In this paper we show that the Internet web, from a user's perspective,
manifests robust scaling properties of the type $P(n)\propto n^{-\tau}$ where n
is the size of the basin connected to a given point, $P$ represents the density
of probability of finding n points downhill and $\tau=1.9 \pm 0.1$ s a
characteristic universal exponent. This scale-free structure is a result of the
spontaneous growth of the web, but is not necessarily the optimal one for
efficient transport. We introduce an appropriate figure of merit and suggest
that a planning of few big links, acting as information highways, may
noticeably increase the efficiency of the net without affecting its robustness.Comment: 6 pages,2 figures, epl style, to be published on Europhysics Letter

De Los Rios P.

Erdös P.

Faloutsos M.

G Caldarelli

Harris T. E.

Huberman B. A.

L Pietronero

Mandelbrot B. B.

Mcmahon T. A.

R Marchetti

Rodriguez-Iturbe I.

English

arXiv

In this paper we show that the Internet web, from a user’s perspective, manifests robust scaling properties of the type P(n)∝n−r where n is the size of the basin connected to a given point, P represents the density of probability of finding a basin of size n connected and τ = 1.9±0.1 is a characteristic universal exponent. The connection between users and providers are studied and modeled as branches of a world spanning tree. This scale-free structure is the result of the spontaneous growth of the web, but is not necessarily the optimal one for efficient transport. We introduce an appropriate figure of merit and suggest that a planning of few big links, acting as information highways, may noticeably increase the efficiency of the net without affecting its robustness

Marchetti, Riccardo

Caldarelli, Guido

Pietronero, Luciano

IMT Institutional Repository

The fractal properties of internet

R. Marchetti

Crossref

ar
X
iv
:c
on
d-
m
at
/0
00
91
78
v1
  [
co
nd
-m
at
.s
ta
t-
m
ec
h]
  1
2 
Se
p 
20
00
Europhysics Letters PREPRINT
The Fractal Properties of Internet
G. Caldarelli1, R. Marchetti1 and L. Pietronero1
1 INFM Unità ROMA1 and Dip. di Fisica - Università di Roma ”La Sapienza”, P.le
A. Moro 2, 00185 Roma, Italy
PACS. 05.10.-a – Computational methods in statistical physics.
PACS. 92.40.Fb – Rivers, runoff, and streamflow.
Abstract. – In this paper we show that the Internet web, from a user’s perspective, manifests
robust scaling properties of the type P (n) ∝ n−τ where n is the size of the basin connected
to a given point, P represents the density of probability of finding n points downhill and
τ = 1.9 ± 0.1 s a characteristic universal exponent. This scale-free structure is a result of the
spontaneous growth of the web, but is not necessarily the optimal one for efficient transport.
We introduce an appropriate figure of merit and suggest that a planning of few big links, acting
as information highways, may noticeably increase the efficiency of the net without affecting its
robustness.
Networks are present in many aspects of everyday life, from the watershed where the rivers
water is collected, to the veins and lymphatic channels that distribute blood and nutrition
in animals and plants [1, 2], to the telephone or electricity or internet webs that transport in
our houses the services we need. In all these cases, the network properties should be such
to optimise some cost function, as for example 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. These results have scientific and technological implications that are briefly
described. In addition 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. The model is inspired to the theory of
river networks [3], 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 [4] 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
c© EDP Sciences
2 EUROPHYSICS LETTERS
by their body. The fractal structure of veins distributes the blood so efficiently 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. The statistical study
of the Internet has already started to attract the interest of the scientific community [5–8].
Differently from these studies we focus here on the physical layer of the net, rather than on
the structure of symbolic links in the web. A striking evidence of the network structure of this
physical layer has been provided by the wonderful maps realised in the “Internet Mapping
Project” by William R. Cheswick, Hal Burch and Steve Branigan at the Bell Laboratories (1).
One of these maps is reproduced in fig. 1, showing the great complexity of this sample of
paths on the physical network of Internet cables. These maps are drawn from a set of data
obtained through a computer instruction that allows to trace the route from one terminal to
any allowed address in the Internet domain. This command records all the nodes through
which the target is reached from the starting point. These paths actually change over time
for the following reasons: firstly the routes reconfigure since the path is variable according to
the traffic at the moment or more generally according to the availability of the connection.
Secondly the whole structure is physically evolving due to new connections that take place.
Furthermore, since the collection of data takes some time, it is possible that sometimes some
loops are formed in the net, since in principle the track through the same point can be different
at different times. We checked that this is an unlikely event and when this happens, most of the
times the different branches of a loop share similar statistical properties. In this perspective,
these sets of data are then suitable to be studied with the same framework introduced for river
networks. In particular one can consider the terminal located in the Bell Laboratories as the
outlet of a river basin, and the path connecting this point to all the possible net addresses can
be considered to form the structure of this basin. It is then interesting to measure this basin
by using 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 for self-similar objects is a power law, that is
P (n) ∝ n−τ . The value of this exponent τ allows one to address and to distinguish between
different physical features. 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 [10, 11]. This universality accounts for the fact that, regardless
the landscape peculiarities, the optimal solution to the drainage problem must be the same
everywhere1. These optimal structures are characterised by an exponent τ = 1.45, different
from that corresponding to a set of random spanning trees (for which τ = 11/8 = 1.375 [12])
and from the extremal ones (for which τ = 3/2 = 1.5 [13]). For the internet data we perform
the same analysis of the statistical measure as shown in fig. 2(a), which characterises the
network’s properties as seen from the perspective of a user. The results show a clear power
law, extending over more than two decades, whose exponent τ is equal to 1.9±0.1, for different
measurements realised from the 18-01-1999 to the 18-04-2000. This value of the exponent
represents a first important difference with the theory of river networks where a maximum
value of τ = 1.5 is supposed [14]. It should be possible to understand such a difference by the
different metric nature of the space considered for points in watershed and sites in space of
the addresses. Since new nodes appear every day and connect to existing providers, the whole
(1)A description of the project, together with maps done by Cheswick B. and Burch H., is available at
http://www.cs.bell-labs.com/who/ches/map/index.html
G. Caldarelli, R. Marchetti and L. Pietronero: The Fractal Properties of Internet3
dimension of the basin is changing with time in a hierarchical rather than spanning way. This
means that space can become oversatured with respect to the river network case. From this
property and from the analysis that has been carried out on the hierarchical structure of the
web pages [5,7] it appears natural to describe this hardware structure in terms of a hierarchical
pattern of the kind of the Cayley tree with some randomness in the process of elementary
bifurcation. This hierarchy of levels is believed to play an important role for the network
properties and it leads to specific features which cannot be reproduced by other models.
For example Barabasi and Albert [6] have shown that models with random connections but
without a hierarchical structure are not fractal in the sense that the probability of finding
highly connected site is decreasing exponentially with the number of connections, thereby
determining a characteristic scale for the network. We test our hypothesis on the hierarchical
nature of connections, by defining a tree-like structure departing from the outlet of the basin
that can be described as a Random Cayley Tree. A Cayley tree branches at each generation
in k different sites. It is easy to check that deterministic Cayley trees have a statistics such
that P (n) ∝ n−1. Random behavior can be introduced by assigning a probability q(s) that a
site will have s sons. Provided that the average number of descendants
〈s〉 =
smax∑
s=1
sq(s) (1)
is greater or equal to one there is a finite probability that the tree will last forever. The
result is dependent on the mean number of sons generated in the evolution, starting from
a value of τ = 3/2 [15] known to be exact when 〈s〉 = 1. One can measure τ = 1.6 ± 0.1
for a q(0) = 0.3, q(1) = q(2) = 0.35 but if one increases the value of 〈s〉 until for example
〈s〉 = 2.75, q(0) = 0, q(1) = .... = q(10) = 0.05 one can measure the value of τ = 1.85 ± 0.1
in good agreement with the real data (fig. 2(b)). Since it is possible to show that in case
of increasing population for stochastic Cayley trees (i.e. 〈s〉 > 1) the expected exponent is
τ = 2 [15], we justify the above variation as finite size effects in our computer simulations.
As a matter of fact, due to these finite size effects the convergence to this exponent is usually
rather slow and this transient behavior may appear as an effective exponent τ lying between
the two exact values 3/2 and 2. A striking clue of the hierarchical structure of the network
can also be witnessed by the value of the exponent characterizing the density function P (n) of
sites uphill a node in the network. Indeed in the case of river networks for a lattice spanning
network with no organization one has to expect τ = 1.375, whilst for a stochastic Cayley tree
(if the system does not die out) the exponent has to be significatively larger. In addition, from
the perspective of our model, we predict that, as the number of connections continues to grow,
one should observe an increase of the exponent τ towards the asymptotic value 2. At this
point we are in condition to evaluate in which way it could be possible to optimise the network
in order (for example) to reduce the mean number of jumps any client has to realise in order
to be connected with all the other points in the net. A possible solution to this requirement
could be represented by a particular type of random graph model known as the ”small-world”
model [16] introduced in order to describe all the social situations where one can get from a
member of a network to any other member via a small number of intermediate acquaintances.
A small world system in the simplest version consists of a 1-dimensional system of L sites with
periodic boundary conditions (a ring). A site is connected to k different neighbours chosen
from the nearest one. Furthermore, m shortcuts are present from different randomly chosen
sites. If for example L = 50, k = 3, and m = 9 one can evaluate numerically that the average
vertex-vertex distance is of the order of 4 ÷ 6 steps. One can see the small-world model in
terms of transportation. The small local bonds represents motion by trains or car, while
4 EUROPHYSICS LETTERS
the big jumps realised by shortcuts correspond to airplane flights. In this respect there are
characteristic lengthscales and the optimal system is not scale invariant. A figure of merit can
be the average number of steps to connect two random points which is typically rather small
in the small-world model. To optimise in such a way the Internet network would be highly
desiderable. Unfortunately, due to the fractal properties we measured on the web, the mean
vertex-vertex distance is much larger (around 15) and furthermore show no characteristic
cutoff. Namely, the optimization in the distance between vertex is linked to the form of the
distribution P (n) that for such system is now peaked around a mean value. In this case then
the P (n) shows no scale-free behaviour (On the other hand, the small world is known to be
characterised by a correlation distance ξ = 1/(φkd)1/d, where φ = m/L is the probability to
have m shortcuts) . Conversely 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 efficiency of the net by planning a certain number of big links which
should play the role of the shortcuts in the small-world model. These information highways
superimposed on the network structure should play the role of the plane transport without
affecting the local structure. We believe that our analysis, providing a quantitative measure
of the physical Internet network could be the starting point of this study. We conclude that,
from the perspective of a single user (the node from which the route has been traced to
all the other nodes), one observes essentially a stochastic Cayley tree whose properties are
robust with respect of the different possible routes used in different tests. This denotes an
underlying hierarchical structure that links providers and users (this is topologically different
from a random spanning tree, because now the space is filled in a ”thicker” way). We are
presently studying this problem in order to check how the present network could be improved
by creating suitable shortcuts that can reduce in a significant way the cost of connections
between different users.
∗ ∗ ∗
We acknowledge fruitful discussions with E. Bonabeau, P. De Los Rios, A. Flammini,
G.Parisi and F. Rossi
Correspondence and requests for materials should be addressed to G.C.
(e-mail: gcalda@pil.phys.uniroma1.it)
REFERENCES
[1] McMahon T.A. and Bonner J.T., On Size and Life (Freeman, New York) 1983
[2] Banavar J.R., Maritan A. and Rinaldo A., Nature, 399 (1999) 130.
[3] Rodriguez-Iturbe I. and Rinaldo A., Fractal River Basins, Chance and Self-Organization
(Cambridge University Press, Cambridge) 1997.
[4] Mandelbrot B.B., The Fractal Geometry of Nature (Freeman, New York) 1983.
[5] Albert R., Jeong H. and Barabasi A.L., Nature, 401 (1999) 130.
[6] Barabasi A.L. and Albert R., Science, 286 (1999) 509.
[7] Huberman B.A., and Adamic L.A., Nature, 401 (1999) 131.
[8] Faloutsos, M., Faloutsos, P. and Faloutsos, C., ACM SIGCOMM ’99. Comput. Commun.
Rev., 29 (1999) 251
[9] Rodriguez-Iturbe I., Rinaldo A., Rigon R., Bras R.L. and Ijjasz-Vasquez E., Geophys.
Res. Lett., 19 (1992) 889.
[10] Caldarelli G., Giacometti A., Maritan A., Rinaldo A. and Rodriguez-Iturbe I., Phys.
Rev. E, 55 (1997) R4865.
G. Caldarelli, R. Marchetti and L. Pietronero: The Fractal Properties of Internet5
Fig. 1 – Picture of the net realised in Bell Laboratories, Courtesy of Lucent Technologies. This map
represents the net as seen by a single user.
[11] Coniglio A., Phys. Rev. Lett., 62 (1989) 3054.
[12] Maritan A., Colaiori F., Flammini A., Cieplak M. and Banavar J., Science, 272 (1996)
984.
[13] Erdös P. and Renyi R., Publ. Math. Inst. Hung. Acad. Sci, 5 (1960) 17.
[14] Harris T.E., The Theory of Branching Processes (Dover Publications New York) 1989.
[15] De Los Rios P., ”Criticality” of Supercritical Random Trees, to be published (1999) .
[16] Watts D.J. and Strogatz S.H., Nature, 393 (1998) 440.
6 EUROPHYSICS LETTERS
10
0
10
1
10
2
10
310
0
10
1
10
2
10
3
10
4
10
5
10
6 q(0)=0.3, q(1)=q(2)=0.35, τ=1.6, <s>=1.05
q(0)=0.5, q(i)=0.1 (i=1,..,4), q(5)=q(6)=0.05, τ=1.8, <s>=1.55
q(0)=0.5, q(i)=0.05 (i=1,....,10), τ =1.85, <s>=2.75
10
0
10
1
10
210
−5
10
−4
10
−3
10
−2
10
−1
10
0
18 January    1999
6 February    1999
16 June       1999
14 September 1999
18 April        2000
(a)
(b)
Fig. 2 – (a) Plot of the density function P (n) of the whole network of 1249747 nodes. This function
gives the probability density of having n descendants (basin of attraction). (b) Plot of the Density
Function P (n) for different Cayley trees, characterised by different distributions of sons per site.
These plots correspond to averages over 500 realizations with a maximum number of generations
equal to 25.


Caldarelli, G

Archivio della ricerca della Scuola IMT Alti Studi Lucca

The fractal properties of Internet

In this paper we show that the Internet web, from a user's perspective, 
manifests robust scaling properties of the type $P(n)\propto n^{-\tau}$, 
where n is the size of the basin connected to a given point, P represents 
the density of probability of finding n points downhill and $\tau=1.9 \pm 0.1$ s a 
characteristic universal exponent. This scale-free structure is a result of 
the spontaneous growth of the web, but is not necessarily the optimal one for 
efficient transport. We introduce an appropriate figure of merit and suggest 
that a planning of few big links, acting as information highways, may 
noticeably increase the efficiency of the net without affecting its robustness

G. Caldarelli

L. Pietronero

EDP Sciences OAI-PMH repository (1.2.0)

The Fractal Properties of Internet

The Fractal Properties of Internet

Abstract

Similar works

Full text

Available Versions

IMT Institutional Repository

Crossref

Archivio della ricerca della Scuola IMT Alti Studi Lucca

Crossref

Crossref

Crossref

Crossref

EDP Sciences OAI-PMH repository (1.2.0)