Fitness model (network theory)

In complex network theory, the fitness model is a model of the evolution of a network: how the links between nodes change over time depends on the fitness of nodes. Fitter nodes attract more links at the expense of less fit nodes.

It has been used to model the network structure of the World Wide Web.

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 ^[1] 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. ^[2] Here a link is created between two vertices ${\displaystyle i,j}$ with a probability given by a linking function ${\displaystyle f(\eta _{i},\eta _{j})}$ of the fitnesses of the vertices involved. The degree of a vertex i is given by: ^[3]

${\displaystyle k(\eta _{i})=N\int _{0}^{\infty }\!\!\!f(\eta _{i},\eta _{j})\rho (\eta _{j})d\eta _{j}}$

If ${\displaystyle k(\eta _{i})}$ is an invertible and increasing function of ${\displaystyle \eta _{i}}$, then the probability distribution ${\displaystyle P(k)}$ is given by

${\displaystyle P(k)=\rho (\eta (k))\cdot \eta '(k)}$

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

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

${\displaystyle f(\eta _{i},\eta _{j})=\Theta (\eta _{i}+\eta _{j}-Z)}$

with ${\displaystyle Z}$ a constant and ${\displaystyle \Theta }$ 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 ${\displaystyle i,j}$ and a linking function of the kind; ^{[4] [5]}

${\displaystyle {\frac {\delta \eta _{i}\eta _{j}}{1+\delta \eta _{i}\eta _{j}}}}$

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, ^[6] 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.

See also

• Bose–Einstein condensation: a network theory approach

References

1. Bianconi G, Barabási AL (May 2001). "Competition and multiscaling in evolving networks" (PDF). Europhysics Letters. 54 (4): 436–442. arXiv: cond-mat/0011029 . Bibcode:2001EL.....54..436B. doi:10.1209/epl/i2001-00260-6. Archived (PDF) from the original on 2017-08-09. Retrieved 2019-12-10.
2. Caldarelli G, Capocci A, De Los Rios P, Muñoz MA (December 2002). "Scale-free networks from varying vertex intrinsic fitness" (PDF). Physical Review Letters. 89 (25): 258702. Bibcode:2002PhRvL..89y8702C. doi:10.1103/PhysRevLett.89.258702. PMID   12484927. Archived (PDF) from the original on 2023-02-04. Retrieved 2019-12-10.
3. Servedio VD, Caldarelli G, Buttà P (November 2004). "Vertex intrinsic fitness: how to produce arbitrary scale-free networks". Physical Review E. 70 (5 Pt 2): 056126. arXiv: cond-mat/0309659 . Bibcode:2004PhRvE..70e6126S. doi:10.1103/PhysRevE.70.056126. PMID   15600711. S2CID   14349707.
4. Garlaschelli D, Loffredo MI (October 2004). "Fitness-dependent topological properties of the world trade web". Physical Review Letters. 93 (18): 188701. arXiv: cond-mat/0403051 . Bibcode:2004PhRvL..93r8701G. doi:10.1103/PhysRevLett.93.188701. PMID   15525215. S2CID   16367275.
5. Cimini G, Squartini T, Garlaschelli D, Gabrielli A (October 2015). "Systemic Risk Analysis on Reconstructed Economic and Financial Networks". Scientific Reports. 5: 15758. arXiv: 1411.7613 . Bibcode:2015NatSR...515758C. doi:10.1038/srep15758. PMC   4623768 . PMID   26507849.
6. Kong JS, Sarshar N, Roychowdhury VP (September 2008). "Experience versus talent shapes the structure of the Web". Proceedings of the National Academy of Sciences of the United States of America. 105 (37): 13724–9. doi: 10.1073/pnas.0805921105 . PMC   2544521 . PMID   18779560.