From: scerir (scerir@libero.it)
Date: Sun Oct 08 2000 - 12:30:57 MDT
http://xxx.lanl.gov/format/cond-mat/0009178
(a long abstract)
Networks are present in many aspects of everyday life.
In all these cases, the network properties should be such
to optimise some cost function, as the number of points
connected with respect to the length of the web.
In this paper we analyze the structure of the Internet web.
The connections between users and providers are studied
and modeled as branches of a world spanning tree.
We propose a model based on a stochastic Cayley tree
which accounts for both qualitative and quantitative
properties and can be used as a prototype model to
explore and optimise the characteristics of the system.
As regards the river basins, for example, striking
similarities can be noted amongst the river structures
in all the world. Namely, the interplay between soil
erosion and drainage network conduces the system towards
a state where the total gravitational energy that is
dissipated is minimal. This universality accounts
for the fact that, regardless the landscape peculiarities,
the optimal solution to the drainage problem must be the
same everywhere. So our model is inspired to the theory
of river networks, which can provide an explanation of
the fractal properties of the net with respect to
the optimization of some thermodynamic potential.
This question is not only of a scientific relevance, but
it also addresses a very important technological question.
Namely, which cost function has to be minimised in order
to improve the net properties both to plan future wiring
of developing countries and to improve the quality of
the net connection for countries already connected.
For network formation, "Nature" often chooses fractal
structures. Fractal objects introduced by Benoit Mandelbrot
are characterised by the property of having similar properties
at all length scales. In this respect they show the same
complexity at different scales without a characteristic
scale or size for their structures. These properties are
defined between a lower and an upper scales which,
for the present case, are the size of a single node and
the total world network. It is exactly this scaling property
that allows animals to survive with a quantity of blood
much smaller than the solid volume occupied by their body.
The fractal structure of veins distributes the blood so
effciently that every cell is reached in a reasonably
short path with the minimum possible structure.
This paper addresses the issue of the characterization
and the design of a rational and optimal web for Internet
by using the examples present in "Nature" for similar
structures.
It is interesting to measure the density function P(n)
expressing the probability that a point in the structure
connects n other points uphill. Such a quantity, also known
as the drainage area, represents the number of points that
lie uphill a certain point in the net. As a signature of
the intrinsic fractal properties of webs this density
function P(n) for self-similar objects is a power law.
The fractal scale-free structure of the present web,
does not guarantee a short number of steps between points,
but instead shows that the probability of a very long path
is small, but finite. The natural conclusion is therefore
that it should be possible to improve the effciency of
the net by planning a certain number of big links.
These information highways superimposed on the network
structure should play the role of the plane transport
without affecting the local structure.
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