph theory)|degree]] <math>k</math> of node <math>i</math>. This assumption involves two hypotheses: first, that <math>\Pi(k)</math> depends on <math>k</math>, in contrast to random graphs in which <math>\Pi(k) = p </math>, and second, that the functional form of <math>\Pi(k)</math> is linear in <math>k</math>.

In non-linear preferential attachment, the form of <math>\Pi(k)</math> is not linear, and recent studies have demonstrated that the degree distribution depends strongly on the shape of the function <math>\Pi(k)</math>

Krapivsky, Redner, and Leyvraz<ref name="Krap"/> demonstrate that the scale-free nature of the network is destroyed for nonlinear preferential attachment. The only case in which the topology of the network is scale free is that in which the preferential attachment is [[asymptotically]] linear, i.e. <math>\Pi(k_i)  \sim a_\infty k_i</math> as <math>k_i \to \infty</math>. In this case the rate equation leads to

: <math> P(k) \sim k^{-\gamma}\text{ with }\gamma = 1 + \frac \mu {a_\infty}.</math>

This way the exponent of the degree distribution can be tuned to any value between 2 and <math>\infty</math>.{{clarify|reason=What is mu?|date=November 2021}}

====Hierarchical network model====
[[Hierarchical network model]]s are, by design, scale free and have high clustering of nodes.<ref name=Rav>{{cite journal | last1 = Ravasz | first1 = E. | last2 = Barabási | year = 2003 | title =  Hierarchical organization in complex networks| journal = Phys. Rev. E | volume = 67 | issue = 2| page = 026112 | doi=10.1103/physreve.67.026112| pmid = 12636753 | arxiv = cond-mat/0206130| bibcode = 2003PhRvE..67b6112R| s2cid = 17777155 }}</ref>

The [[Iterative hierarchy|iterative]] construction leads to a hierarchical network. Starting from a fully connected cluster of five nodes, we create four identical replicas connecting the peripheral nodes of each cluster to the central node of the original cluster. From this, we get a network of 25 nodes (''N''&nbsp;=&nbsp;25).
Repeating the same process, we can create four more replicas of the original cluster – the four peripheral nodes of each one connect to the central node of the nodes created in the first step. This gives ''N''&nbsp;=&nbsp;125, and the process can continue indefinitely.

====Fitness model====
The idea is that the link between two vertices is assigned not randomly with a probability ''p'' equal for all the couple of vertices. Rather, for 
every vertex ''j'' there is an intrinsic ''fitness'' ''x''<sub>''j''</sub> and a link between vertex ''i'' and ''j'' is created with a probability 
<math> p(x_i,x_j)</math>.<ref name=Cal>{{cite journal | last1 = Caldarelli | first1 = G. | display-authors = etal | year = 2002 | title =  Scale-free networks from varying vertex intrinsic fitness| doi = 10.1103/physrevlett.89.258702 | journal = Phys. Rev. Lett. | volume = 89 | issue = 25| page = 258702 | pmid=12484927| bibcode = 2002PhRvL..89y8702C | url = http://eprints.imtlucca.it/1137/1/PhysRevLett_Caldarelli_02.pdf }}</ref> 
In the case of World Trade Web it is possible to reconstruct all the properties by using as fitnesses of the country their GDP, and taking

: <math> p(x_i,x_j)=\frac {\delta x_ix_j}{1+\delta x_ix_j}. </math><ref name=Gar>{{cite journal | last1 = Garlaschelli | first1 = D. | display-authors = etal | year = 2004 | title =  Fitness-Dependent Topological Properties of the World Trade Web| doi =10.1103/physrevlett.93.188701 | journal = Phys. Rev. Lett. | volume = 93 | issue = 18| page = 188701 | pmid = 15525215 | bibcode=2004PhRvL..93r8701G| arxiv = cond-mat/0403051 | s2cid = 16367275 }}</ref>

====Hyperbolic geometric graphs====
{{Main|Hyperbolic geometric graph}}
Assuming that a network has an underlying hyperbolic geometry, one can use the framework of [[spatial network]]s to generate scale-free degree distributions. This heterogeneous degree distribution then simply reflects the negative curvature and metric properties of the underlying hyperbolic geometry.<ref name=RefDimaPre>{{cite journal|last1=Krioukov|first1=Dmitri|last2=Papadopoulos|first2=Fragkiskos|last3=Kitsak|first3=Maksim|last4=Vahdat|first4=Amin|last5=Boguñá|first5=Marián|title=Hyperbolic geometry of complex networks|journal=Physical Review E|volume=82|issue=3|doi=10.1103/PhysRevE.82.036106|year=2010|arxiv=1006.5169|bibcode=2010PhRvE..82c6106K|pmid=21230138|page=036106|s2cid=6451908}}</ref>

====Edge dual transformation to generate scale free graphs with desired properties====

Starting with scale free graphs with low degree correlation and clustering coefficient, one can generate new graphs with much higher degree correlations and clustering coefficients by applying edge-dual transformation.<ref name="journals.aps.org"/>

====Uniform-preferential-attachment model (UPA model)====

[[UPA model]] is a variant of the preferential attachment model (proposed by Pachon et al.) which takes into account two different attachment rules: a preferential attachment mechanism (with probability 1−p) that stresses the rich get richer system, and a uniform choice (with probability p) for the most recent nodes. This modification is interesting to study the robustness of the scale-free behavior of the degree distribution. It is proved analytically that the asymptotically power-law degree distribution is preserved.<ref name=PSY />