Large-scale topological and dynamical properties of the ...vazquez/publications/internet.pre.pdf · Large-scale topological and dynamical properties of ... We study the large-scale

PHYSICAL REVIEW E, VOLUME 65, 066130

Large-scale topological and dynamical properties of the Internet

Alexei Vazquez,1 Romualdo Pastor-Satorras,2 and Alessandro Vespignani3

1International School for Advanced Studies SISSA/ISAS, via Beirut 4, 34014 Trieste, Italy2Departament de Fı´sica i Enginyeria Nuclear, Universitat Polite`cnica de Catalunya, Campus Nord, Mo`dul B4,

08034 Barcelona, Spain3The Abdus Salam International Centre for Theoretical Physics (ICTP), P.O. Box 586, 34100 Trieste, Italy

~Received 21 December 2001; published 28 June 2002!

We study the large-scale topological and dynamical properties of real Internet maps at the autonomoussystem level, collected in a 3-yr time interval. We find that the connectivity structure of the Internet presentsstatistical distributions settled in a well-defined stationary state. The large-scale properties are characterized bya scale-free topology consistent with previous observations. Correlation functions and clustering coefficientsexhibit a remarkable structure due to the underlying hierarchical organization of the Internet. The study of theInternet time evolution shows a growth dynamics with aging features typical of recently proposed growingnetwork models. We compare the properties of growing network models with the present real Internet dataanalysis.

DOI: 10.1103/PhysRevE.65.066130 PACS number~s!: 89.75.2k, 87.23.Ge, 05.70.Ln

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I. INTRODUCTION

The Internet is a capital example of growing complnetwork @1,2# interconnecting large numbers of computearound the world. Growing networks exhibit a high degreewiring entanglement that takes place during their dynamevolution. This feature, at the heart of the proposed andteresting topological properties recently observed in grownetwork systems@3,4#, has triggered the attention of the rsearch community to the study of the large-scale properof router-level maps of the Internet@5–7#. The statisticalanalysis performed so far has focused on several quanexhibiting nontrivial properties: wiring redundancy and clutering, @8–11#, the distribution of shortest path length@5,10#, and the eigenvalue spectra of the connectivity ma@10#. Noteworthy, the presence of a power-law connectivdistribution @8,10–13# makes the Internet an example of threcently identified class of scale-free networks@14,15#. Thisevidence implies the absence of any charactericonnectivity—large connectivity fluctuations—and a hiheterogeneity of the network structure.

As widely pointed out in the literature@13,16,17#, adeeper empirical understanding of the topological properof the Internet is fundamental in the developing of realisInternet map generators, that on their turn are used to tesoptimize Internet protocols. In fact, the Internet topology ha great influence on the dynamics that data traffic carrieson top of it. Hence, a better understanding of the Interstructure is of primary importance in the design of routi@16,17# and searching algorithms@18,19#, and to protect fromvirus spreading@20# and node failures@21–23#. In this per-spective, the direct measurement and statistical charactetion of real Internet maps are of crucial importance in tidentification of the basic mechanisms that rule the Interstructure and dynamics.

In this work, we shall consider the evolution of real Intenet maps from 1997 to 2000, collected by the National Laratory for Applied Network Research~NLANR! @5#, in orderto study the underlying dynamical processes leading to

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Internet structure and topology. We provide a statistianalysis of several average properties. In particular, we csider the average connectivity, clustering coefficient, plength, and betweenness. These quantities will provide aliminary test of the stationarity of the network. The scale-frnature of the Internet has been pointed out by inspectingconnectivity probability distribution, and it implies that thfluctuations around the average connectivity arebounded. In order to provide a full characterization of tscale-free properties of the Internet, we analyze the conntivity and betweenness probability distributions for differetime snapshot of the Internet maps. We observe that thdistributions exhibit an algebraic behavior and are characized by scaling exponents that are stationary in time. Tshortest path length between pairs of nodes, on the ohand, appears to be sharply peaked around its average vproviding a striking evidence for the presence of wedefined small-world properties@24#. A more detailed pictureof the Internet can be achieved by studying higher orcorrelation functions of the network. In this sense, we shthat the Internet hierarchical structure is reflected in ntrivial scale-free betweenness and connectivity correlatfunctions. Finally, we study several quantities related togrowth dynamics of the network. The analysis points outpresence of two distinct wiring processes: the first concenewly added nodes, while the second is related to alreexisting nodes increasing their interconnections. We confithat newly added nodes establish new links with the linpreferential attachment rule often used in modeling grownetworks@14#. In addition, a study of the connectivity evolution of a single node shows a rich dynamical behavior waging properties. The present study could provide some hfor a more realistic modeling of the Internet evolution, awith this purpose in mind we provide a discussion of somethe existing growing network models in the light of our finings. A short account of these results appeared in Ref.@25#.

The paper is organized as follows. In Sec. II we descrthe Internet maps used in our study. Section III is devotedthe study of average quantities as a function of time. In S

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VAZQUEZ, PASTOR-SATORRAS, AND VESPIGNANI PHYSICAL REVIEW E65 066130

IV we provide the analysis of the statistical distributiocharacterizing the Internet topology. We obtain evidencethe scale-free nature of this network as well as for thetionarity in time of this property. In Sec. V we characterithe hierarchical structure of the Internet by the statistianalysis of the betweenness and connectivity correlafunctions. Section VI reports the study of dynamical propties such as the preferential attachment and the evolutiothe average connectivity of newly added nodes. These perties, which show aging features, are the basis for theveloping of Internet dynamical models. Section VII is dvoted to a detailed discussion of some Internet modelscompared with the presented real data analysis. FinallySec. VIII we draw our conclusions and perspectives.

II. MAPPING THE INTERNET

Several Internet mapping projects are currently devoteobtain high-quality router-level maps of the Internet. In mocases, the map is constructed by using a hop-limited pr~such as theUNIX traceroutetool! from a single location inthe network. In this case the result is a ‘‘directed,’’ mapseen from a specific location on the Internet@7#. This ap-proach does not correspond to a complete map of the Intebecause cross-links and other technical problems~such asmultiple Internet provider aliases! are not fully consideredHeuristic methods to take into account these problems hbeen proposed~see, for instance, Ref.@26#!.

A different representation of the Internet is obtainedmapping the autonomous systems~AS! topology. The Inter-net can be considered as a collection of subnetworks thaconnected together. Within each subnetwork the informais routed using an internal algorithm that may differ from osubnetwork to another. Thus, each subnet is an indepenunit of the Internet and it is often referred as an AS. TheAS communicate between them using a specific routinggorithm, the border gateway protocol. Each AS numberproximately maps to an Internet service provider~ISP! andtheir links are inter-ISP connections. In this case it is possto collect data from several probing stations to obtain intconnectivity maps~see Refs.@5,6# for a technical descriptionof these projects!. In particular, the NLANR project is col-lecting data since November 1997, and it provides topolocal as well as dynamical information on a consistent subof the Internet. The first November 1997 map contains 31AS, and it has grown in time until the December 1999 msurement, consisting of 6374 AS. In the following we wconsider the graph whose nodes represent the AS and wlinks represent the adjacencies~interconnections! betweenAS. In particular we will focus on three different snapshocorresponding to 8 November 1997, 1998, and 1999,will be referenced as AS97, AS98, and AS99, respective

The NLANR connectivity maps are collected with a reslution of one day and are changing from day to day. Thchanges are due to the addition~birth! and deletion~death! ofnodes and links, but also to the flickering of connections,that a node may appear to be isolated~not mapped! fromtime to time. A simple test, however, shows that flickeringappreciable just in nodes with low connectivity. We compu

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the ratio r between the number of days in which a nodeobserved in the NLANR maps and the total number of daafter the first appearance of the node, averaged over all nin the maps. The analysis reveals thatr .1 andr .0.65 fornodes with connectivityk>10, and k,10, respectively.Hence, nodes withk,10 have fluctuations that must btaken into account. In order to shed light on this point,inspect the incidence of deletion events with respect tocreation of new nodes. We consider a deletion event onlynode is not observed in the map during a 1-yr time intervIn Table I we show the total number of deletion events inyear, for 1997, 1998, and 1999, in comparison with the tonumber of new nodes created. It can be seen that the Abirth rate appears to be larger by a factor of 2 than the dtion rate. More interestingly, if we restrict the analysisnodes with connectivityk.10, the deletion rate is reduced ta few percent of the birth rate. This clearly indicates thonly poorly connected nodes have an appreciable probabto disappear. This fact is easily understandable in termsthe market competition among ISP’s, where small newcoers are the ones which more likely go out of business.

III. AVERAGE PROPERTIES AND STATIONARITY

The growth rate of AS maps reveals that the Internet irapidly evolving network. Thus, it is extremely importantknow whether or not it has reached a stationary state whaverage properties are time independent. This will imthat, despite the continuous increase of nodes and contions in the system, the network’s topological propertiesnot appreciably changing in time. As a first step, we haanalyzed the behavior in time of several average magnituthe average connectivityk&, the clustering coefficientc&,the average path lengthl &, and the average betweenne^b&.

The connectivityki of a nodei is defined as the number oconnections of this node with other nodes in the netwoand ^k& is the average ofki over all nodes in the networkSince each connection contributes to the connectivity of tnodes, we have that^k&52E/N, whereE is the total numberof connections andN is the number of nodes. BothE andNare increasing with time but their ratio remains almost costant. The average connectivity for the years 1997, 1998,1999~averaged over all the AS maps available for that ye!is shown in Table II. In average each node has three to fconnections, which is a small number compared with thaa fully connected network of the same size (^k&5N21;103). The average connectivity gives information about tnumber of connections of any node but not about the ove

TABLE I. Total number of new (Nnew) and deleted (Ndel) nodesin the years 1997, 1998, and 1999. We also report the numbedeleted nodes with connectivityk.10.

Year 1997 1998 1999

Nnew 309 1990 3410Ndel 129 887 1713Ndel(k.10) 0 14 68

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LARGE-SCALE TOPOLOGICAL AND DYNAMICAL . . . PHYSICAL REVIEW E 65 066130

structure of these connections. More information can betained using the clustering coefficient introduced in R@24#. The number of neighbors of a nodei is given by itsconnectivityki . On their turn, these neighbors can be conected among them forming a triangle with nodei. The clus-tering coefficientci is then defined as the ratio between tnumber of connections among theki neighbors of a givennode i and its maximum possible value,ki(ki21)/2. Theaverage clustering coefficient^c& is the average ofci over allnodes in the network. The clustering coefficient thus pvides a measure of how well locally interconnected areneighbors of any node. The maximum value of^c& is 1,corresponding to a fully connected network. For randgraphs@27#, which are constructed by connecting nodesrandom with a fixed probabilityp, the clustering coefficiendecreases with the network sizeN as^c& rand5^k&/N. On thecontrary, it remains constant for regular lattices. The averclustering coefficient obtained for the years 1997, 1998,1999 is shown in Table II. As it can be seen, the clustercoefficient of the AS maps increases slowly with increasN and takes valuesc&.0.2, two orders of magnitudes largethan ^c& rand.1023, corresponding to a random graph withe same number of nodes and average connectivity. Thfore, the AS maps are far from being a random graphfeature that can be naively understood using the followargument: In AS maps the connections among nodesequivalent, but they are actually characterized by a real splength corresponding to the actual length of the physical cnection between AS’s. The larger this length is, the higthe costs of installation and maintenance of the line, favortherefore the connections between nearby nodes. It islikely that nodes within the same geographical region whave a large number of connection among them, increain this way the local clustering coefficient.

With this reasoning one might be led to the conclusthat the Internet topology is close to a regular twdimensional lattice. The analysis of the shortest path lenbetween nodes, however, reveals that this is not the cTwo nodesi andj are said to be connected if one can go fronode i to j following the connections in the network. Thpath fromi to j may not be unique and its length is given bthe number of nodes visited. The average path length^l & isdefined as the shortest path length between two nodesi andj,

TABLE II. Average properties of the Internet for three differeyears.N, number of nodes;E, number of connections;^k&, averageconnectivity;^c&, average clustering coefficient;^l &, average pathlength; ^b&, average betweenness. Figures in parentheses indthe statistical uncertainty from averaging the values of the cosponding months in each year.

Year 1997 1998 1999

N 3112 3834 5287E 5450 6990 10100^k& 3.5~1! 3.6~1! 3.8~1!

^c& 0.18~3! 0.21~3! 0.24~3!

^l & 3.8~1! 3.8~1! 3.7~1!

^b&/N 2.4~1! 2.3~1! 2.2~1!

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l i j , averaged over every pair of nodes in the network. Fregular lattices, l &D;N1/D, whereD is the spatial dimen-sion. As it can be seen from Table II, for the AS maps^l &.3.7, which is smaller than the expected value for a regutwo-dimensional lattice of the same size. The Internet stingly exhibits what is known as the ‘‘small-world’’ effec@24,28#: in average one can go from one node to any othethe system passing through a very small number of interdiate nodes. This necessarily implies that besides the slocal connections that contribute to the large clustering coficient, there are some hubs and backbones that connecferent regional networks, strongly decreasing the averpath length. Another measure of this feature is given bynumber of minimal paths that pass by each node. To go frone node in the network to another following the shortpath, a sequence of nodes is visited. If we do this for evpair of nodes in the network, there will be a certain numbof key nodes that will be visited more often than others. Sunodes will be of great importance for the transmissioninformation along the network. This fact can be quantitively measured by means of the betweennessbi , defined bythe total number of shortest paths between any two nodethe network that pass thorough the nodei. The average be-tweennessb& is the average value ofbi over all nodes in thenetwork. The betweenness has been introduced in the ansis of social networks in Ref.@29# and more recently it hasbeen studied in scale-free networks, with the name of lo@30#. Moreover, an algorithm to compute the betweennhas been given in Ref.@29#. For a star network the betweenness takes its maximum valueN(N21)/2 at the central nodeand its minimum valueN21 at the vertices of the star. Thaverage betweenness of the three AS maps analyzed heshown in Table II. Its value is between 2N and 3N, which isquite small in comparison with its maximum possible valN(N21)/2;107.

The present analysis makes clear that the Internet isdominated by a very few highly connected nodes similarlystar-shaped architectures. As well, simple average measments rule out the possibility of a random graph structurea regular grid architecture. This evidence hints towardpeculiar topology that will be fully identified by looking athe detailed probability distributions of several quantities.nally, it is important to stress that despite the network sizemore than doubled in the 3-yr period considered, the averquantities suffer variations of a few percent~see Table II!.This points out that the system seems to have reached a fwell-defined stationary state, as we shall confirm in the flowing section by analyzing the detailed statistical propertof the Internet.

IV. FLUCTUATIONS AND SCALE-FREE PROPERTIES

In order to get a deeper understanding of the netwtopology we look at the probability distributionspk(k) andpb(b) that any given node in the network has a connectivk and a betweennessb, respectively. The study of these proability distributions will allow us to probe the extent of fluctuations and heterogeneity present in the network. We ssee that the strong scale-free nature of the Internet, pr

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pk~k!;k2g, ~1!

extending over three orders of magnitude. In Fig. 1 we repthe integrated connectivity distribution

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corresponding to the AS97, AS98, and AS99 maps. Thetegrated distribution, which expresses the probability thanode has connectivity larger than or equal tok, scales as

Pk~k!;k12g, ~3!

and it has the advantage of being considerably less noisythe original distribution. In all maps we find a clear powelaw behavior with slope close to21.2 ~see Fig. 1!, yieldinga connectivity exponentg52.260.1. The distribution cutoffis fixed by the maximum connectivity of the system andrelated to the overall size of the Internet map. We see thamore recent maps the cutoff is slightly increasing, aspected due to the Internet growth. On the other hand,

FIG. 1. Integrated connectivity distribution for the AS97, AS9and AS99 maps. The power-law behavior is characterized bslope21.2, which yields a connectivity exponentg52.260.1.

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connectivity exponentg seems to be independent of time ain good agreement with previous measurements@10#.

The betweenness distributionpb(b) ~i.e., the probabilitythat any given node is passed over byb shortest paths! showsalso scale-free properties, with a power-law distribution

pb~b!;b2d ~4!

extending over three decades. As shown in Fig. 2~a!, theintegrated betweenness distribution measured in the AS mis evidently stable in the 3-yr period analyzed and followspower-law decay

Pb~b!5Eb

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pb~b8!db8;b12d, ~5!

where the betweenness exponent isd52.160.2. The con-nectivity and betweenness exponents can be simply relatone assumes that the number of shortest pathsbk passingover a node of connectivityk follows the scaling form

bk;kb. ~6!

By inserting the latter relation in the integrated betweenndistribution Eq.~5! we obtain

Pk~k!;kb(12d). ~7!

Since we have thatPk(k);k12g, we obtain the scaling relation

b5g21

d21. ~8!

The measuredg and d have approximately the same valufor the AS maps data and we expect to recoverb'1.0. Thisis corroborated in Fig. 2~b!, where we report the direct measurement of the average betweenness of a node as a funof its connectivityk. It is also worth remarking the study othe betweenness distribution in scale-free networks madRef. @30#. From a numerical study of both static and dynamscale-free network models with different values ofg, it wasfound in Ref.@30# that the betweenness distribution followspower-law decay with an estimated exponentd52.260.1.The authors argued that this fact represents a universal perty, independent of the connectivity exponent, for all sca

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FIG. 2. ~a! Integrated between-ness distribution for the AS97AS98, and AS99 maps. Thepower-law behavior is characterized by a slope 21.1, whichyields a betweenness exponentd52.160.2. ~b! Betweennessbk asa function of the node’s connectivity k. The full line correspondsto the expected behaviorbk;k.Errors bars take into account statistical fluctuations over differentnodes with the same connectivity

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free networks with 2,g<3. Our results on the AS mappresent further support to the universality claim made in R@30#.

Another quantity of interest is the probability distributioof the clustering coefficient of the nodes. In our analysisdo not find definitive evidence for a power-law behaviorthis distribution. However, still useful information can bgathered from studying the clustering coefficientck as afunction of the node connectivity. In this case the local clutering coefficient of each nodeci is averaged over all nodewith the same connectivityk. The plots for the AS97, AS98and AS99 maps are shown in Fig. 3. Also in this case, msurements yield a power-law behaviorck;k2v with v50.7560.03, extending over three orders of magnitude. Texponent 0.75 has been computed as an average overegressions of the individual data sets. This fact implies tnodes with a small number of connections have larger loclustering coefficients than those with a large connectivThis behavior is consistent with the picture previously dscribed in Sec. III of highly clustered regional networsparsely interconnected by national backbones and intetional connections. The regional clusters of AS are probaformed by a large number of nodes with small connectivbut large clustering coefficients. Moreover, they also shocontain nodes with large connectivities that are connec

FIG. 3. Clustering coefficientck as a function of the connectivity k for the AS97, AS98, and AS99 maps. The best fitting powlaw behavior is characterized by a slope20.75. Errors bars takeinto account statistical fluctuations over different nodes withsame connectivity.

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with the other regional clusters. These large connectivnodes will be on their turn connected to nodes in differeclusters that are not interconnected and, therefore, will ha small local clustering coefficient. This picture also shothe existence of some hierarchy in the network that will bcome more evident in the following section.

A different behavior is followed by the shortest palength l between two nodes, which does not show singufluctuations from one pair of nodes to another. This canshown by means of the probability distributionpl (l ) ofshortest path lengthsl between pairs of nodes, reportedFig. 4~a!. This distribution is characterized by a sharp pearound its average value and its shape remains essenunchanged from the AS97 to the AS99 maps. Associatethe shortest path length distribution we have the hop pintroduced in Ref.@10#. The hop plot is defined as the aveage fraction of nodesM (l )/N within a distance less than oequal tol from a given node. Atl 50 we find the startingnode and, therefore,M (0)51. At l 51 we find the startingnode plus its neighbors and thusM (1)5^k&11. If the net-work is made up by a single cluster, forl 5l M , wherel Mis the maximum shortest path length, we haveM (l M)5N.For regularD-dimensional lattices,M (l );l D, and in thiscaseM can be interpreted as the mass. The hop plot is relato the distribution of shortest path lengths through the flowing relation:

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The hop plots for the AS97, AS98 and AS99 maps are shoin Fig. 4~b!. In this case the shortest path length barely spa decade (l M511). Most importantly,M (l ) practicallyreaches its maximum valueN at l 55. Hence, the shortespath length does not show strong fluctuations, as alrenoticed from the shortest path length distribution. In R@10# it was argued that the increase ofM (l ) for small lfollows a power-law behavior. This observation is not cosistent with the present data, that yield a very abrupt incretaking place in a very narrow range, as shown in Fig. 4~b!.

Finally, it is important to stress again that all the measudistributions are characterized by scaling exponents orhaviors that are not changing in time. This implies that tstatistical properties characterizing the Internet are time

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FIG. 4. ~a! Distribution ofshortest path lengthspl (l ) forthe AS97, AS98, and AS99 maps~b! Hop plotsM (l ) for the samemaps. See text for definitions.

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FIG. 5. ~a! Average connectiv-ity ^knn& of the nearest neighborof a node as a function of the connectivity k for the AS97, AS98,and AS99 maps. The full line haa slope 20.5. ~b! Average be-tweenness^bnn& of the nearestneighbors of a node as a functioof its betweennessb for the samemaps. The full line has a slope20.4.

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dependent, providing a further test to the network stationity; i.e., the Internet is self-organized in a stationary stacharacterized by scale-free fluctuations.

V. HIERARCHY AND CORRELATIONS

Due to installation costs, the Internet has been desigwith a hierarchical structure. The primary known structudifference between Internet nodes is the distinction betwstuband transit domains. Nodes in stub domains have linthat go only through the domain itself. Stub domains, onother hand, are connected via a gateway node to transitmains that, on the contrary, are fairly well interconnectedmany paths. This hierarchy can be schematically dividedinternational connections, national backbones, regionalworks, and local area networks. Nodes providing accesinternational connections or national backbones are of coon top level of this hierarchy, since they make possiblecommunication between regional and local area netwoMoreover, in this way, a small average path length canachieved with a small average connectivity.

Very likely the hierarchical structure will introduce somcorrelations in the network topology. We can explore thierarchical structure of the Internet by means of the contional probability pc(k8uk) that a link belonging to a nodewith connectivityk points to a node with connectivityk8. Ifthis conditional probability is independent ofk, we are inpresence of a topology without any correlation amongnodes’ connectivity. In this case,pc(k8uk)5pc(k8);k8pk(k8), in view of the fact that any link points to nodewith a probability proportional to their connectivity. On thcontrary, the explicit dependence onk is a signature of non-trivial correlations among the nodes’ connectivity, and tpresence of a hierarchical structure in the network topoloA direct measurement of thepc(k8uk) function is a rathercomplex task due to large statistical fluctuations. More clindications can be extracted by studying the quantity

^knn&5(k8

k8pc~k8uk!, ~10!

i.e., the nearest-neighbors average connectivity of nodesconnectivityk. In Fig. 5~a! we show the results obtained fothe AS97, AS98, and AS99 maps, that again exhibit a cpower-law dependence on the connectivity degree,

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with an exponentnk50.560.1. This observation clearly implies that the connectivity correlation function has a markdependence uponk, suggesting nontrivial correlation propeties for the Internet. In practice, this result indicates thhighly connected nodes are more likely pointing to less cnected nodes, emphasizing the presence of a hierarchwhich smaller providers connect to larger ones and soclimbing different levels of connectivity.

Similarly, it is expected that nodes with high betweenne~that is, carrying a heavy load of traffic!, and consequently alarge connectivity, will be connected to nodes with smalbetweenness, less load and, therefore, small connectivitsimple way to measure this effect is to compute the averbetweennessbnn& of the neighbors of the nodes with a givebetweennessb. The plot of ^bnn& for the AS97, AS98, andAS99 maps, represented in Fig. 5~b!, shows that the averagneighbor betweenness exhibits a clear power-law depdence on the node betweennessb,

^bnn&;b2nb, ~12!

with an exponentnb50.460.1, evidencing that the morloaded nodes~backbones! are more frequently connectewith less loaded nodes~local networks!.

These hierarchical properties of the Internet are likdriven by several additional factors such as the space loity, economical resources, and the market demand. Antempt to relate and study some of these aspects can be fin Ref. @13#, where the geographical distribution of popultion and Internet access are studied. In Sec. VII we scompare a few of the existing models for the generationscale-free networks with our data analysis, in an attempidentify some relevant features in the Internet modeling.

VI. DYNAMICS AND GROWTH

In order to inspect the Internet dynamics, we focus oattention on the addition of new nodes and links into tmaps. In the 3-yr range considered, we keep track ofnumber of linksLnew appearing between a newly introducenode and an already existing node. We also monitor theof appearance of linksLold between already existing nodeIn Table III we can observe that the creation of new links

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governed by these two processes at the same time. Spcally, the largest contribution to the growth is given by tappearance of links between already existing nodes. Tclearly points out that the Internet growth is strongly drivby the need of redundancy in the wiring and an increaneed of available bandwidth for data transmission.

A customarily measured quantity in the case of grownetworks is the average connectivity^ki(t)& of new nodes asa function of their aget. In Refs.@15,31,32# it is shown that^ki(t)& is a scaling function of botht and the absolute time obirth of the nodet0. We thus consider the total numbernodes born within a small observation windowDt0, such thatt0.const with respect to the absolute time scale that isInternet lifetime. For these nodes, we measure the aveconnectivity as a function of the timet elapsed since theibirth. The data for two different time windows are reportin Fig. 6, where it is possible to distinguish two differedynamical regimes: At early times, the connectivity is neaconstant with a very slow increase. Later on, connectivgrows rapidly approaching what appears to be a power-or faster growth regime. While reliable fits or exponent esmates are affected by noise and limited time window effethe crossover between two distinct dynamical regimescompatible with the general aging form obtained in the ctext of growing networks in Refs.@31,32#.

A very important issue in the modeling of growing neworks concerns the understanding of the growth mechaniat the origin of the developing of new links. As we shall smore in detail in the following section, the basic ingredienin the modeling of scale-free growing networks is the prerential attachment hypothesis@14#. In general, all growing

TABLE III. Monthly rate of new links connecting existingnodes to new (Lnew) and old (Lold) nodes.

Year 1997 1998 1999

Lnew 183~9! 170~8! 231~11!

Lold 546~35! 350~9! 450~29!

Lnew/Lold 0.34~2! 0.48~2! 0.53~3!

FIG. 6. Average connectivity of nodes borne within a small timwindow Dt0, after a timet elapsed since their appearance. Timet ismeasured in days.

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network algorithms define models in which the rateP(k)with which a node withk connections receives new linksproportional toka ~see Ref.@14# and Sec. VII!. The inspec-tion of the exact value ofa in real networks is an importanissue since the connectivity properties strongly dependthis exponent@31–33#. Here we use a simple recipe thallows to extract the value ofa by studying the appearancof new links. We focus on links emanating from newly apeared nodes in different time windows ranging from onethree years. We consider the frequencym(k) of links thatconnect to nodes with connectivityk. By using the preferen-tial attachment hypothesis, this effective probability ism(k);kapk(k). Since we know thatpk(k);k2g, we expect tofind a power-law behaviorm(k);ka2g for the frequency. InFig. 7 we report the obtained results which show for tintegrated frequencymcum(k)5*k

`m(k8)dk8 a behavior com-patible with an algebraic dependencem(k);k21.2. By usingthe independently obtained valueg52.2 we find a preferen-tial attachment exponenta.1.0, in good agreement with thresult obtained with a different analysis in Ref.@33#. Weperformed a similar analysis also for links emanated byisting nodes, recovering the same form of preferential attament~see Fig. 7!. The present analysis confirms the validiof the preferential attachment hypothesis, but leaves openquestion of the interplay with several other factors, suchthe nodes’ hierarchy, space locality, and resource constra

VII. MODELING THE INTERNET

In the preceding section we have presented a thoroanalysis of the AS maps topology. Apart from providing usful empirical data to understand the behavior of the Internour analysis is of great relevance in order to test the validof models of the Internet topology. The Internet topology ha great influence on the information traffic carried on topit, including routing algorithms@16,17#, searching algorithms@18,19#, virus spreading@20#, and resilience to node failure@21–23#. Thus, designing network models that accuratelyproduce the Internet topology is of capital importancecarry out simulations on top of these networks.

FIG. 7. Frequency of links emanating from new and existinodes that attach to nodes with connectivityk. The full line corre-sponds to a slope21.2, which yields an exponenta.1.0. The flattails are originated from the poor statistics at very highk values.

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Early works considered the Erdo¨s-Renyi @34# model orhierarchical networks as models of the Internet@35#. How-ever, they yield connectivity distributions with a fast~expo-nential! decay for large connectivities, in disagreement wthe power-law decay observed in real data. Only recentlyInternet modeling benefited of the major advance providethe field of growing networks by the introduction of thBarabasi-Albert ~BA! model @14,15,36#, which is related to1955 Simon’s model@37–39#. The main ingredients of thismodel are the growing nature of the network and a prefertial attachment rule, in which the probability of establishinew links toward a given node grows linearly with its conectivity. The BA model is constructed using the followinalgorithm@14#: We start from a small numberm0 of discon-nected nodes; every time step a new node is added, wimlinks that are connected to an old nodei with probability

PBA~ki !5ki

(j

kj

, ~13!

whereki is the connectivity of thei th node. After iteratingthis procedureN times, we obtain a network with a connetivity distribution pk(k);k23 and average connectivityk&52m. In this model, heavily connected nodes will increatheir connectivity at a larger rate than less connected noa phenomenon that is known as the ‘‘rich-get-richer’’ effe@14#. It is worth remarking, however, that more general stuies @4,31,32# have revealed that nonlinear attachment ratethe formP(k);ka with aÞ1 have as an outcome connetivity distributions that depart form the power-law behavioThe BA model has been successively modified with thetroduction of several ingredients in order to account for cnectivity distribution with 2,g,3 @31,32,40#, local geo-graphical factors@41#, wiring among existing nodes@42#, andage effects@43#.

In the preceding section we have analyzed different msures that characterize the structure of AS maps. Sinceeral models are able to reproduce the right power law behior for the connectivity distribution, the analysis obtainedthe previous sections can provide the effective tools to stinize the different models at a deeper level. In particular,perform a data comparison for three different models tgenerate networks with power-law connectivity distributionFirst we have considered a random graph with a power-connectivity distribution, constructed using the Molloy aReed ~MR! algorithm @44,45#. Secondly, we have studietwo variations of the BA model, that yield connectivity eponents compatible with the one measured in the Interthe generalized Baraba´si-Albert ~GBA! model @40#, whichincludes the possibility of connection rewiring, and the finess model@46#, that implements a weighting of the nodesthe preferential attachment probability. The models arefined as follows:

MR model. In the construction of this model@4,44,45,47#we start assigning to each nodei in a set ofN nodes a ran-dom connectivityki drawn from the probability distributionpk(k);k2g, with m<ki,N, imposing the constraint thathe sum( iki must be even. The graph is completed by ra

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domly connecting the nodes with( iki /2 links, respecting theassigned connectivities. The results presented here aretained usingm51 and a connectivity exponentg52.2,equal to that found in the AS maps. Clearly this constructalgorithm does not take into account any correlations ornamical features of the Internet and it can be consideredfirst order approximation that focuses only on the connecity properties.

GBA model. It is defined by starting withm0 nodes con-nected in a ring@40#: At each time step one of the followingoperations is performed:

~i! With probability q we rewire m links. For each ofthem, we randomly select a nodei and a linkl i j connected toit. This link is removed and replaced by a new linkl i 8 j con-necting the nodej to a new nodei 8 selected with probabilityP(ki 8) where

PGBA~ki !5ki11

(j

~kj11!

. ~14!

~ii ! With probability p we addm new links. For each ofthem, one end of the link is selected at random, whileother is selected with probability as in Eq.~14!.

~iii ! With probability 12q2p we add a new node withmlinks that are connected to nodes already present with pability as in Eq.~14!.

The preferential attachment probability Eq.~14! leads to apower-law distributed connectivity, whose exponent depeon the parametersq and p. In the particular casep50, theconnectivity exponent is given by@40#

g511~12q!~2m11!

m. ~15!

Hence, changing the value ofm and q we can obtain thedesired connectivity exponentg. In the present simulationswe use the valuesm52 andq513/25, that yield the expo-nent g52.2. The GBA model embeds both the rich-gericher paradigm and the growing nature of the Internet; hoever, it does not take into account any possible differencehierarchies in newly appearing nodes.

Fitness model. This network model introduces an externcompetence among nodes to gain links, that is controlleda random~fixed! fitness parameterh i that is assigned to eacnodei from a probability distributionr(h). In this case, wealso start withm0 nodes connected in a ring and at each timstep we add a new nodei 8 with m links that are connected tonodes already present on the network with probability

Pfitness~ki !5h iki

(j

h j kj

. ~16!

The newly added node is assigned a fitnessh i 8 . The resultspresented here are obtained usingm52 and a probabilityr(h) uniformly distributed in the interval@0,1#, which yieldsa connectivity distributionpk(k);k2g/ ln k with g'2.26@46#. The fitness model adds to the growing dynamics w

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preferential attachment a stochastic parameter, the fitnthat embeds all the properties, other than the connectithat may influence the probability of gaining new links.

We have performed simulations of these three modwith the parameters mentioned above and using sizes oN.4000 nodes, in analogy with the size of the AS maps alyzed. In each case we perform averages over 1000 differealizations of the networks. It is worth remarking that whthe fitness model generates a connected network, bothGBA and the MR model yield disconnected networks. Thisdue to the rewiring process in the GBA model, while tdisconnect nature of the graph in the MR model is an inhent consequence of the connectivity exponent being lathat 2 @47#. In these two cases we therefore work with grapwhose giant component~that is, the largest cluster of connected nodes in the network@27#! has a size of the orderN.It is important to remind the reader that we are working wnetworks of a relatively small size, chosen so as to fit the sof the Internet maps analyzed in the previous sections. Inperspective, all the numerical analysis that we shall perfoin the following serve only to check the validity of the moels as representations of the Internet as we know it, andnot refer to the intrinsic properties of the models in the thmodynamic limitN→`.

As a first check of the connectivity properties of the moels, in Fig. 8 we have plotted the respective integrated cnectivity distributions. For the MR model we recover thexpected exponentgMR.2.20, since it was imposed in thvery definition of the model. For the GBA model we obtanumericallygGBA.2.19 for the giant component, in excelent agreement with the value predicted by Eq.~15! for theasymptotic network. For the fitness model, on the other haa numerical regression of the integrated connectivity disbution yields an effective exponentgfitness.2.4. This value islarger than the theoretical prediction 2.26 obtained formodel@46#. The discrepancy is mainly due to the logarithmcorrections present in the connectivity distribution of thmodel. These corrections are more evident in the relativsmall-sized networks used in this work and become progsively smaller for larger network sizes.

In Table IV we report the average values of the conn

FIG. 8. Integrated connectivity distribution for the MR, GBAand fitness models, compared with the result from the AS98 mThe full line has slope21.2.

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tivity, clustering coefficient, path length, and betweennessthe three models, compared with the respective values cputed for the Internet during 1998. From the examinationthis table, one could surprisingly conclude that the Mmodel, which neglects by construction any correlatiamong nodes, yields the average values in better agreewith the Internet data. As we can observe, the fitness moprovides too small a value for the average clustering coecient, while the GBA model clearly fails for the average palength and the betweenness. A more crucial test aboutmodels is however provided by the analysis of the full dtribution of the various quantities, that should reproducescale-free features of the real Internet.

The betweenness distributionpb(b) of the three modelsgive qualitatively similar results. The integrated betweenndistribution Pb(b) obtained is plotted in Fig. 9~a!. Both theMR and the fitness models follow a power-law decpb(b);b2d with an exponentd.2, in agreement with thevalue obtained from the AS maps. The GBA model showsappreciable bending that, nevertheless, is compatible wthe experimental Internet behavior. These results areagreement with the numerical prediction in Ref.@30# andsupport the conjecture that the exponentd.2.2 is a universalquantity in all scale-free networks with 2,g,3. In order tofurther inspect the betweenness properties, we plot in9~b! the average betweennessbk as a function of the connectivity. In this case, the MR and GBA models yield an expnent b.1, compatible with the AS maps, while the fitnemodel exhibits a somewhat larger exponent, close toAlso in this case, we have that the finite size logarithmcorrections present in the fitness model could play a deminant role in this discrepancy.

While properties related to the betweenness do not apto pinpoint a major difference among the models, the mstriking test is provided by analyzing the correlation propties of the models. In Figs. 10 and 11, we report the averclustering coefficient as a function of the connectivity,ck ,and the average connectivity of the neighbors,^knn&, respec-tively. The data from the Internet maps show a nontriviakstructure that, as discussed in previous sections, is duscale-free correlation properties among nodes. These proties depend on their turn upon the underlying hierarchythe Internet structure. The only model that renders resultqualitative agreement with the Internet maps is the fitnmodel. On the contrary, the MR and GBA models complet

p.

TABLE IV. Average properties of the MR, GBA, and fitnesmodels, compared with the values from the Internet in 1998.^k&,average connectivity;c&, average clustering coefficient;^l &, aver-age path length;b&, average betweenness. Figures in parentheindicate the statistical uncertainty from the average of 1000 realtions of the models.

MR GBA Fitness 1998

^k& 4.8~1! 5.4~1! 4.00~1! 3.6~1!

^c& 0.16~1! 0.12~1! 0.02~1! 0.21~3!

^l & 3.1~1! 1.8~1! 4.0~1! 3.8~1!

^b&/N 2.2~1! 1.9~1! 2.1~1! 2.3~1!

0-9

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FIG. 9. ~a! Integrated between-ness distribution for the MR,GBA, and fitness models, compared with the result from theAS98 map. The full line has aslope21.1, corresponding to theInternet map.~b! Betweennessbk

as a function of the node’s connectivity k corresponding to theprevious results. The full line hasa slope 1.0.

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fail, producing quantities that are almost independent onk.The reason of this striking difference can be traced backthe lack of correlations among nodes, which in the Mmodel is imposed by construction~the model is a randomnetwork with fixed connectivity distribution!, and in theGBA model it is due to the destruction of correlations by trandom rewiring mechanism implemented. The general alytic study of connectivity correlations in growing networkmodels has been discussed in Ref.@32#, and the conditionalprobability pc(k8uk) has been computed for a determinisscale-free network model@48#. However, it is worth noticingthat ak structure in correlation functions, as probed by tquantity^knn&, does not arise in all growing network modeIn this perspective we can use correlation properties asof the discriminating feature among various models tshow the same scale-free connectivity exponent. Interingly, a stochastic network model@49# has been recently proposed, in the spirit of the scenario advanced in Ref.@50#, thatappears to capture the correlation function propertiessented here. This model is defined in terms of three elemtary rules. At each time step:~i! The number of nodes isincreased by a constant fraction of the nodes present inprevious time step; the newly added nodes are connecteone or two previously present nodes.~ii ! Each vertex in-creases its connectivity by a constant factor, the new libeing connected following the preferential attachment ruEq. ~13!. ~iii ! Each vertex randomly disconnects existi

FIG. 10. Clustering coefficientck as a function of the connectivity k for the MR, GBA, and fitness models, compared with tresult from the AS98 map. The full line has a slope20.75.

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links or connects new links, following in this last case tpreferential attachment rule. With these three elements,model described in Ref.@49# recovers a connectivity exponent and clustering coefficient comparable with the valufound in the present work, while yielding a function^knn&decreasing withk as a power-law, in close analogy with thbehavior we have reported for real AS maps.

The fitness model is able to reproduce the nontrivial crelation properties because of the fitness parameter of enode that mimics the different hierarchical, economical, ageographical constraints of Internet growth. Since the mois embedding many features in one single parameter,have to consider it just as a very first step towards a mrealistic modeling of the Internet. In this perspective, modin which the attachment rate depends on both the conneity and the real space distance between two nodes hasstudied in@13,41#. These models seem to give a better dscription of the Internet topology. In particular, the modelRef. @13# includes a new element, the inclusion of geograpcal constraints, that was not considered previously. Tmodel describes the Internet in terms of an evolving netwin which the added nodes have a geographical positforming a scale-invariant fractal set with a fractal dimensicompatible with the value found in a real router-level maAlso, the probability of the addition of new links is regulate

FIG. 11. Average connectivity of the nearest neighbors of a nas a function of the connectivityk for the MR, GBA, and fitnessmodels, compared with the result from the AS98 map. The ASdata have been binned for the sake of clarity. The full line haslope20.5.

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by two competing mechanisms, being directly proportioto the connectivity of the nodes and inversely proportionathe physical distance between nodes. While the path opeby this model seems quite promising, a comparison with rdata is more difficult because Internet maps at the AS legenerally lack geographical and economical information.

VIII. SUMMARY AND CONCLUSIONS

In summary, we have shown that the Internet maps exha stationary scale-free topology, characterized by nontriconnectivity correlations. An investigation of the Internet dnamics confirms the presence of a preferential attachmbehaving linearly with the nodes’ connectivity and identifitwo different dynamical regimes during the nodes’ evolutioWe have compared several models of scale-free networkthe experimental data obtained from the AS maps. Whilethe models seem to capture the scale-free connectivity dibution, correlation and clustering properties are captu

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only in models that take into account several other ingreents, such as the nodes’ hierarchy, resource constraintsgeographical location. Other ingredients that should becluded in the Internet modeling concern the possibilityincluding the wiring among existing nodes and age effethat our analysis show to be an appreciable feature ofInternet evolution. The results presented in this work shthat the understanding and modeling of the Internet isinteresting and stimulating problem that needs the cooptive efforts of data analysis and theoretical modeling.

ACKNOWLEDGMENTS

This work has been partially supported by the EuropeCommission — Fet Open Project No. COSIN IST-20033555. R.P.-S. acknowledges financial support from the Misterio de Ciencia y Tecnologı´a ~Spain! and from the AbdusSalam International Center for Theoretical Physics~ICTP!,where part of this work was done.

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