scale-free network#Generalized scale-free model

In studies of citations between scientific papers, Derek de Solla Price showed in 1965 that the number of citations a paper receives had a heavy-tailed distribution following a Pareto distribution or power law. In a later paper in 1976, Price also proposed a mechanism to explain the occurrence of power laws in citation networks, which he called "cumulative advantage." However, both treated citations are scalar quantities, rather than a fundamental feature of a new class of networks.

The interest in scale-free networks started in 1999 with work by Albert-László Barabási and Réka Albert at the University of Notre Dame who mapped the topology of a portion of the World Wide Web,{{Cite journal |last=Albert |first=Réka |last2=Jeong |first2=Hawoong |last3=Barabási |first3=Albert-László |date=9 September 1999 |title=Diameter of the World-Wide Web |url=https://www.nature.com/articles/43601 |journal=Nature |language=en |volume=401 |issue=6749 |pages=130–131 |doi=10.1038/43601 |issn=1476-4687|arxiv=cond-mat/9907038 }} finding that some nodes, which they called "hubs", had many more connections than others and that the network as a whole had a power-law distribution of the number of links connecting to a node. In a subsequent paper{{cite journal |last1=Barabási |first1=Albert-László |author-link1=Albert-László Barabási |last2=Albert |first2=Réka. |date=15 October 1999 |title=Emergence of scaling in random networks |journal=Science |volume=286 |issue=5439 |pages=509–512 |arxiv=cond-mat/9910332 |bibcode=1999Sci...286..509B |doi=10.1126/science.286.5439.509 |mr=2091634 |pmid=10521342 |s2cid=524106}} Barabási and Albert showed that the power laws are not a unique property of the WWW, but the feature is present in a few real networks, prompting them to coin the term "scale-free network" to describe the class of networks that exhibit a power-law degree distribution.

Barabási and Réka Albert proposed a generative mechanism to explain the appearance of power-law distributions, which they called "preferential attachment". Analytic solutions for this mechanism were presented in 2000 by Dorogovtsev, Mendes and Samukhin{{Cite journal | last1 = Dorogovtsev | first1 = S. | last2 = Mendes | first2 = J. | last3 = Samukhin | first3 = A. | doi = 10.1103/PhysRevLett.85.4633 | title = Structure of Growing Networks with Preferential Linking | journal = Physical Review Letters | volume = 85 | issue = 21 | pages = 4633–4636 | year = 2000 | pmid = 11082614|arxiv = cond-mat/0004434 |bibcode = 2000PhRvL..85.4633D | s2cid = 118876189 }} and independently by Krapivsky, Redner, and Leyvraz, and later rigorously proved by mathematician Béla Bollobás.{{Cite journal | last1 = Bollobás | first1 = B. |author-link1 = Béla Bollobás| last2 = Riordan | first2 = O. | last3 = Spencer | first3 = J. | last4 = Tusnády | first4 = G.| title = The degree sequence of a scale-free random graph process | journal = Random Structures and Algorithms | volume = 18 | issue = 3| pages = 279–290| year = 2001 | doi = 10.1002/rsa.1009 | mr = 1824277| s2cid = 1486779 }}

When the concept of "scale-free" was initially introduced in the context of networks, it primarily referred to a specific trait: a power-law distribution for a given variable $k$, expressed as $f(k)\backslash propto\; k^\{-\backslash gamma\}$. This property maintains its form when subjected to a continuous scale transformation $k\backslash to\; k+\backslash epsilon\; k$, evoking parallels with the renormalization group techniques in statistical field theory.{{Cite book

| last1 = Itzykson

| first1 = Claude

| last2 = Drouffe

| first2 = Jean-Michel

| title = Statistical Field Theory: Volume 1, From Brownian Motion to Renormalization and Lattice Gauge Theory

| year = 1989

| edition = 1st

| publisher = Cambridge University Press

| location = New York

| isbn = 978-0-521-34058-8

| language = English

}}

{{Cite book

| last1 = Itzykson

| first1 = Claude

| last2 = Drouffe

| first2 = Jean-Michel

| title = Statistical Field Theory: Volume 2, Strong Coupling, Monte Carlo Methods, Conformal Field Theory and Random Systems

| year = 1989

| edition = 1st

| publisher = Cambridge University Press

| location = New York

| isbn = 978-0-521-37012-7

| language = English

}}

However, there's a key difference. In statistical field theory, the term "scale" often pertains to system size. In the realm of networks, "scale" $k$ is a measure of connectivity, generally quantified by a node's degree—that is, the number of links attached to it. Networks featuring a higher number of high-degree nodes are deemed to have greater connectivity.

The power-law degree distribution enables us to make "scale-free" assertions about the prevalence of high-degree nodes.{{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

}} For instance, we can say that "nodes with triple the average connectivity occur half as frequently as nodes with average connectivity". The specific numerical value of what constitutes "average connectivity" becomes irrelevant, whether it's a hundred or a million.{{Cite journal

| last = Tanaka

| first = Reiko

| title = Scale-Rich Metabolic Networks

| year = 2005

| journal = Phys. Rev. Lett.

| volume = 94

| issue = 16

| pages = 168101

| doi = 10.1103/PhysRevLett.94.168101

| pmid = 15904266

| bibcode = 2005PhRvL..94p8101T

}}

Image:Scale-free network sample.svg

File:Complex network degree distribution of random and scale-free.png

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_max~ log N, where N is the network size, a very slow dependence. In contrast, in scale-free networks the largest hub scales as k_max~ ~N^1/(γ−1) 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= 2>/, which governs multiple network-based processes, from network robustness to epidemic spreading and network synchronization. While for a random network κ= + 1, i.e. the ration is independent of the network size N, for a scale-free network we have κ~ N^{(3−γ)/(γ−1)}, increasing with the network size, indicating that for these networks the degree heterogeneity increases.

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 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.

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$c$ 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. 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.{{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 }}

Examples of networks found to be scale-free include:

• Some Social networks, including collaboration networks. Two examples that have been studied extensively are the collaboration of movie actors in films and the co-authorship by mathematicians of papers.
• Many kinds of computer networks, including the internet and the webgraph of the World Wide Web.
• Some financial networks such as interbank payment networks {{cite journal|title=Fitness model for the Italian interbank money market|journal=Physical Review E|year=2006|first=Giulia|last=De Masi |display-authors=etal |volume=74|issue=6|pages=066112|doi= 10.1103/PhysRevE.74.066112|pmid=17280126|arxiv=physics/0610108|bibcode=2006PhRvE..74f6112D|s2cid=30814484}}{{cite journal|title=The topology of interbank payment flows|journal=Physica A: Statistical Mechanics and Its Applications|year=2007|first=Kimmo|last=Soramäki |display-authors=etal |volume=379|issue=1|pages=317–333|doi= 10.1016/j.physa.2006.11.093|bibcode = 2007PhyA..379..317S |hdl=10419/60649|hdl-access=free}}
• Protein–protein interaction networks.
• Semantic networks.{{cite journal|title=The Large-Scale Structure of Semantic Networks: Statistical Analyses and a Model of Semantic Growth|journal=Cognitive Science|year=2005|first=Mark|last=Steyvers|author2=Joshua B. Tenenbaum |volume=29|issue=1|pages=41–78|doi= 10.1207/s15516709cog2901_3|pmid=21702767|arxiv=cond-mat/0110012|s2cid=6000627}}
• Airline networks.

Scale free topology has been also found in high temperature superconductors.{{cite journal |doi=10.1038/nature09260 |last1=Fratini |first1=Michela |last2=Poccia |first2=Nicola |last3=Ricci |first3=Alessandro |last4=Campi |first4=Gaetano |last5=Burghammer |first5=Manfred |last6=Aeppli |first6=Gabriel |last7=Bianconi |first7=Antonio |title=Scale-free structural organization of oxygen interstitials in La2CuO4+y |journal=Nature |volume=466 |issue=7308 |pages=841–4 |year=2010 |pmid=20703301|arxiv = 1008.2015 |bibcode = 2010Natur.466..841F |s2cid=4405620 }} The qualities of a high-temperature superconductor — a compound in which electrons obey the laws of quantum physics, and flow in perfect synchrony, without friction — appear linked to the fractal arrangements of seemingly random oxygen atoms and lattice distortion.{{cite journal |doi=10.1073/pnas.1208492109 |last1=Poccia |first1=Nicola |last2=Ricci |first2=Alessandro |last3=Campi |first3=Gaetano |last4=Fratini |first4=Michela |last5=Puri |first5=Alessandro |last6=Di Gioacchino |first6=Daniele |last7=Marcelli |first7=Augusto |last8=Reynolds |first8=Michael |last9=Burghammer |first9=Manfred |last10=Saini |first10=Naurang L. |last11=Aeppli |first11=Gabriel |last12=Bianconi |first12=Antonio |title=Optimum inhomogeneity of local lattice distortions in La2CuO4+y |journal=PNAS |volume=109 |issue=39 |pages=15685–15690 |year=2012 |pmid=22961255|pmc=3465392 |arxiv = 1208.0101 |bibcode =2012PNAS..10915685P |doi-access=free }}

Scale-free networks do not arise by chance alone. 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 graphs 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. For a review see the book by Dorogovtsev and Mendes.{{Citation needed|date=December 2020}} Some mechanisms such as 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.

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.{{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 }} (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.{{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 }}

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.

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.

However, the growth of the networks (adding new nodes) is not a necessary condition for creating a scale-free network (see

Dangalchev{{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 }}). 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.

There has been a burst of activity in the modeling of scale-free complex networks. The recipe of Barabási and AlbertBarabási, A.-L. and R. Albert, Science 286, 509 (1999). has been followed by several variations and generalizationsR. Albert, and A.L. Barabási, Phys. Rev. Lett. 85, 5234(2000).S. N. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhim, cond-mat/0011115.P.L. Krapivsky, S. Redner, and F. Leyvraz, Phys. Rev. Lett. 85, 4629 (2000).B. Tadic, Physica A 293, 273(2001).{{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}} and the revamping of previous mathematical works.S. Bomholdt and H. Ebel, cond-mat/0008465; H.A. Simon, Bimetrika 42, 425(1955).

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.{{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

}}

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 nodes. Usually we concentrate on growing the network, i.e. adding nodes.

2. Preferential attachment: The probability $\backslash Pi$ that new nodes will be connected to the "old" node.

Note that some models (see

Dangalchev 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.

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

The Barabási–Albert model, an undirected version of Price's model has a linear preferential attachment $\backslash Pi(k\_i)=\backslash frac\{k\_i\}\{\backslash sum\_j\; k\_j\}$ and adds one new node at every time step.

(Note, another general feature of $\backslash Pi(k)$ in real networks is that $\backslash Pi(0)\backslash neq\; 0$, i.e. there is a nonzero probability that a

new node attaches to an isolated node. Thus in general $\backslash Pi(k)$ has the form $\backslash Pi(k)=A\; +k^\backslash alpha$, where $A$ is the initial attractiveness of the node.)

Dangalchev (see ) 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.

: $\backslash Pi(k\_i)=\backslash frac\{k\_i\; +\; C\; \backslash sum\_\{(i,j)\}\; k\_j\}\{\backslash sum\_j\; k\_j\; +\; C\; \backslash sum\_j\; k\_j^2\},$

where C is a coefficient between 0 and 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. 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.

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

: $\backslash Pi(i)\; =\; \backslash frac\{k\_i\}\; N\; \backslash frac\{\backslash sum\_\{j=1\}^\{k\_i\}\; \backslash frac\; 1\; \{k\_j\}\; \}\{k\_i\}.$

The factor $\backslash frac\{\backslash sum\_\{j=1\}^\{k\_i\}\; \backslash frac\; 1\; \{k\_j\}\; \}\{k\_i\}$ is the inverse of the harmonic mean

(IHM) of degrees of the $k\_i$ neighbors of a node $i$. Extensive numerical investigation suggest that for approximately $m>\; 14$ the mean IHM value in the large $N$ limit becomes a constant which means $\backslash Pi(i)\; \backslash propto\; k\_i$. 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.{{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 }}

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

{{See also|Non-linear preferential attachment}}

The Barabási–Albert model assumes that the probability $\backslash Pi(k)$ that a node attaches to node $i$ is proportional to the degree $k$ of node $i$. This assumption involves two hypotheses: first, that $\backslash Pi(k)$ depends on $k$, in contrast to random graphs in which $\backslash Pi(k)\; =\; p$, and second, that the functional form of $\backslash Pi(k)$ is linear in $k$.

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

Krapivsky, Redner, and Leyvraz 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. $\backslash Pi(k\_i)\; \backslash sim\; a\_\backslash infty\; k\_i$ as $k\_i\; \backslash to\; \backslash infty$. In this case the rate equation leads to

: $P(k)\; \backslash sim\; k^\{-\backslash gamma\}\backslash text\{\; with\; \}\backslash gamma\; =\; 1\; +\; \backslash frac\; \backslash mu\; \{a\_\backslash infty\}.$

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

Hierarchical network models are, by design, scale free and have high clustering of nodes.{{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 }}

The 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 = 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 = 125, and the process can continue indefinitely.

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_j and a link between vertex i and j is created with a probability

$p(x\_i,x\_j)$.{{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 }}

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

: $p(x\_i,x\_j)=\backslash frac\; \{\backslash delta\; x\_ix\_j\}\{1+\backslash delta\; x\_ix\_j\}.${{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 }}

{{Main|Hyperbolic geometric graph}}

Assuming that a network has an underlying hyperbolic geometry, one can use the framework of spatial networks to generate scale-free degree distributions. This heterogeneous degree distribution then simply reflects the negative curvature and metric properties of the underlying hyperbolic geometry.{{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}}

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.

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.

In the context of network theory a scale-free ideal network is a random network with a degree distribution following the scale-free ideal gas density distribution. These networks are able to reproduce city-size distributions and electoral results by unraveling the size distribution of social groups with information theory on complex networks when a competitive cluster growth process is applied to the network.{{cite arXiv |author1=A. Hernando |author2=D. Villuendas |author3=C. Vesperinas |author4=M. Abad |author5=A. Plastino |eprint=0905.3704 |class=physics.soc-ph |title=Unravelling the size distribution of social groups with information theory on complex networks |year=2009 }}, submitted to European Physical Journal B

{{cite journal |author1=André A. Moreira |author2=Demétrius R. Paula |author3=Raimundo N. Costa Filho |author4=José S. Andrade, Jr. |arxiv=cond-mat/0603272 |title=Competitive cluster growth in complex networks |year=2006 |doi=10.1103/PhysRevE.73.065101 |volume=73 |issue=6 |journal=Physical Review E|bibcode=2006PhRvE..73f5101M |pmid=16906890 |page=065101|s2cid=45651735 }} In models of scale-free ideal networks it is possible to demonstrate that Dunbar's number is the cause of the phenomenon known as the 'six degrees of separation'.

For a scale-free network with $n$ nodes and power-law exponent $\backslash gamma>3$, the induced subgraph constructed by vertices with degrees larger than $\backslash log\{n\}\backslash times\; \backslash log^\{*\}\{n\}$ is a scale-free network with $\backslash gamma\text{'}=2$, almost surely.{{cite arXiv |author=Heydari, H. |author2=Taheri, S.M. |author3=Kaveh, K. |title=Distributed Maximal Independent Set on Scale-Free Networks | year=2018 |eprint=1804.02513 |class=cs.DC }}

On a theoretical level, refinements to the abstract definition of scale-free have been proposed. For example, Li et al. (2005) offered a potentially more precise "scale-free metric". Briefly, let G be a graph with edge set E, and denote the degree of a vertex $v$ (that is, the number of edges incident to $v$) by $\backslash deg(v)$. Define

: $s(G)\; =\; \backslash sum\_\{(u,v)\; \backslash in\; E\}\; \backslash deg(u)\; \backslash cdot\; \backslash deg(v).$

This is maximized when high-degree nodes are connected to other high-degree nodes. Now define

: $S(G)\; =\; \backslash frac\{s(G)\}\{s\_\backslash max\},$

where s_max is the maximum value of s(H) for H in the set of all graphs with degree distribution identical to that of G. This gives a metric between 0 and 1, where a graph G with small S(G) is "scale-rich", and a graph G with S(G) close to 1 is "scale-free". This definition captures the notion of self-similarity implied in the name "scale-free".

Estimating the power-law exponent $\backslash gamma$ of a scale-free network is typically done by using the maximum likelihood estimation with the degrees of a few uniformly sampled nodes.{{Cite journal |last=Clauset |first=Aaron |author2=Cosma Rohilla Shalizi |author3=M. E. J Newman |year=2009 |title=Power-law distributions in empirical data |journal=SIAM Review |volume=51 |issue=4 |pages=661–703 |arxiv=0706.1062 |bibcode=2009SIAMR..51..661C |doi=10.1137/070710111 |s2cid=9155618}} However, since uniform sampling does not obtain enough samples from the important heavy-tail of the power law degree distribution, this method can yield a large bias and a variance. It has been recently proposed to sample random friends (i.e., random ends of random links) who are more likely come from the tail of the degree distribution as a result of the friendship paradox.{{Cite journal |last1=Eom |first1=Young-Ho |last2=Jo |first2=Hang-Hyun |date=2015-05-11 |title=Tail-scope: Using friends to estimate heavy tails of degree distributions in large-scale complex networks |journal=Scientific Reports |volume=5 |issue=1 |page=9752 |doi=10.1038/srep09752 |pmid=25959097 |pmc=4426729 |arxiv=1411.6871 |bibcode=2015NatSR...5.9752E |issn=2045-2322|doi-access=free }}{{Cite journal |last1=Nettasinghe |first1=Buddhika |last2=Krishnamurthy |first2=Vikram |date=2021-05-19 |title=Maximum Likelihood Estimation of Power-law Degree Distributions via Friendship Paradox-based Sampling |journal=ACM Transactions on Knowledge Discovery from Data |volume=15 |issue=6 |pages=1–28 |doi=10.1145/3451166 |issn=1556-4681|doi-access=free |arxiv=1908.00310 }} Theoretically, maximum likelihood estimation with random friends lead to a smaller bias and a smaller variance compared to classical approach based on uniform sampling.

• {{annotated link|Random graph}}
• {{annotated link|Erdős–Rényi model}}
• {{annotated link|Non-linear preferential attachment}}
• {{annotated link|Bose–Einstein condensation (network theory)}}
• {{annotated link|Scale invariance}}
• {{annotated link|Complex network}}
• {{annotated link|Webgraph}}
• {{annotated link|Barabási–Albert model}}
• {{annotated link|Bianconi–Barabási model}}

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{{DEFAULTSORT:Scale-Free Network}}

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