|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{Network Science}}
| | I'm Luther but I by no means truly liked that title. Playing croquet is something I will never give up. I am a production and distribution officer. Some time ago he chose to reside in Idaho.<br><br>Take a look at my site ... [http://Christianculturecenter.org/ActivityFeed/MyProfile/tabid/61/userId/34117/Default.aspx Christianculturecenter.org] |
| The '''Barabási–Albert (BA) model''' is an algorithm for generating random [[scale-free network|scale-free]] [[complex network|networks]] using a [[preferential attachment]] mechanism. Scale-free networks are widely observed in natural and human-made systems, including the [[Internet]], the [[world wide web]], [[citation analysis|citation networks]], and some [[social networks]]. The algorithm is named for its inventors [[Albert-László Barabási]] and [[Réka Albert]].
| |
| | |
| ==Concepts==
| |
| Many observed networks fall into the class of [[scale-free networks]], meaning that they have [[power law|power-law]] (or scale-free) degree distributions, while random graph models such as the [[Erdős–Rényi model|Erdős–Rényi (ER) model]] and the [[Watts and Strogatz model|Watts–Strogatz (WS) model]] do not exhibit power laws. The Barabási–Albert model is one of several proposed models that generates scale-free networks. It incorporates two important general concepts: growth and [[preferential attachment]]. Both growth and preferential attachment exist widely in real networks.
| |
| | |
| Growth means that the number of nodes in the network increases over time.
| |
| | |
| [[Preferential attachment]] means that the more connected a node is, the more likely it is to receive new links. Nodes with higher [[degree (graph theory)|degree]] have stronger ability to grab links added to the network. Intuitively, the preferential attachment can be understood if we think in terms of [[social networks]] connecting people. Here a link from A to B means that person A "knows" or "is acquainted with" person B. Heavily linked nodes represent well-known people with lots of relations. When a newcomer enters the community, s/he is more likely to become acquainted with one of those more visible people rather than with a relative unknown. Similarly, on the web, new pages link preferentially to hubs, i.e. very well known sites such as [[Google]] or [[Wikipedia]], rather than to pages that hardly anyone knows. If someone selects a new page to link to by randomly choosing an existing link, the probability of selecting a particular page would be proportional to its degree. This explains the preferential attachment probability rule.
| |
| | |
| Preferential attachment is an example of a [[positive feedback]] cycle where initially random variations (one node initially having more links or having started accumulating links earlier than another) are automatically reinforced, thus greatly magnifying differences. This is also sometimes called the [[Matthew effect (sociology)|Matthew effect]], "the [[rich get richer]]", and in chemistry [[autocatalysis]].
| |
| | |
| ==Algorithm==
| |
| [[File:Barabasi Albert model.gif|thumb|300px|The steps of the growth of the network according to the Barabasi–Albert model (<math>m_0=m=2</math>)]]
| |
| The network begins with an initial connected network of <math>m_0</math> nodes.
| |
| | |
| New nodes are added to the network one at a time. Each new node is connected to <math>m \le m_0</math> existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability <math>p_i</math> that the new node is connected to node <math>i</math> is<ref name=RMP>{{Cite journal
| |
| | url = http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/StatisticalMechanics_Rev%20of%20Modern%20Physics%2074,%2047%20(2002).pdf
| |
| | author1 = R. Albert
| |
| | author2 = A.-L. Barabási
| |
| | title = Statistical mechanics of complex networks
| |
| | journal = [[Reviews of Modern Physics]]
| |
| | volume = 74
| |
| | pages = 47–97
| |
| | year = 2002
| |
| | doi = 10.1103/RevModPhys.74.47
| |
| | bibcode=2002RvMP...74...47A
| |
| |arxiv = cond-mat/0106096 }}</ref>
| |
| | |
| : <math>p_i = \frac{k_i}{\sum_j k_j},</math>
| |
| | |
| where <math>k_i</math> is the degree of node <math>i</math> and the sum is made over all pre-existing nodes <math>j</math> (i.e. the denominator results in the current number of edges in the network). Heavily linked nodes ("hubs") tend to quickly accumulate even more links, while nodes with only a few links are unlikely to be chosen as the destination for a new link. The new nodes have a "preference" to attach themselves to the already heavily linked nodes.
| |
| | |
| [[File:Barabasi Albert generated network.jpg|thumb|A network generated according to the Barabasi Albert model. The network is made of 50 vertices with initial degrees <math>m_0=1</math>.]]
| |
| | |
| ==Properties==
| |
| | |
| ===Degree distribution===
| |
| [[File:Barabasi-albert model degree distribution.svg|thumb|The degree distribution of the BA Model, which follows a power law. In loglog scale the power law function is a straight line.<ref name=Barabasi1999 />]]
| |
| The degree distribution resulting from the BA model is scale free, in particular, it is a power law of the form
| |
| | |
| : <math>P\left(k\right)\sim k^{-3} \, </math>
| |
| | |
| ===Average path length===
| |
| The [[average path length]] of the BA model increases approximately logarithmically with the size of the network. The actual form has a double logarithmic correction<ref name=RMP/> and goes as
| |
| | |
| : <math>\ell\sim\frac{\ln N}{\ln \ln N}.</math>
| |
| | |
| The BA model has a systematically shorter average path length than a random graph.
| |
| | |
| ===Node degree correlations===
| |
| Correlations between the degrees of connected nodes develop spontaneously in the BA model because of the way the network evolves. The probability, <math>n_{k\ell}</math>, of finding a link between nodes of degrees <math>k</math> and <math>\ell</math> in the BA model when <math>m=1</math> is given by
| |
| | |
| : <math>n_{k\ell}=\frac{4\left(\ell-1\right)}{k\left(k+1\right)\left(k+\ell\right)\left(k+\ell+1\right)\left(k+\ell+2\right)}+\frac{12\left(\ell-1\right)}{k\left(k+\ell-1\right)\left(k+\ell\right)\left(k+\ell+1\right)\left(k+\ell+2\right)}.</math>
| |
| | |
| This is certainly not the result expected if the distributions were uncorrelated, <math>n_{k\ell}=k^{-3}\ell^{-3}</math><ref name=RMP/>
| |
| | |
| ===Clustering coefficient===
| |
| | |
| <!-- Image with unknown copyright status removed: [[File:ClusteringCoefficientBAModel.jpg|thumb|right|The BA Model is more clustered than a random graph, and the clustering decreases more slowly with network size than for the random graph.<ref name=RMP/>]] -->
| |
| While there is no analytical result for the [[clustering coefficient]] of the BA model, the empirically determined clustering coefficients are generally significantly higher for the BA model than for random networks. The clustering coefficient also scales with network size following approximately a power law
| |
| | |
| : <math>C\sim N^{-0.75}. \, </math>
| |
| | |
| This behavior is still distinct from the behavior of small-world networks where clustering is independent of system size.
| |
| In the case of hierarchical networks, clustering as a function of node degree also follows a power-law,
| |
| | |
| : <math>C(k) = k^{-1}. \, </math>
| |
| | |
| This result was obtained analytically by Dorogovtsev, Goltsev and Mendes.<ref>S.N. Dorogovtsev, A.V. Goltsev, and J.F.F. Mendes, e-print cond-mat/0112143.</ref>
| |
| | |
| ===Spectral properties===
| |
| | |
| The spectral density of BA model has a different shape from the semicircular spectral density of random graph. It has a triangle-like shape with the top lying well above the semicircle and edges decaying as a power law.
| |
| <!--
| |
| Image with unknown copyright status removed: [[File:SpectralDensityOfBA.jpg|thumb|right|Rescaled spectral density of BA networks, which has a different shape from the semicircular spectral density of random graph.<ref name=RMP/>]] -->
| |
| | |
| ==Limiting cases==
| |
| | |
| ===Model A===
| |
| | |
| Model A retains growth but does not include preferential attachment. The probability of a new node connecting to any pre-existing node is equal. The resulting degree distribution in this limit is geometric,<ref>{{cite journal|last=Pekoz|first=Erol|coauthors=A. Rollin, N. Ross|title=Total variation and local limit error bounds for geometric approximation|journal=Bernoulli|year=2012|url=http://www.e-publications.org/ims/submission/index.php/BEJ/user/submissionFile/10315?confirm=c40442a0}}</ref> indicating that growth alone is not sufficient to produce a scale-free structure.
| |
| | |
| ===Model B===
| |
| | |
| Model B retains preferential attachment but eliminates growth. The model begins with a fixed number of disconnected nodes and adds links, preferentially choosing high degree nodes as link destinations. Though the degree distribution early in the simulation looks scale-free, the distribution is not stable, and it eventually becomes nearly Gaussian as the network nears saturation. So preferential attachment alone is not sufficient to produce a scale-free structure.
| |
| <!--
| |
| Image with unknown copyright status removed: [[File:Limiting_Case_A.GIF|thumb|left|The degree distribution for Model A is an exponential distribution<ref name=RMP/>]] -->
| |
| <!-- Image with unknown copyright status removed: [[File:Limiting_Case_B.jpg|thumb|right|The degree distribution for Model B becomes a Gaussian around its mean value<ref name=RMP/>]] -->
| |
| | |
| The failure of models A and B to lead to a scale-free distribution indicates that growth and preferential attachment are needed simultaneously to reproduce the stationary power-law distribution observed in real networks.<ref name=RMP/>
| |
| | |
| == History ==
| |
| | |
| The first use of a preferential attachment mechanism to explain power-law distributions appears to have been by [[Udny Yule|Yule]] in 1925,<ref>{{cite journal
| |
| | author = [[Udny Yule]]
| |
| | last2 = Yule
| |
| | first2 = G. Udny
| |
| | title = A Mathematical Theory of Evolution Based on the Conclusions of Dr. J. C. Willis, F.R.S.
| |
| | jstor = 2341419
| |
| | journal = Journal of the Royal Statistical Society
| |
| | volume = 88
| |
| | issue = 3
| |
| | pages = 433–436
| |
| | year = 1925
| |
| | doi = 10.2307/2341419
| |
| }}</ref> although Yule's mathematical treatment is opaque by modern standards because of the lack of appropriate tools for analyzing stochastic processes. The modern master equation method, which yields a more transparent derivation, was applied to the problem by [[Herbert A. Simon]] in 1955<ref>{{Cite journal
| |
| | author = [[Herbert A. Simon]]
| |
| | title = On a Class of Skew Distribution Functions
| |
| | journal = [[Biometrika]]
| |
| | volume = 42
| |
| | issue = 3–4
| |
| | pages = 425–440
| |
| |date=December 1955
| |
| | doi = 10.1093/biomet/42.3-4.425
| |
| }}</ref> in the course of studies of the sizes of cities and other phenomena. It was first applied to the growth of networks by [[Derek J. de Solla Price|Derek de Solla Price]] in 1976<ref>{{Cite journal
| |
| | author = [[D.J. de Solla Price]]
| |
| | title = A general theory of bibliometric and other cumulative advantage processes
| |
| | journal = [[Journal of the American Society for Information Science]]
| |
| | volume = 27
| |
| | pages = 292–306
| |
| | year = 1976
| |
| | doi = 10.1002/asi.4630270505
| |
| }}</ref>
| |
| who was interested in the networks of citation between scientific papers. The name "preferential attachment" and the present popularity of scale-free network models is due to the work of [[Albert-László Barabási]] and [[Réka Albert]], who rediscovered the process independently in 1999 and applied it to degree distributions on the web.<ref name=Barabasi1999>{{Cite journal
| |
| | url = http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/EmergenceRandom_Science%20286,%20509-512%20(1999).pdf
| |
| | author = [[Albert-László Barabási]] & [[Réka Albert]]
| |
| | title = Emergence of scaling in random networks
| |
| | journal = [[Science (journal)|Science]]
| |
| | volume = 286
| |
| | pages = 509–512
| |
| |date=October 1999
| |
| | doi = 10.1126/science.286.5439.509
| |
| | issue=5439
| |
| | pmid=10521342
| |
| |arxiv = cond-mat/9910332 |bibcode = 1999Sci...286..509B }}</ref>
| |
| | |
| ==See also==
| |
| * [[Erdős–Rényi model|Erdős–Rényi (ER) model]]
| |
| * [[Watts and Strogatz Model]]
| |
| * [[Small-world network]]
| |
| | |
| ==References==
| |
| | |
| <references />
| |
| | |
| ==External links==
| |
| *[http://www.popsci.com/science/article/2011-10/man-could-rule-world "This Man Could Rule the World"]
| |
| | |
| {{DEFAULTSORT:Barabasi-Albert Model}}
| |
| [[Category:Social networks]]
| |
| [[Category:Graph algorithms]]
| |
| [[Category:Random graphs]]
| |