Editing Scale-free network (section)

==Generative models==
Scale-free networks do not arise by chance alone. [[Paul Erdős|Erdős]] and Rényi (1960) studied a model of growth for graphs in which, at each step, two nodes are chosen uniformly at random and a link is inserted between them. The properties of these [[random graph]]s are different from the properties found in scale-free networks, and therefore a model for this growth process is needed.

The most widely known generative model for a subset of scale-free networks is Barabási and Albert's (1999) [[rich get richer]] generative model in which each new Web page creates links to existing Web pages with a probability distribution which is not uniform, but
proportional to the current in-degree of Web pages.  According to this process, a page with many in-links will attract more in-links than a regular page. This generates a power-law but the resulting graph differs from the actual Web graph in other properties such as the presence of small tightly connected communities. More general models and network characteristics have been proposed and studied. For example, Pachon et al. (2018) proposed a variant of the [[rich get richer]] generative model which takes into account two different attachment rules: a preferential attachment mechanism and a uniform choice only for the most recent nodes.<ref name="PSY"/> For a review see the book by Dorogovtsev and [[José Fernando Ferreira Mendes|Mendes]].{{Citation needed|date=December 2020}} Some mechanisms such as [[Non-linear preferential attachment|super-linear preferential attachment]] and second neighbour attachment generate networks which are transiently scale-free, but deviate from a power law as networks grow large.<ref name="Scale-free networks as preasymptoti"/><ref name="Transient"/>

A somewhat different generative model for Web links has been suggested by Pennock et al. (2002). They examined communities with interests in a specific topic such as the home pages of universities, public companies, newspapers or scientists, and discarded the major hubs of the Web. In this case, the distribution of links was no longer a power law but resembled a [[normal distribution]]. Based on these observations, the authors proposed a generative model that mixes preferential attachment with a baseline probability of gaining a link.

Another generative model is the '''copy''' model studied by Kumar et al.<ref>{{cite conference |last1=Kumar |first1=Ravi |last2=Raghavan |first2=Prabhakar |title=Stochastic Models for the Web Graph |year=2000 |conference=Foundations of Computer Science, 41st Annual Symposium on |pages=57–65 |url=http://cs.brown.edu/research/webagent/focs-2000.pdf |doi=10.1109/SFCS.2000.892065 |access-date=2016-02-10 |archive-date=2016-03-03 |archive-url=https://web.archive.org/web/20160303181517/http://cs.brown.edu/research/webagent/focs-2000.pdf |url-status=live }}</ref> (2000),
in which new nodes choose an existent node at random and copy a fraction of the links of the existent node. This also generates a power law.

There are two major components that explain the emergence of the power-law distribution in the [[Barabási–Albert model]]: the growth and the preferential attachment.<ref name="NETWORK BIOLOGY: UNDERSTANDING THE">{{cite journal
 |last1=Barabási |first1=Albert-László |author-link1=Albert-László Barabási
 |last2=Zoltán N. |first2= Oltvai.
 |title=Network biology: understanding the cell's functional organization
 |doi=10.1038/nrg1272
 |volume=5
 |year=2004
 |journal=Nature Reviews Genetics
 |issue=2 |pages=101–113
 |pmid=14735121
 |s2cid=10950726 }}</ref> 
By "growth" is meant a growth process where, over an extended period of time, new nodes join an already existing system, a network (like the World Wide Web which has grown by billions of web pages over 10 years). Finally, by "preferential attachment" is meant that new nodes prefer to connect to nodes that already have a high number of links with others. Thus, there is a higher probability that more and more nodes will link themselves to that one which has already many links, leading this node to a hub ''in-fine''.<ref name="Emergence of scaling in random netw"/>
Depending on the network, the hubs might either be assortative or disassortative. Assortativity would be found in social networks in which well-connected/famous people would tend to know better each other. Disassortativity would be found in technological (Internet, World Wide Web) and biological (protein interaction, metabolism) networks.<ref name="NETWORK BIOLOGY: UNDERSTANDING THE"/>

However, the ''growth'' of the networks (adding new nodes) is not a necessary condition for creating a scale-free network (see 
Dangalchev<ref name = "CAD">{{cite journal |last1=Dangalchev |first1=Chavdar |title=Generation models for scale-free networks |journal=Physica A: Statistical Mechanics and Its Applications |date=July 2004 |volume=338 |issue=3–4 |pages=659–671 |doi=10.1016/j.physa.2004.01.056|bibcode=2004PhyA..338..659D |url=https://zenodo.org/record/1259307 }}</ref>). One possibility (Caldarelli et al. 2002) is to consider the structure as static and draw a link between vertices according to a particular property of the two vertices involved. Once specified the statistical distribution for these vertex properties (fitnesses), it turns out that in some circumstances also static networks develop scale-free properties.