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,

$$\Pi_i = \frac{\eta_i k_i}{\sum_j \eta_{j}k_j}.$$

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.