In
complex network
In the context of network theory, a complex network is a graph (network) with non-trivial topological features—features that do not occur in simple networks such as lattices or random graphs but often occur in networks representing real ...
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
The World Wide Web (WWW), commonly known as the Web, is an information system enabling documents and other web resources to be accessed over the Internet.
Documents and downloadable media are made available to the network through 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
Ginestra Bianconi is a network scientist and mathematical physicist, known for her work on statistical mechanics, network theory, multilayer and higher-order networks, and in particular for the Bianconi–Barabási model of growing of complex n ...
and
Albert-László Barabási
Albert-László Barabási (born March 30, 1967) is a Romanian-born Hungarian-American physicist, best known for his discoveries in network science and network medicine.
He is Distinguished University Professor and Robert Gray Professor of Netw ...
proposed a new model called
Bianconi-Barabási model, a variant to the Barabási-Albert model (
BA model
BA, Ba, or ba may refer to:
Businesses and organizations
* Bangladesh Army
* Bibliotheca Alexandrina, an Egyptian library and cultural center
* Boeing (NYSE stock symbol BA)
* Booksellers Association of the UK and Ireland
* Boston Acoustics, ...
), 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
with a probability given by a linking function
of the fitnesses of the vertices involved.
The degree of a vertex i is given by:
:
If
is an invertible and increasing function of
, then
the probability distribution
is given by
:
As a result if the fitnesses
are distributed as a power law, then also the node degree does.
Less intuitively with a fast decaying probability distribution as
together with a linking function of the kind
:
with
a constant and
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
and a linking function of the kind;
:
Fitness model and the evolution of the Web
The fitness model has been used to model the network structure of the
World Wide Web
The World Wide Web (WWW), commonly known as the Web, is an information system enabling documents and other web resources to be accessed over the Internet.
Documents and downloadable media are made available to the network through 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
A preferential attachment process is any of a class of processes in which some quantity, typically some form of wealth or credit, is distributed among a number of individuals or objects according to how much they already have, so that those who ...
mechanism, leading to a
heavy-tailed distribution
In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded: that is, they have heavier tails than the exponential distribution. In many applications it is the right tail of the distr ...
of incoming links on the Web.
See also
*
Bose–Einstein condensation: a network theory approach
References
{{Reflist, 30em
Network theory