User:Nurbudapest/sandbox/The Bianconi-Barabási model

The Bianconi-Barabási model is a model in Network Science which explains the growth of complex networks and also can generate a complex network. This model is named after Ginestra Bianconi and Albert-László Barabási. This model is a variant of the Barabási–Albert model.

Concepts

The Barabási–Albert (BA) model uses two concepts growth and preferential attachment, the Bianconi-Barabási model uses these two and another fitness parameter.

The Bianconi-Barabási model uses an analogy with evolutionary models. This model assigns an intrinsic fitness value to each node which embodies all the properties other than the degree . The higher the fitness the higher the probability of attracting new edges.

While the Barabási–Albert (BA) model explains the first mover advantage but Bianconi-Barabási model explains how late comers can also win. In a network, where fitness is an attribute, a node with higher fitness will acquire links at a higher rate than less fit nodes. This model explains that age is not the best predictor of a node's success, late comers also have chance to attract links.

The Bianconi-Barabási model can reproduce the degree correlations of the Internet Autonomous Systems . This model can also show condensation phase transitions in the evolution of complex network .

Algorithm

The fitness network begins with a fix number of interconnected nodes. As they have different fitness which can be described with fitness parameter, $\eta _{j}$ which is chosen from a fitness distribution ρ(η).

Growth

Here assumption is that a node’s fitness is independent of time and fixed. If a new node j with m links and a fitness $\eta _{j}$ is added with each time-step.

Preferential Attachment

The probability Πi that a new node connects to one of existing links to a node i in the network depends on the number of edges, $k_{i}$, and on the fitness $\eta _{i}$ of node i, such that,

$\Pi _{i}={\frac {\eta _{i}k_{i}}{\sum _{j}\eta _{j}k_{j}}}.$

Properties

Equal Fitnesses

If all fitnesses are equal in a fitness network the Bianconi-Barabási model becomes the Barabási-Albert model.

The probability $p_{i}$ that the new node is connected to node $i$ is :

$\Pi _{i}={\frac {k_{i}}{\sum _{j}k_{j}}}.$

Here $k_{i}$ is the degree of node $i$.

The degree distribution

The degree distribution of the Bianconi-Barabási model depends on the fitness distribution ρ(η). There are two scenarios that can happen based on the probability distribution. If the fitness distribution has a finite domain then just like BA model degree distribution will have a power-law. In second case, if the fitness distribution has an infinite domain then a nodes with highest fitness value will attract large number of nodes and show a winners-take-all scenario .

History

In 1999, Albert-László Barabási requested his student Bianconi to investigate evolving network where nodes have a fitness parameter. Barabási was interested in finding out how Google a late comer in the search engine market became a top player. Bianconi's work showed that when fitness parameter is present "early bird" is not always the winner .

In 2001 Ginestra Bianconi and Albert-László Barabási in Europhysics Letters published this model .

See also

• Barabási–Albert model
• Bose–Einstein condensation: a network theory approach

External links

• Networks: A Very Short Introduction
• Advance Network Dynamics