Bianconi–Barabási model

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

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

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

Each node’s evolution with time can be predicted using the continuum theory. If initial number of node is m, then the degree of node i changes at the rate:

${\displaystyle {\frac {\partial k_{i}}{\partial t}}=m{\frac {\eta _{i}k_{i}}{\sum _{j}\eta _{j}k_{j}}}}$

Assuming the evolution of ${\displaystyle k_{i}}$ follows a power law with a fitness exponent ${\displaystyle \beta (\eta _{i})}$

${\displaystyle k(t,t_{i},\eta _{i})=m\left({\frac {t}{t_{i}}}\right)^{\beta (\eta _{i})}}$,

where ${\displaystyle t_{i}}$ is the time since the creation of node ${\displaystyle i}$.

Here, ${\displaystyle \beta (\eta )={\frac {\eta }{C}}{\text{ and }}C=\int \rho (\eta ){\frac {\eta }{1-\beta (\eta )}}\,d\eta .}$

If all fitnesses are equal in a fitness network, the Bianconi–Barabási model reduces to the Barabási–Albert model, when the degree is not considered, the model reduces to the fitness model (network theory).

When fitnesses are equal, the probability ${\displaystyle \Pi _{i}}$ that the new node is connected to node ${\displaystyle i}$ when ${\displaystyle k_{i}}$ is the degree of node ${\displaystyle i}$ is,

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

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 the degree distribution will have a power-law just like the BA model. In the second case, if the fitness distribution has an infinite domain, then the node with the highest fitness value will attract a large number of nodes and show a winners-take-all scenario.[8]

There are various statistical methods to measure node fitnesses ${\displaystyle \eta _{i}}$ in the Bianconi–Barabási model from real-world network data.[9][10] From the measurement, one can investigate the fitness distribution ρ(η) or compare the Bianconi–Barabási model with various competing network models in that particular network.[10]

The Bianconi–Barabási model has been extended to weighted networks [11] displaying linear and superlinear scaling of the strength with the degree of the nodes as observed in real network data.[12] This weighted model can lead to condensation of the weights of the network when few links acquire a finite fraction of the weight of the entire network.[11] Recently it has been shown that the Bianconi–Barabási model can be interpreted as a limit case of the model for emergent hyperbolic network geometry [13] called Network Geometry with Flavor.[14] The Bianconi–Barabási model can be also modified to study static networks where the number of nodes is fixed.[15]