The Barabási–Albert (BA) model is an algorithm for generating random scale-free networks using a preferential attachment mechanism. Several natural and human-made systems, including the Internet, the world wide web, citation networks, and some social networks are thought to be approximately scale-free and certainly contain few nodes (called hubs) with unusually high degree as compared to the other nodes of the network. The BA model tries to explain the existence of such nodes in real networks. The algorithm is named for its inventors Albert-László Barabási and Réka Albert and is a special case of an earlier and more general model called Price's model.
Many observed networks (at least approximately) 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 generate 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 a higher degree have a 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, they are more likely to become acquainted with one of those more visible people rather than with a relative unknown. The BA model was proposed by assuming that in the World Wide Web, new pages link preferentially to hubs, i.e. very well known sites such as Google, 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. The BA model claims that this explains the preferential attachment probability rule. However, in spite of being quite a useful model, empirical evidence suggests that the mechanism in its simplest form does not apply to the World Wide Web as shown in "Technical Comment to 'Emergence of Scaling in Random Networks'".
Later, the Bianconi–Barabási model works to address this issue by introducing a "fitness" parameter. 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". See also autocatalysis.
The network begins with an initial connected network of nodes.
New nodes are added to the network one at a time. Each new node is connected to existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability that the new node is connected to node is
where is the degree of node and the sum is made over all pre-existing nodes (i.e. the denominator results in twice 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.
The degree distribution resulting from the BA model is scale free, in particular, it is a power law of the form
Hirsch index distribution
Furthermore, an analytic result for the density of nodes with h-index 1 can be obtained in the case where
Average path length
The BA model has a systematically shorter average path length than a random graph.
The percolation threshold of BA model has been found to be pc=0. The meaning is that when removing randomly nodes in the BA network, any fraction of nodes will not break the network. On the other hand removing only a relatively small fraction of highest degree nodes will cause the network to collapse.
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, , of finding a link that connects a node of degree to an ancestor node of degree in the BA model for the special case of (BA tree) is given by
This confirms the existence of degree correlations, because if the distributions were uncorrelated, we would get .
For general , the fraction of links who connect a node of degree to a node of degree is
Also, the nearest-neighbor degree distribution , that is, the degree distribution of the neighbors of a node with degree , is given by
In other words, if we select a node with degree , and then select one of its neighbors randomly, the probability that this randomly selected neighbor will have degree is given by the expression above.
An analytical result for the clustering coefficient of the BA model was obtained by Klemm and Eguíluz and proven by Bollobás. A mean-field approach to study the clustering coefficient was applied by Fronczak, Fronczak and Holyst.
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,
This result was obtained analytically by Dorogovtsev, Goltsev and Mendes.
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. 
By definition, the BA model describes a time developing phenomenon and hence, besides its scale-free property, one could also look for its dynamic scaling property. In the BA network nodes can also be characterized by generalized degree , the product of the square root of the birth time of each node and their corresponding degree , instead of the degree alone since the time of birth matters in the BA network. We find that the generalized degree distribution has some non-trivial features and exhibits dynamic scaling
It implies that the distinct plots of vs would collapse into a universal curve if we plots vs .
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, indicating that growth alone is not sufficient to produce a scale-free structure.
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.
Non-linear preferential attachment
The BA model can be thought of as a specific case of the more general non-linear preferential attachment (NLPA) model. The NLPA algorithm is identical to the BA model with the attachment probability replaced by the more general form
where is a constant positive exponent. If , NLPA reduces to the BA model and is referred to as "linear". If , NLPA is referred to as "sub-linear" and the degree distribution of the network tends to a stretched exponential distribution. If , NLPA is referred to as "super-linear" and a small number of nodes connect to almost all other nodes in the network. For both and , the scale-free property of the network is broken in the limit of infinite system size. However, if is only slightly larger than , NLPA may result in degree distributions which appear to be transiently scale free.
Preferential attachment made its first appearance in 1923 in the celebrated urn model of the Hungarian mathematician György Pólya in 1923. The modern master equation method, which yields a more transparent derivation, was applied to the problem by Herbert A. Simon in 1955 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. Price was interested in the networks of citation between scientific papers and the Price model used "cummulative advantage" (his name for preferential attachment) to produce a directed network so the Barabási-Albert model is an undirected version of Price's model. 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.
- Bianconi–Barabási model
- Chinese restaurant process
- Complex networks
- Erdős–Rényi (ER) model
- Price's model
- Percolation theory
- Scale-free network
- Small-world network
- Watts and Strogatz model
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