Fitness model (network theory)

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{{cite journal | vauthors = Bianconi G, Barabási AL | title = Competition and multiscaling in evolving networks. | journal = Europhysics Letters | date = May 2001 | volume = 54 | issue = 4 | pages = 436–442 | url = http://www.uvm.edu/pdodds/files/papers/others/everything/bianconi2001a.pdf | doi = 10.1209/epl/i2001-00260-6 | bibcode = 2001EL.....54..436B | arxiv = cond-mat/0011029 | access-date = 2019-12-10 | archive-date = 2017-08-09 | archive-url = https://web.archive.org/web/20170809130220/http://www.uvm.edu/pdodds/files/papers/others/everything/bianconi2001a.pdf | url-status = live }} 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.{{cite journal | vauthors = Caldarelli G, Capocci A, De Los Rios P, Muñoz MA | title = Scale-free networks from varying vertex intrinsic fitness | journal = Physical Review Letters | volume = 89 | issue = 25 | pages = 258702 | date = December 2002 | pmid = 12484927 | doi = 10.1103/PhysRevLett.89.258702 | bibcode = 2002PhRvL..89y8702C | url = http://eprints.imtlucca.it/1137/1/PhysRevLett_Caldarelli_02.pdf | access-date = 2019-12-10 | archive-date = 2023-02-04 | archive-url = https://web.archive.org/web/20230204160224/http://eprints.imtlucca.it/1137/1/PhysRevLett_Caldarelli_02.pdf | url-status = live }}

Here a link is created between two vertices $i,j$ with a probability given by a linking function $f(\backslash eta\_i,\backslash eta\_j)$ of the fitnesses of the vertices involved.

The degree of a vertex i is given by:{{cite journal | vauthors = Servedio VD, Caldarelli G, Buttà P | title = Vertex intrinsic fitness: how to produce arbitrary scale-free networks | journal = Physical Review E | volume = 70 | issue = 5 Pt 2 | pages = 056126 | date = November 2004 | pmid = 15600711 | doi = 10.1103/PhysRevE.70.056126 | bibcode = 2004PhRvE..70e6126S | arxiv = cond-mat/0309659 | s2cid = 14349707 }}

:$k(\backslash eta\_i)=N\backslash int\_0^\backslash infty\; \backslash !\backslash !\backslash !\; f(\backslash eta\_i,\backslash eta\_j)\; \backslash rho(\backslash eta\_j)\; d\backslash eta\_j$

If $k(\backslash eta\_i)$ is an invertible and increasing function of $\backslash eta\_i$, then

the probability distribution $P(k)$ is given by

:$P(k)=\backslash rho(\backslash eta(k))\; \backslash cdot\; \backslash eta\text{'}(k)$

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

Less intuitively with a fast decaying probability distribution as

$\backslash rho(\backslash eta)=e^\{-\backslash eta\}$ together with a linking function of the kind

:$f(\backslash eta\_i,\backslash eta\_j)=\backslash Theta(\backslash eta\_i+\backslash eta\_j-Z)$

with $Z$ a constant and $\backslash Theta$ 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;{{cite journal | vauthors = Garlaschelli D, Loffredo MI | title = Fitness-dependent topological properties of the world trade web | journal = Physical Review Letters | volume = 93 | issue = 18 | pages = 188701 | date = October 2004 | pmid = 15525215 | doi = 10.1103/PhysRevLett.93.188701 | bibcode = 2004PhRvL..93r8701G | arxiv = cond-mat/0403051 | s2cid = 16367275 }}{{cite journal | vauthors = Cimini G, Squartini T, Garlaschelli D, Gabrielli A | title = Systemic Risk Analysis on Reconstructed Economic and Financial Networks | journal = Scientific Reports | volume = 5 | pages = 15758 | date = October 2015 | pmid = 26507849 | pmc = 4623768 | doi = 10.1038/srep15758 | bibcode = 2015NatSR...515758C | arxiv = 1411.7613 }}

:$\backslash frac\{\backslash delta\; \backslash eta\_i\backslash eta\_j\}\{1+\; \backslash delta\; \backslash eta\_i\backslash eta\_j\}$

The fitness model has been used to model the network structure of the World Wide Web. In a PNAS article,{{cite journal | vauthors = Kong JS, Sarshar N, Roychowdhury VP | title = Experience versus talent shapes the structure of the Web | journal = Proceedings of the National Academy of Sciences of the United States of America | volume = 105 | issue = 37 | pages = 13724–9 | date = September 2008 | pmid = 18779560 | pmc = 2544521 | doi = 10.1073/pnas.0805921105 | doi-access = free }} 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.

• Bose–Einstein condensation: a network theory approach

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Category:Network theory