BA model

The Barabási–Albert (BA) model is an algorithm for generating random scale-free networks using a preferential attachment mechanism. Scale-free networks are widely observed in natural and man-made systems, including the Internet, the world wide web, citation networks, and some social networks.

Concepts

Many observed networks fall into the class of scale-free networks, meaning that they have power-law (or scale-free) degree distributions, while random graph models such as the Erdős–Rényi (ER) model and the 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 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, "the rich get richer", and in chemistry autocatalysis.

Algorithm

The network begins with an initial network of $m\_0$ nodes. It should be noted that $m\_0geq\; 2$ and the degree of each node in the initial network should be at least 1, otherwise it will always remain disconnected from the rest of the network.

New nodes are added to the network one at a time. Each new node is connected to $m$ of the existing with a probability that is biased so that it is proportional to the number of links that the existing node already has. Formally, the probability $p\_i$ that the new node is connected to node "i" is [http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/StatisticalMechanics_Rev%20of%20Modern%20Physics%2074,%2047%20(2002).pdf R. Albert and A.-L. Barabási, Statistical mechanics of complex networks, Reviews of Modern Physics, Vol 74, page 47-97, 2002.] ]

: $p\_i\; =\; frac\{k\_i\}\{displaystylesum\_j\; k\_j\}$,

where $k\_i$ is the degree of node "i". 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.

Properties

Degree distribution

The degree distribution resulting from the BA model is scale free, in particular, it is a power law of the form

: $Pleft(k\; ight)sim\; k^\{-3\}$

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 and goes as

: $lsimfrac\{ln\{N\{ln\{ln\{N\}.$

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, $n\_\{kl\}$, of finding a link between nodes of degrees $k$ and $l$ in the BA model is given by

: $n\_\{kl\}=frac\{4left(l-1\; ight)\}\{kleft(k+1\; ight)left(k+l\; ight)left(k+l+1\; ight)left(k+l+2\; ight)\}+frac\{12left(l-1\; ight)\}\{kleft(k+l-1\; ight)left(k+l\; ight)left(k+l+1\; ight)left(k+l+2\; ight)\}.$

This is certainly not the result expected if the distributions were uncorrelated, $n\_\{kl\}=k^\{-3\}l^\{-3\}$

Clustering coefficient

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

: $Csim\; N^\{-0.75\}.$

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,

: $C(k)\; =\; k^\{-1\}.$

This result was obtained analytically by Dorogovtsev, Goltsev and Mendes [S.N. Dorogovtsev, A.V. Goltsev, and J.F.F. Mendes, e-print cond-mat/0112143.] .

pectral 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.

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 exponential, 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.

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.

History

The first use of a preferential attachment mechanism to explain power-law distributions appears to have been by Yule in 1925 [cite journal
author = Udny Yule
title = A Mathematical Theory of Evolution Based on the Conclusions of Dr. J. C. Willis, F.R.S.
journal = Journal of the Royal Statistical Society
volume = 88
pages = 433–436
year = 1925
doi = 10.2307/2341419] , 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 Simon in 1955 [Cite journal
author = Herbert A. Simon
title = On a Class of Skew Distribution Functions
journal = Biometrika
volume = 42
issue = 3-4
pages = 425–440
month = December
year = 1955
doi = 10.1093/biomet/42.3-4.425] in the course of studies of the sizes of cities and other phenomena. It was first applied to the growth of networks by Derek de Solla Price in 1976 [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] 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 webCite 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
volume = 286
pages = 509–512
month = October
year = 1999
doi = 10.1126/science.286.5439.509] .

References