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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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