Dynamical and correlation properties of the Internet

4 Pages·0.18 MB·English

by  Romualdo Pastor-Satorras

Dynamical and correlation properties of the Internet
Romualdo Pastor-Satorras,1 Alexei Va´zquez,2 and Alessandro Vespignani3
1Departament deF´ısicaiEnginyeriaNuclear, Universitat Polit`ecnicadeCatalunya, CampusNord B4,08034 Barcelona, Spain
2International School for Advanced Studies SISSA/ISAS, via Beirut 4, 34014 Trieste, Italy
3The Abdus Salam International Centre for Theoretical Physics (ICTP), P.O. Box 586, 34100 Trieste, Italy
2
(Dated: February 1, 2008)
0
0 The description of the Internet topology is an important open problem, recently tackled with
2 the introduction of scale-free networks. In this paper we focus on the topological and dynamical
n properties of real Internet maps in a three years time interval. We study higher order correlation
a functionsaswellasthedynamicsofseveralquantities. WefindthattheInternetischaracterizedby
J non-trivialcorrelationsamongnodesanddifferentdynamicalregimes. Wepointouttheimportance
9 ofnodehierarchyandagingintheInternetstructureandgrowth. Ourresultsprovidehintstowards
therealistic modeling of theInternetevolution.

]
h PACSnumbers: PACSnumbers: 89.75.-k,87.23.Ge,05.70.Ln
c
e
m
Complex networks play an important role in the un- ternet. In most cases, the map is constructed by using
- derstanding of many natural systems [1, 2]. A network a hop-limited probe (such as the UNIX trace-route tool)
t
a is a set of nodes and links, representing individuals and from a single location in the network. In this case the
t
s the interactions among them, respectively. Despite this result is a “directed” map as seen from a specific point
t. simple definition, growing networks can exhibits an high on the Internet [5]. This approach does not correspond
a
degree of complexity, due to the inherent wiring entan- toacompletemapoftheInternetbecausecross-linksand
m
glement occurring during their growth. The Internet is othertechnicalproblems(suchasmultipleIPaliases)are
- a capital example of growing network with technologi- not considered. Heuristic methods to take into account
d
n cal and economical relevance; however, the recollection theseproblemshavebeenproposed[9]. However,itisnot
o of router-level maps of the Internet has received the at- cleartheirreliabilityandthecorrespondingcompleteness
c tention of the research community only very recently of maps constructed in this way. A different representa-
[
[3,4,5]. Thestatisticalanalysisperformedsofarhavere- tion of Internet is obtained by mapping the autonomous

2 vealedthattheInternetexhibitsseveralnon-trivialtopo- systems (AS) topology. Each AS number approximately
v logical properties. (wiring redundancy, clustering, etc.). maps to an Internet Service Provider (ISP) and their
1
Among them, the presence of a power-law connectivity links areinter-ISPconnections. Inthis caseit is possible
6
distribution [6, 7] makes the Internet an example of the to collect data from several probing stations to obtain
1
5 recently identified class of scale-free networks [8]. completeinterconnectivitymaps[3,4]. Inparticular,the
0 Inthis Letter,wefocusonthe dynamicalpropertiesof NLANR project is collecting data since Nov. 1997, and
1 the Internet. We shall consider the evolution of real In- it provides topological as well as dynamical information
0
/ ternetmapsfrom1997to2000,collectedbytheNational on a consistent subset of the Internet. The first Nov.
at Laboratoryfor AppliedNetworkResearch(NLANR) [3]. 1997 map contains 3180 AS, and it has grown in time
m In particular, we will inspect the correlation properties untilthe Dec. 1999measurement,consistingof6374AS.
- of nodes’ connectivity, as well as the time behavior of In the following we will consider the graph whose nodes
d severalquantities relatedto the growthdynamics ofnew represent AS and whose links represent the connections
n nodes. Ouranalysisshowsadynamicalbehaviorwithdif- between AS.
o
c ferent growth regimes depending on the node’s age and In dealingwith the Internet as anevolvingnetwork,it
v: connectivity. The analysispoints out twodistinct wiring is important to discern whether or not it has reached
processes; the first one concerns newly added nodes,
i a stationary state whose average properties are time-
X while the second is related to already existing nodes in- independent. As a first step, we have analyzed the be-
r creasing their interconnections. A feature introduced in
havior in time of several average quantities such as the
a
thepresentworkreferstotheInternethierarchicalstruc- connectivity hki, the clustering coefficient hCi and the
ture,reflectedinanon-trivialscale-freeconnectivitycor- average minimum path distance hdi of the network [10].
relation function. Finally, we discuss recent models for
The first two quantities (see Table I) show a very slow
the generation of scale-free networks in the light of the
tendency to increase in time, while the average mini-
present analysis of real Internet maps. The results pre-
mum path distance is slowly decreasing with time. A
sentedinthisLettercouldhelpdevelopingmoreaccurate
more clear-cut characterization of the topological prop-
models of the Internet. erties of the network is given by the connectivity distri-
Several Internet mapping projects are currently de- bution, P(k). In Fig. 1 we show the probability P(k)
voted to obtain high-quality router-levelmaps of the In- that a given node has k links to other nodes. We report
2
the distributionfor snapshots ofthe Internetatdifferent 100
times. In all cases, we found a clear power law behavior
P(k)∼k−γ with γ =2.2±0.1. The distribution cut-off 1997
1998
is fixed by the maximum connectivity of the system and 10−1
1999
is related to the overallsize of the Internet map. On the
other hand, the power law exponent γ seems to be in- )
dependent of time and in good agreement with previous (k 10−2
m
measurements[6]. This evidenceseemsto pointoutthat
u
P
the Internet’s topologicalproperties havealreadysettled
to a rather well-defined stationary state. 10−3
Initially, the modeling of Internet considered algo-
rithms based on its static topological properties [11].
However, since the Internet is the natural outcome of 10−4
100 101 102 103
a complex growth process, the understanding of the dy-
namical processes leading to its present structure must
k
be consideredas a fundamentalgoal. Fromthis perspec-
FIG. 1: Cumulated connectivity distribution for the 1997,
tive, the Baraba´si-Albert (BA) model, Ref. [8, 12], can
1998, and 1999 snapshots of the Internet. The power law
beconsideredasamajorstepforwardintheunderstand-
behaviorischaracterizedbyaslope−1.2,whichyieldsacon-
ingofevolvingnetworks. UnderlyingtheBAmodelisthe nectivity exponentγ =2.2.
preferential attachment rule [8]; i.e., new nodes will link
with higher probability to nodes with an already large
103
connectivity. Thisfeatureisquantitativelyaccountedfor
Internet 1998
by postulating that the probability of a new link to at-
Generalized BA γ=2.2
tach to an old node with connectivity k , Π(k ), is lin-
i i Fitness model γ=2.25
early proportional to k , Π(k ) ∼ k . This is an intu-
i i i
itive feature of the Internet growth where large provider
hubsaremorelikelytoestablishconnectionsthansmaller > 102
providers. TheBAmodelhasbeensuccessivelymodified nn
k
with the introduction of several ingredients in order to <
account for connectivity distribution with 2 < γ < 3
[13,14],localgeographicalfactors[15],wiringamongex-
isting nodes [16], and age effects [17]. While all these 101
models reproduce the scale-free behavior of the connec-
tivity distribution, it is interesting to inspect deeper the
100 101 102 103
Internet’s topology to eventually find a few discriminat-
ing features of the dynamical processes at the basis of
k
the Internet growth.
A first step in a more detailed characterization of the FIG.2: Averageconnectivityhknniofthenearestneighborsof
anodedependingonitsconnectivityk forthe1998 snapshot
Internetconcernstheexplorationoftheconnectivitycor-
of Internet, the generalized BA model with γ =2.2, Ref. [8],
relations. This factor is best represented by the condi-
andthefitnessmodel,Ref.[18]. Thefulllinehasaslope−0.5.
tional probability Pc(k′|k) that a link belonging to node The scattered results for very large k are due to statistical
fluctuations.
Year 1997 1998 1999
with connectivity k points to a node with connectiv-
hki 3.47(4) 3.62(5) 3.82(6)
ity k′. If this conditional probability is independent of
hCi 0.18(1) 0.21(2) 0.24(1)
k, we are in presence of a topology without any cor-
hdi 3.77(1) 3.76(2) 3.72(1)
relation among the nodes’ connectivity. In this case,
P (k′|k)=P (k′)∼k′P(k′), in view of the fact that any
c c
TABLE I:Average properties for threedifferent years. hki is
link points to nodes with a probability proportional to
the average connectivity. hdi is the minimum path distance
their connectivity. On the contrary, the explicit depen-
dij averagedovereverypairofnodes(i,j). hCiisthecluster-
ingcoefficientCiaveragedoverallnodesi,whereCiisdefined denceonkisasignatureofnon-trivialcorrelationsamong
as the ratio between the number of links between the neigh- the nodes’ connectivity, and the possible presence of a
borsofianditsmaximumpossiblevalueki(ki−1). Figuresin hierarchical structure in the network topology. A direct
parenthesisindicatethestatisticaluncertaintyfromaveraging measurement of the P (k′|k) function is a rather com-
c
thevalues of the corresponding months in each year. plextaskduetolargestatisticalfluctuations. Moreclear
3
indications can be extracted by studying the quantity 0.6
hk i = P k′P (k′|k); i.e. the nearest neighbors aver-
nn k′ c (cid:1)t0 =7 days 0:5
age connectivity of nodes with connectivity k. In Fig. 2, t
(cid:1)t0 =30 days
weshowtheresultsobtainedfortheInternetmapof1998,
that strikingly exhibit a clear power law dependence on 0.4
i
the connectivity degree hknni∼k−ν, with ν ≃0.5. This (t)
result clearly implies the existence of non-trivial corre- ki
h
lation properties for the Internet. The primary known 10
g
structural difference between Internet nodes is the dis- o 0.2
l
tinctionbetweenstubandtransitdomains. Nodesinstub
domains have links that go only through the domain it-
0:1
self. Stub domains, on the other hand, are connected t
via a gateway node to transit domains that, on the con- 0.0
0.5 1.5 2.5
trary, are fairly well interconnected via many paths. In
other words, there is a hierarchy imposed on nodes that
log10t
is very likely at the basis of the above correlation prop-
erties. As instructive examples, we report in Fig. 2 the FIG. 3: Average connectivity of nodes borne within a small
averagenearest-neighborconnectivityforthegeneralized time window ∆t0, after a time t elapsed since their appear-
ance. Timetismeasuredindays. Asacomparisonwereport
BA model with γ = 2.2 [13] and the fitness model de-
thelines corresponding to t0.1 and t0.5.
scribedin Ref. [18], with γ =2.25,fornetworkswith the
same size than the Internet snapshot considered. While
in the first case we do not observe any noticeable struc-
Fig. 3, where it is possible to distinguish two different
ture with respect to the connectivity k, in the latter we
dynamical regimes: At early times, the connectivity is
obtainapower-lawdependencesimilartotheexperimen- nearlyconstantwithaveryslowincrease(hk (t)i∼t0.1).
i
tal findings. The general analytic study of connectivity
Later on, the behavior approaches a power law growth
correlationsin growingnetworks models can be found in hk (t)i ∼ t0.5. While exponent estimates are affected by
i
Ref. [19]. A detailed discussion of different models is be-
noise and limited time window effects, the crossover be-
yondthe scopeofthe presentpaper; however,itisworth
tweentwodistinct dynamicalregimesis compatible with
noticing that a k-structure in correlation functions, as
the general aging form obtained in Ref. [19]. In partic-
probed by the quantity hk i, does not arise in allgrow-
nn ular both the generalized BA model [13] and the fitness
ing network models.
model[18]presentagingeffects similartothoseabtained
In order to inspect the Internet dynamics, we focus
in real data. A more detailed comparison would require
our attention on the addition of new nodes and links
aquantitativeknowledgeoftheparameterstobeusedin
into the maps. In the three-years range considered, we
the models and will be reported elsewhere.
have kept track of the number of links ℓ appearing
new Abasicissueinthemodelingofgrowingnetworkscon-
betweenanewlyintroducednodeandanalreadyexisting
cerns the preferential attachment hypothesis [8]. Gener-
node. We have also monitored the rate of appearance of
alizing the BA model algorithm it is possible to define
links ℓ between alreadyexisting nodes. In Table II we
old models in which the rate Π(k) with which a node with
see that the creation of new links is governed by these
k links receive new nodes is proportional to kα. The
two processes at the same time. Specifically, the largest
inspection of the exact value of α in real networks is an
contributionto the growthis givenby the appearanceof
importantissuesincetheconnectivitypropertiesstrongly
links betweenalreadyexistingnodes. This clearlypoints
depend on this exponent [14, 20]. Here we use a simple
out that the Internet growth is strongly driven by the
recipe that allows to extract the value of α by studying
need of a redundancy wiring and an increased need of
the appearance of new links. We focus on links emanat-
available bandwidth for data transmission.
ingfromnewlyappearednodesindifferenttimewindows
A customarily measured quantity in the case of grow-
ranging from one to three years. We consider the fre-
ing networks is the average connectivity hk (t)i of new
i
nodes as a function of their age t. In Refs. [8, 19] it
is shown that hk (t)i is a scaling function of both t and
i Year 1997 1998 1999
the absolute time of birth of the node t0. We thus con-
sider the total number of nodes born within an small ℓnew 183(9) 170(8) 231(11)
ℓ 546(35) 350(9) 450(29)
observation window ∆t0, such that t0 ≃ const. with re- old
spect to the absolute time scale that is the Internet life- ℓnew/ℓold 0.34(2) 0.48(2) 0.53(3)
time. For these nodes, we measure the average connec-
tivityasafunctionofthetimetelapsedsincetheirbirth. TABLE II: Monthly rate of new links connecting existing
The data for two different time windows are reported in nodes to new(ℓnew) and old (ℓold) nodes.
4
PB97-0693. We thank M.-C. Miguel, Y. Moreno-Vega,
links from new nodes
−1 and R. V. Sol´e for helpful comments and discussions.
links from old nodes
−2
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1
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o
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−4
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−5
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This work has been partially supported by the Eu- 86, 3200 (2001).
ropean Network Contract No. ERBFMRXCT980183.
RP-S also acknowledges support from the grant CICYT