Editing Scale-free network (section)

==Characteristics==
[[Image:Scale-free network sample.svg|thumb|right|Random network (a) and scale-free network (b)]]
[[File:Complex network degree distribution of random and scale-free.png|thumb|Complex network degree distribution of random and scale-free]]

The most notable characteristic in a scale-free network is the relative commonness of vertices with a degree that greatly exceeds the average. The highest-degree nodes are often called "hubs", and are thought to serve specific purposes in their networks, although this depends greatly on the domain. In a random network the maximum degree, or the expected largest hub, scales as ''k<sub>max</sub>~ log N'', where ''N'' is the network size, a very slow dependence. In contrast, in scale-free networks the largest hub scales as ''k<sub>max</sub>~ ~N<sup>1/(γ−1)</sup>'' indicating that the hubs increase polynomically with the size of the network. 

A key  feature of scale-free networks is their high degree heterogeneity,  κ= ''<k<sup>2</sup>>/<k>'', which governs multiple network-based processes, from network robustness to epidemic spreading and network synchronization. While for a random network  κ= ''<k> + 1,'' i.e. the ration is independent of the network size ''N'', for a scale-free network we have  κ''~ N<sup>(3−γ)/(γ−1)</sup>'', increasing with the network size, indicating that for these networks the degree heterogeneity increases.

===Clustering===
Another important characteristic of scale-free networks is the [[clustering coefficient]] distribution, which decreases as the node degree increases. This distribution also follows a power law. This implies that the low-degree nodes belong to very dense sub-graphs and those sub-graphs are connected to each other through hubs. Consider a social network in which nodes are people and links are acquaintance relationships between people. It is easy to see that people tend to form communities, i.e., small groups in which everyone knows everyone (one can think of such community as a [[complete graph]]). In addition, the members of a community also have a few acquaintance relationships to people outside that community. Some people, however, are connected to a large number of communities (e.g., celebrities, politicians). Those people may be considered the hubs responsible for the [[small-world phenomenon]].

At present, the more specific characteristics of scale-free networks vary with the generative mechanism used to create them. For instance, networks generated by preferential attachment typically place the high-degree vertices in the middle of the network, connecting them together to form a core, with progressively lower-degree nodes making up the regions between the core and the periphery. The random removal of even a large fraction of vertices impacts the overall connectedness of the network very little, suggesting that such topologies could be useful for [[network security|security]], while targeted attacks destroys the connectedness very quickly. Other scale-free networks, which place the high-degree vertices at the periphery, do not exhibit these properties. Similarly, the clustering coefficient of scale-free networks can vary significantly depending on other topological details.

===Immunization===
The question of how to immunize efficiently scale free networks which represent realistic networks such as the Internet and social networks has been studied extensively. One such strategy is to immunize the largest degree nodes, i.e., targeted (intentional) attacks since for this case p<math>c</math> is relatively high and less nodes are needed to be immunized. 
However, in many realistic cases the global structure is not available and the largest degree nodes are not known.

Properties of random graph may change or remain invariant under graph transformations. [[Alireza Mashaghi|Mashaghi A.]] et al., for example, demonstrated that a transformation which converts random graphs to their edge-dual graphs (or line graphs) produces an ensemble of graphs with nearly the same degree distribution, but with degree correlations and a significantly higher clustering coefficient. Scale free graphs, as such, remain scale free under such transformations.<ref name="journals.aps.org">{{cite journal | last1 = Ramezanpour | first1 = A. | last2 = Karimipour | first2 = V. | last3 = Mashaghi | first3 = A. | year = 2003 | title = Generating correlated networks from uncorrelated ones | journal = Phys. Rev. E | volume = 67 | issue = 4| page = 046107 | doi=10.1103/PhysRevE.67.046107| arxiv = cond-mat/0212469 | bibcode = 2003PhRvE..67d6107R | pmid=12786436| s2cid = 33054818 }}</ref>