Scale-free network

From Free net encyclopedia

A scale-free network is a specific kind of complex network that has attracted attention since many real-world networks fall into this category. In scale-free networks, some nodes act as "highly connected hubs" (high degree), although most nodes are of low degree.

Contents

[edit]

History

Using a Web crawler, physicist Albert-Laszlo Barabasi and his colleagues at the University of Notre Dame in Indiana, USA, in 1999 mapped the connectedness of the Web. To their surprise, the web did not have an even distribution of connectivity (so-called "random connectivity"). Instead, a very few network nodes (called "hubs") were far more connected than other nodes. In general, they found that the probability p(k) that a node in the network connects with k other nodes was, in a given network, proportional to k−γ. The degree exponent γ is not universal and depends on the detail of network structure. Numerical values of the exponent γ for various systems are diverse but most of them are in the range 2 < γ ≤ 3. At the same time a similar observation was obtained to the Internet by the Faloutsos brothers (1999). In this form, essentially all graphs with a power law degree distribution were grouped together as "scale-free". Several revisions of this definition have been suggested. (see below)

[edit]

Formal definition

Although no one definition is universally accepted, Li et al (2005) defined a "scale-free metric". Let g be a graph with edge-set ε, and let the degree (number of edges) at a vertex i be <math>d_i</math>. Define

<math> s(g) = \sum_{(i,j) \in \epsilon}d_id_j </math>

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

<math> S(g) = \frac{s(g)}{s_{max}} </math>

where smax is the maximum value of s(h) for h in the set of all graphs with an identical degree distribution to g. This gives a metric between 0 and 1, such that graphs with low S(g) are "scale-rich", and graphs with S(g) close to 1 are "scale-free". This definition captures the notion of self-similarity implied in the name "scale-free".

[edit]

Characteristics and examples

Image:Scale-free network sample.png

Scale-free networks tend to contain centrally located and interconnected high degree "hubs", which dramatically influences the way a network operates. For example, random node failures have very little effect on a scale-free network's connectivity or effectiveness, however deliberate attacks on such a network's hubs can dismantle a network with alarming ease. Thus, the realization that certain networks are scale-free is important to security.

These networks also exhibit the Small world phenomenon, in which two average nodes are separated by a very small number of connections. Also, scale-free networks generally have high clustering coefficients.

A multitude of real-world networks have been shown to be scale-free, including:

[edit]

Generative models

These scale-free networks do not arise by chance alone. Erdős and Renyi (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 not consistent with the properties observed in scale-free networks, and therefore a model for this growth process is needed.

The scale-free properties of the Web have been studied, and its distribution of links is very close to a power law, because there are a few Web sites with huge numbers of links, which benefit from a good placement in search engines and an established presence on the Web. Those sites are the ones that attract more of the new links. This has been called the winners take all phenomenon.

The mostly widely accepted generative model is Barabasi and Albert's (1999) rich get richer generative model in which each new Web page creates link to existent Web pages with a probability distribution which is not uniform, but proportional to the current in-degree of Web pages. This model was originally discovered by Derek de Solla Price in 1965 under the term cumulative advantage, but did not reach popularity until Barabasi rediscovered the results under its current name. 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 networks characteristics have been proposed and studied (for a review see the book by Dorogovtsev and Mendes).

A different generative model is the copy model studied by Kumar et al. (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.

However, if we look at communities of interests in a specific topic, discarding the major hubs of the Web, the distribution of links is no longer a power law but resembles more a normal distribution, as observed by Pennock et al. (2002) in the communities of the home pages of universities, public companies, newspapers and scientists. Based on these observations, the authors propose a generative model that mixes preferential attachment with a baseline probability of gaining a link.

Recently, Manev and Manev (Med. Hypotheses, 2005) proposed that small world networks may be operative in adult brain neurogenesis. Adult neurogenesis has been observed in mammalian brain including human but a question remains: how do new neurons become functional in the adult brain? It is proposed that the random addition of only a few new neurons functions as a maintenance system for the brain's "small-world" networks. Randomly added to an orderly network, new links enhance signal propagation speed and synchronizability. Newly generated neurons are ideally suited to become such links: they are immature, form more new connections compared to mature ones, and their number but not their precise location may be maintained by continuous proliferation and dying off. Similarly, it is envisaged that the treatment of brain pathologies by cell transplantation would also create new random links in small-world networks and that even a small number of successfully incorporated new neurons may be functionally important.

[edit]

References

it:Rete a invarianza di scala

Retrieved from "http://www.netipedia.com/index.php/Scale-free_network"