Fitness model (network theory)

Description of the model

The model is based on the idea of fitness, an inherent competitive factor that nodes may have, capable of affecting the network's evolution. According to this idea, the nodes' intrinsic ability to attract links in the network varies from node to node, the most efficient (or "fit") being able to gather more edges in the expense of others. In that sense, not all nodes are identical to each other, and they claim their degree increase according to the fitness they possess every time. The fitness factors of all the nodes composing the network may form a distribution ρ(η) characteristic of the system been studied.

Ginestra Bianconi and Albert-László Barabási proposed a new model called Bianconi-Barabási model, a variant to the Barabási-Albert model (BA model), where the probability for a node to connect to another one is supplied with a term expressing the fitness of the node involved. The fitness parameter is time-independent and is multiplicative to the probability.

Fitness model where fitnesses are not coupled to preferential attachment has been introduced by Caldarelli et al. Here a link is created between two vertices i,j with a probability given by a linking function f(\eta_i,\eta_j) of the fitnesses of the vertices involved. The degree of a vertex i is given by:

:k(\eta_i)=N\int_0^\infty !!! f(\eta_i,\eta_j) \rho(\eta_j) d\eta_j

If k(\eta_i) is an invertible and increasing function of \eta_i, then the probability distribution P(k) is given by

:P(k)=\rho(\eta(k)) \cdot \eta'(k)

As a result if the fitnesses \eta are distributed as a power law, then also the node degree does.

Less intuitively with a fast decaying probability distribution as \rho(\eta)=e^{-\eta} together with a linking function of the kind

: f(\eta_i,\eta_j)=\Theta(\eta_i+\eta_j-Z)

with Z a constant and \Theta the Heavyside function, we also obtain scale-free networks.

Such model has been successfully applied to describe trade between nations by using GDP as fitness for the various nodes i,j and a linking function of the kind;

: \frac{\delta \eta_i\eta_j}{1+ \delta \eta_i\eta_j}

Fitness model and the evolution of the Web

The fitness model has been used to model the network structure of the World Wide Web. In a PNAS article, Kong et al. extended the fitness model to include random node deletion, a common phenomena in the Web. When the deletion rate of the web pages are accounted for, they found that the overall fitness distribution is exponential. Nonetheless, even this small variance in the fitness is amplified through the preferential attachment mechanism, leading to a heavy-tailed distribution of incoming links on the Web.

References

References

1. (May 2001). "Competition and multiscaling in evolving networks.". Europhysics Letters.
2. (December 2002). "Scale-free networks from varying vertex intrinsic fitness". Physical Review Letters.
3. (November 2004). "Vertex intrinsic fitness: how to produce arbitrary scale-free networks". Physical Review E.
4. (October 2004). "Fitness-dependent topological properties of the world trade web". Physical Review Letters.
5. (October 2015). "Systemic Risk Analysis on Reconstructed Economic and Financial Networks". Scientific Reports.
6. (September 2008). "Experience versus talent shapes the structure of the Web". Proceedings of the National Academy of Sciences of the United States of America.