Submetido para TEMA
Topology and Shortest Path Length Evolution of
The Internet Autonomous Systems
Interconnectivity
N. ALVES Jr., M. P. de ALBUQUERQUE e M. P. de ALBUQUERQUE1, Centro
Brasileiro de Pesquisas Fı́sicas - CBPF/MCT,Rua Dr. Xavier Sigaud, 150, Urca,
Rio de Janeiro, RJ, CEP 22290-180 - Brazil,
J. T. de ASSIS2, Universidade do Estado do Rio de Janeiro, Instituto Politécnico,
UERJ/IPRJ. Rua Alberto Rangel, s/n, Vila Nova, Nova Friburgo, RJ, CEP 28610974 Brazil
Abstract. Connection networks are observed in many areas of human knowledge.
The characterization and topological studies of these networks may be performed
through distribution of connectivity degrees, rank properties, shortest path length
between nodes, adjacency matrix etc. This paper characterizes the Internet connections evolution over the last 10 years at the Autonomous Systems (AS) level
analyzing the complete BGP data set from Oregon route server. We present the
behavior of the power law function and the shortest path length of the Internet for
each year. Simulations of the connections network were carried out by a proposed
model developed from continuous growth premises, possibilities of new and rearranging connections. This model was based on the concept of potential preferable
connection showing a stable exponential factor that reproduces the true shortest
path parameter over the years.
Keywords. Autonomous Systems, Shortest Path, Adjacency Matrix, Complex
Networks, Scale Free Networks.
1.
Introduction
A complex dynamic system is constituted by many elements that interact among
themselves and with the environment. Recently, connection networks have been
described, studied, characterized and represented by parameters using concepts of
Complex Systems domain. These networks may be natural such as the neuron
system of the human brain, the chemic molecular connections in protein conglomerates, the system of virus propagation in an epidemic situation etc. Or artificial,
such as the lines of energy distribution in a country, the tangled social contacts
1 (naj,mpa,marcelo)@cbpf.br
2 [email protected]
2
Alves, Albuquerque, Albuquerque e Assis
that we make through our lives or the Internet, which is one of the most interesting
examples and certainly the most modern one [4][6][13].
The Internet may be seen under several levels of reach and complexity considering different basic units. The most intuitive view and of easy understanding
to the general public, is to suppose that these units are computers connected by
wires, optical fibers or even using wireless technology through the whole planet. Another vision, a little bit wider, defines basic unit as a cluster of computers, servers,
printers, switchers and routers which form a Local Area Network − LAN. Each
institution, company or residential building, would have its LAN which would be
connected to others LANs and so on.
A third vision would be to consider the Internet basic element as an Autonomous
System − AS. A straight definition would be: an AS is defined as a cluster of
LANs or routers submitted to the same policy of usage, connectivity and technically
administrated by the same network management group.
Each ASs registered at a regional control organism receives a number ASN between 1 and 65536 that will be a digital print to the hole Internet and will be used to
configure the routers. The IP prefixes of LANs’ addresses of an AS are represented
by one ASN. The number of autonomous systems increases in a daily basis, in 1998
there were around 2550 and nowadays about 21000 ASNs are used. AS numbers
are not contiguous because some are reserved, others dedicated and some are no
longer used due end of operation.
Internet (e-mail, web links and AS connectivity) has been widely used as a model
for the comprehension of networks properties. Internet data can be obtained at the
router level between Autonomous System relationships, analyzed and studied by
Complex Networks theory. Nevertheless, the study of network topologies remains an
open problem. This study can be useful for new protocols design, security policies
definition for institutional data networks, network connectivity policies, and new
technologies that uses Internet as framework. The complex network considered in
this work is composed by Autonomous Systems (vertices) and the established traffic
connection (edges) between them obtained from the BGP routing table. Many
interesting property of this networks is analyzed, e.g. degree distribution (the rank
and outdegree exponents) from 1998 to 2007 and the shortest path length (L),
obtained by a proposed computational method (Friburgo algorithm) among each
pair of ASs represented in the adjacency matrix.
This work is divided in three main parts. In section 2 the real data obtained
from Oregon University database [22] in the period from 1998 to 2007 are treated
and classified in rank of connectivity and distribution of connection probabilities
according to the connectivity coefficient. Section 3 presents some fundamental
concepts of Shortest Path Length parameter concerning its average calculation by
Friburgo algorithm. Session 4 presents the analysis of the real data and a growth
model describing the property. Section 5 is dedicated to presenting some concluding
remarks.
Topology and Shortest Path Evolution of Internet
2.
3
Topology of the Internet Complex Network
In this work Internet is studied through the extent level of the Autonomous Systems
- AS. The communication between these systems depends on the BGP protocol,
which is a protocol that announces network prefixes among ASs. The relationships
occur after the establishment of a traffic exchange agreement with or without financial commitment (peering) between two or more ASs. In any peering model
adopted, it is necessary that each AS determines which networks will be announced
to a neighboring ASs. Once these announcements are defined, this information is
customized in the border routers of each AS that communicate through the BGP
protocol with its neighbor border routers. When the border routers of all ASs exchange information, tables with networks prefixes and its respective routes in ASs
are assembled, establishing an ASs path, from the AS origin router to the AS destiny one. If the table has all network prefixes, i.e. all LANs can be reached by an
ASs, it is classified as Full Routing Table − FRT. Nowadays the full routing of the
global Internet table has more than 190000 networks prefixes and more than 21000
ASs.
We used the complete table (FRT) obtained from the University of Oregon
Route Views Project which has a great historic patrimony of BGP tables. It was
considered here the FRTs of January of each year from 1998 to 2007. Each of these
tables are treated and converted to the equivalent Adjacency Matrix − AM, which
contains a map of the established ASs connectivity paths for each year.
We calculate for each year the degree distribution, of ASs. In figure 1 we present
the probability distribution P(k) for 1998 and 2007. ASs networks is often studied
and represented as a scale-free network, where the degree distribution is a power
law
P (k) ∝ k γ
(2.1)
The exponent γ of these networks observed at the AS level from 1998 to 2007
is −2.00 ± 0.06 (slope line of figure 1). The complete data distribution concerning others years, were removed for a simpler visual, however we can say that the
superposition at the initial part of the graph, for the small connectivity values, is
practically the same as shown in the figure. We can also observe in this figure
a change in the basis of both graphs, as the total number of ASs increases the
probability of ASs hubs reduces.
Another interpretation of Internet data is the rank of a node which consists in
its position on a list of all nodes sorted in decreasing degree. In figure 2 we present
the annual evolution from 1998 to 2007 of this data set. It can be observed, in
despite of the increase in the number of neighbors in the vertical axis, expected
by the increase of ASs, that the linear characteristic in a log − log graph remains
the same. We can observe also a non linear behavior in the first positions of the
rank graphs, interpreted here by the low statistics of high-degrees ASs, high values.
These first ASs positions are called ASs Hubs or high-degrees ASs. Those consist
on highly connected ASs, which prefer to attach to other high-degrees one. Zhou
and Mondragón [21] classified a cluster of these high degrees ASs as a rich club
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Alves, Albuquerque, Albuquerque e Assis
Figure 1: Probabilities of number of neighbors for 1998 and 2007. The AS level
Internet topology was obtained from a complete BGP data set. The inset list shows
the adjustment of the power law function exponent γ for each year.
that are responsible for a “super” traffic of the network. Those with low values are
called ASs leafs, edge ASs, or low-degrees ASs. On the right side of figure 2, there
is an expected degeneration of the rank position for the low degrees ASs, once it is
bigger the number of ASs with few neighbors. These degenerations were taken back
for considering only one point, the first position of each degenerated value, keeping
the absolute rank position for subsequent points.
Finally, in figure 2 we present all values for the linear adjustments in a log − log
scale for the studied interval (the average value of the angular coefficient a was
−0.93 ± 0.02). We can observe a power law behavior even with a continuous growth
of ASs.
3.
Shortest Path Length
This section presents a proposed algorithm to compute the Shortest Path Length
−L, introducing the features of the method and the dynamic used.
Topology and Shortest Path Evolution of Internet
5
Figure 2: Log − Log plot of the rank evolution of connectivity degree ki . The
legend shows the annual evolution of the angular coefficients a of the log − log
linear adjustments of the curves.
3.1.
Shortest Path Calculation
The minimum or shortest path length problem consists in finding the shortest way
considering time, price, distance, etc. (or a convenience parameter), between two
points inside a network. This kind of problem applies in telecommunication, transportation, electronics, biomedical fields, etc. More formally, in Graphs Theory, the
shortest path problem is the problem of finding the way between two vertices or
nodes, that the sum of the connections weight is minimized. The average L used in
this work is the mean of all shortest paths between each AS with all others.
The L parameter calculation is a classic problem studied by several authors that
developed its own algorithms. Each one has its own importance and performance,
being appropriate when some features are present, e.g. Dijkstra [10] for non-negative
connection weights, and Bellman [5] and Ford [14] independently for negative and
positive weights, Dantzig et. al. [9] for sparse graphs, etc.
In the case of the complex network of the Internet ASs connections, all weights
are defined as “1” and, for this reason, one start the method with the Dijkstra
algorithm, where the temporal complexity is O(N 2 ) with N meaning the number
of nodes in the network.
The characteristics of the program developed by Escardó [12] imposed restric-
6
Alves, Albuquerque, Albuquerque e Assis
tions for its utilization in this work. In a 64 bits High Performance Computing
System - HPCS, the execution time for the smaller connection networks of ASs for
the year 1998 (concerning 2,500 ASs) were grater than 8 days and for more recent
years, we compute for more than 30 days without a final result.
Figure 3: Diagram of the shortest path length calculating algorithm
Moreover the computing restrictions to calculate bigger networks (around 21,000
ASs in 2007) required the development of a new algorithm adapted for our needs
and for our computing resources. The new algorithm was named Friburgo Method
because of the city where it was conceived. To understand it, lets consider a practical
example:
Consider a connection network composed of 6 connected nodes as shown in figure
3. An edge is represented by a bidirectional connection: (A,B), (A,D), (A,E), (B,E),
Topology and Shortest Path Evolution of Internet
7
(B,F) and (C,F). This figure also shows the adjacency matrix - AM of this example
network. The method requires a repetitive procedure for each node or matrix line.
In each line of the AM an X character represents a node and the symbols ‘1’ and
‘0’ represent the existence, or not, of a connection between them respectively. The
basic concept of the proposed method is to search missing connections in the AM
lines on those points where a direct connection between nodes already exists. For
example, starting from node A the algorithm search for connections with C and
F only in B, D and E node lines. If we observe node B line, there exist three
connections (B,A), (B,E) and (B,F). The first two are not considered, because node
A is already connected to itself and to node E. The connection (B,F) indicates
an indirect connection (A,B,F) considered as a level 2 one. Assuming this same
procedure for nodes D and E one can easy observe any new level 2 connection.
We have still a missing indirect connection (A,...,C) represented by the 0 in the
second line of the scheme of node A. The process is repeated up to node F line
observing two connections (two symbols ‘1’) in it. The (F,B) edge doesn’t add any
new information to the algorithm but (F,C) edge is part of (A,B,F,C) path, which is
a level 3 one because it came from the level 2 path (A,B,F) which, for its turn, came
from the initial edge (A,B), level 1. Finally, we do not have any more connections
and the algorithm passes to next node, B. In this case, the searching for indirect
connections of node B to nodes C and D can be seen in lines of nodes F and A,
respectively. Both are connections of level 2 to the algorithm.
Considering node C, one can see that there is only one direct connection (‘1’),
to node F. The node F line is a level 2 connection with B, due to (F, B) edge. The
node B line shows a level 3 connection due to (B,A) and (B, E) edges. Finally,
from node A line, one observe a level 4 connection due to (A,D) edge. Note that
there are many connections that actually do not contribute, either because they are
higher level ones or because they have already been considered before.
The procedure repeats to each node. The Lnode is computed by the
Paverage of
last line for each node schema. The parameter L is calculated by L =
Lnode /N ,
considering the average for all N Lnode , where N means the total number of ASs.
In fact we calculate the average L for the Internet.
The efficiency of the proposed method is better observed for a network with
great number of nodes. This method needs to search all other nodes looking for
connections to each of them, but one should note that there are variations in the
number of iterations, figure 3. In fact the search is unique and only for the nodes
with a lower neighborhood order.
Table 3.1. shows the time performance to calculate the L parameter of the
connection network among Internet ASs ranging from 1998 to 2007 by Friburgo
algorithm. The processing of year 1998 data set was achieved in 35 seconds which is
very superior when compared with 8.4 days obtained by Escardó’s implementation.
Part of this superior performance was due to the search for missing or indirect
connections only for those of n-order type (n ≥ 2).
The implementation of the Friburgo Method was successfully tested for various
networks maps, such as the proposed methodology of Watts and Strogatz [19].
Although it has not been developed as search graph algorithm, the Friburgo
8
Alves, Albuquerque, Albuquerque e Assis
Table 1: Friburgo method computing time as function of the analyzed number of
nodes.
Year
1998
1999
2000
2001
2002
# of nodes
2541
4452
6397
7897 12376
Time(s)
35
152
420
793
2979
Year
2003
2004
2005
2006
2007
# of nodes 14370 16761 18772 20951 20989
Time(s)
8601
7552 11570 15525 14076
Method can be easily confused with one of them. Actually, some similarity with
those methods exists mainly with those that use a uniform search and therefore do
not use a specific heuristic, as for example the Breadth-First Search − BFS method
[7]. There are also similarities with those using a proper heuristic, such as Best-First
Search [16][17] which is an optimization of the BFS and the Depth-First Search −
DFS [8]. The Friburgo method determines the distance of a node to all others ones
composing the net, the average shortest path length. This is the central difference
between the Friburgo method and those that are only dedicated to find the smaller
distance between two nodes. In other words, Friburgo Method is optimized and
specific to compute the average path length of a node to the entire network.
4.
Experimental Results and Shortest Path Modeling
This section is divided in into two parts. The first one shows the experimental results
of the calculation of the average Shortest Path parameter −L of ASs networks from
1998 to 2007. In the second part we present a growth model of a connection network
reproducing the observed ASs topology. In both cases, Friburgo method was used
to determine L.
4.1.
Experimental Results
Just like the topology studies, presented in section 2, we used the FRTs data from
Oregon project for the same period. We treat this experimental data obtained from
BGP tables, which, consist in selecting the relevant AS neighborhood from the
table (pre-treatment) assembling the adjacency matrix to a posterior calculation of
average shortest path length. The results present an almost stable behavior of L
value, as seen in figure 4. This characteristic is an indicator of the appearance of
new ASs nodes concentrating its new connections on those considered high-degree
ones, even for an increasing node topology condition (Table 2).
Topology and Shortest Path Evolution of Internet
9
Figure 4: Shortest Path Length L behavior for the network of connections of the
Internet Autonomous Systems −ASs in the period from 1998 to 2007.
4.2.
Model Results
In 1999, Barabasi and Albert outlined the Internet through Web links [1]. In its
article they demonstrated that the Web structure was not in agreement with the
model of normal distributed connectivity. In fact, the Web structure presents some
few links with high probabilities of connections (majority one or two) and many
others with few probabilities. It did not have a typically average value of connections
which is a characteristic of random nets. This type of behavior can be better
described with power law functions as one can see in the study of phase transitions,
hierarchic structures, fractals, etc. Barabási and Albert were, probably, the firsts
to observe the Internet as a scale-free system, i.e. networks without scale or simple
free-scale networks, the probability of a node to be connected to another node is
given by eq.(2.1). They demonstrated that these networks depend on two basic
principles: continuous growth of the number of nodes and preferential attachment
connections. A preferential attachment connection is when nodes which already
have a high number of connections (hubs) have bigger probability of receiving new
connections. The probability of new connections is given by
10
Alves, Albuquerque, Albuquerque e Assis
P (ki ) =
ki
N
P
j=1
(4.1)
kj
Based on these two principles, Barabási and Albert [2] proposed a model starting
with nodes and without connections. At each step a new node is added with m new
connections to different nodes that already exist. They shown that there is an
independence in the value of m and that the free scale network obtained had the
exponential coefficient γ = −3. Barabási and Albert model is based only on the
attachment of new nodes. However, the appearance of new internal links among old
nodes has also been observed in the evolution of the Internet. One should expect
for the Internet topology a different value for the γ exponent, (γ = −2.00), and it
will be necessary to add other rearrangements mechanisms to the basic model, to
get closer to reality.
Two local mechanisms of a network are those which represent the possibility of
adding new connections among the already existing nodes and of removing already
existing connections. The end of operation of an AS is quite rare, representing few
cases in which AS ceases to exist being normally absorbed by another, case of fusion
between two or more Internet Service Providers − ISPs. In the real observed Internet network it is much more common a rearrangement which consists in removal
of a link followed by a new connection, representing a neighborhood adjustment
due to cost-benefit. The adding mechanism increases the connection density, and is
different from the rearrangement one, that keeps the system with the same global
characteristics. These mechanisms were introduced in the base models by Barabási
and Albert [3] and afterwards also by Dorogovtsev and Mendes in [11]. Simulations
with these models were carried out and did not show the expected agreement with
the experimental data, suggesting that another mechanism should be added to the
model. Dorogovtsev e Mendes (2000) [11] and Krapivsky et. al (2000) [15] for connectivities networks and in a more recent studies Zhou and Mondragón (2004) [21]
for the Internet topology and White et al. (2006) [20] for chemical autocatalytic
reactions and corporative alliance networks proposed a way of privileging or intensifying the basic mechanism of preferential connection, adopting expression (4.2)
where there is an exponent α creating a nonlinear preferential probability [18][21]:
kα
P (ki ) = P i α ,
kj
α>1
(4.2)
j
Note that if, and only if, the condition will be satisfied, the exponent will make
the modeling to be more coherent with the experimental data. In this case, the
exponent increases the preferential connection making those hubs ASs to have even
more preferences in the sort connections when we compare with equation (4.1). This
consideration is based on empirical observation of the Internet ASs connections,
where 2 (1998) and 4 (2007) ASs have 10% of the connections.
In the proposed model, the basic unit of the temporal cycle is the inclusion
of a new AS. It also considers, with p and q probability, the possibilities of new
Topology and Shortest Path Evolution of Internet
11
connections and rearrangement among the already existing ASs, respectively.
At each cycle, besides the inclusion of m new connections of a new AS, a probability is cast, 0 < prob < 1. If prob is smaller than p, the model adds a new
connection among existing ASs. If the probability is between p and p + q, a rearrangement operation takes place. Finally, if the probability is bigger than p + q,
nothing besides the inclusion of this new AS and its connections occurs.
The p and q parameters have its values adjusted in each year simulations in
order to obtain the total number of nodes and connections of the resulted network
compatible with those of the experimental data files.
Table 2: Comparison between experimental results and simulations. The new connections probability p was 33.6% on average (see text). The rearrangement probability q was 5%. The α exponent stands to the non linear preferential probability
of the simulations.
Ano
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
#ASs
2,541
4,452
6,397
7,897
12,376
14,370
16,761
18,772
20,951
20,989
Exp.
6,125
11,799
17,198
21,018
33,076
37,645
45,189
51,138
56,395
58,836
Conexões
Sim.
Var %
6,140
0.24
11,816
0.14
17,263
0.38
21,107
0.42
33,110
0.10
37,645
0.34
45,137
0.12
51,275
0.27
56,531
0.24
58,982
0.06
p(%)
21
33
35
34
34
31
35
37
35
41
Exp.
4.51
4.12
4.09
4.09
4.00
4.28
4.16
4.20
4.25
4.16
MCM - L
Sim. Var.%
4.44
1.49
4.21
2.33
3.97
2.91
4.13
1.03
3.96
1.00
4.27
0.30
4.03
3.10
4.17
0.69
4.22
0.64
4.24
1.92
α
1.20
1.21
1.21
1.21
1.21
1.19
1.20
1.20
1.19
1.19
In Table 2, we present the values for the experimental data and the simulations
results of the main parameters for the proposed model. In the first column (Years)
the temporal interval of the study is listed. In the second column (# Nodes) we show
the number of ASs, N, existing in the experimental data and that was maintained
in the simulations procedures. The third column (# Connections) was divided
into three other columns with experimental and simulated values of the number of
connections and the variation of the absolute value between them. In the fourth
column (p(%)) we show the value for the relative probability used to control the
adding mechanism of new connections among already existing ASs. The fifth column
(L) was also divided into three columns showing the obtained results with the
experimental data, simulations and the variation between them. Finally the sixth
column presents the behavior of the α exponent for the simulated conditions over
the years.
The value of the new connections probability p is an average over 20 simulations. We try to reproduce in the simulations the same number of nodes of the
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Alves, Albuquerque, Albuquerque e Assis
Table 3: Annual evolution of the gamma and angular coefficients (a) of the log − log
linear adjustments of the model simulations.
Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
γ
-2.03
-2.08
-2.06
-2.09
-2.22
-2.13
-2.21
-2.21
-2.16
-2.14
Simulated
σγ
a
0.07 -0.90
0.05 -0.91
0.06 -0.90
0.06 -0.93
0.06 -0.97
0.05 -0.92
0.05 -0.93
0.06 -0.94
0.04 -0.91
0.04 -0.91
σa
0.05
0.041
0.05
0.05
0.08
0.05
0.07
0.06
0.03
0.05
real observations. It can be observed that the average value of the new connections probabilities is 34.3%, not considering the year of 1998, appearing to be an
atypical year. In these simulations it was verified that q probability that responds
for the rearrangement mechanism, slightly attenuates the effect of the preferential
connection. We empirically adopted, based in our experience in the operational
administration of the state of Rio de Janeiro academic network (ASN 2715), the
constant value of 5%.
After defined the annual probabilities p and q we reproduce the total number
of connections observed in the experimental data to determine the α parameter.
This was made by varying α within an interval and selecting those that lead us to
a similar value of the experimental L. Due to randomness of the simulation, the
considered values are averaged over 10 independent runs. One can observe that the
average value of α remains constant, approximately 1.20, over the years.
Finally in Table 3 we present the coefficients evolution for the simulation of the
probability distribution and the rank. The networks generated using the proposed
model have the same power-law relationship for the probability distribution of ASs
and between degree and rank position. The mean value obtained for the γ exponent
of the power law of the probability distribution of ASs over the years was −2.13±0.07
and for the rank was −0.92 ± 0.02.
5.
Conclusions
In this article we study and characterize the evolution of the Internet topology,
analyzing the real data set of the last ten years and we proposed a model to describe
this evolution by a nonlinear preferential probability attachment rule. The ASs
rank observed from 1998 to 2007 shows the same behavior over the years (with an
Topology and Shortest Path Evolution of Internet
13
average exponent of −0.93). The degree exponent of the power law observed for all
probability distributions in the period was −2.00.
The Friburgo algorithm developed to calculate the average shortest path length
is conceptually simple and based on the adjacency matrix of the connection network.
The method was turned out to be extremely fast enabling the search of the L
parameter even for big connection networks, e.g. those with more than 20000 nodes.
The behavior of the L parameter obtained among ASs from 1998 to 2007 is
relatively stable and with an average value of 4.19 ± 0.13, despite a growth of a
factor of 10 of the connection network in this period (2, 541 ASs in 1998 to 20, 989
in 2007), this growth follows the same pattern of preferential connectivity with a
exponent of 1.20.
The behavior of the Barabási-Albert model appeared to be inadequate to simulate, to the Internet network, the number of connections and the correct shortest
path length between ASs. In our proposed model, the probabilities of new connections and rearrangement among already existing ASs showed different importance,
mainly because it was mandatory to keep the total number of ASs for each year.
The exponential factor demonstrated to be fundamental and also practically stable
(equal to 1.20) to reproduce the values of the L value.
The proposed model can also be characterized by analyzing the evolution of
the clustering coefficient of ASs as function of new connections and rearrangement
mechanisms. The Friburgo method can be generalized to work with negative and
different weights and compared to other methods. Finally we consider also completing the studied data set, while actual AS-level map is quite complete, the BGP
measurements can only see a portion of the network. This work can be improved
if we augment the collected information from additional public route servers and
Looking Glass sites.
6.
Acknowlegments
This work was developed in cooperation with the state of Rio de Janeiro academic
network (Rede-Rio) a special Project of the State of Rio de Janeiro Research Foundation (FAPERJ), the support from the National Council for Scientific and Technological Development (CNPq) of the Brazilian Ministry of Science and Technology
and also the University of Oregon which made the most part of the experimental
data available to us.
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