Zhou, S

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180 IEEE COMMUNICATIONS LETTERS, VOL. 8, NO. 3, MARCH 2004The Rich-Club Phenomenon in the Internet TopologyShi Zhou and Raúl J. MondragónAbstract—We show that the Internet topology at the au-tonomous system (AS) level has a rich-club phenomenon. The richnodes, which are a small number of nodes with large numbersof links, are very well connected to each other. The rich-club isa core tier that we measured using the rich-club connectivityand the node-node link distribution. We obtained this core tierwithout any heuristic assumption between the ASs. The rich-clubphenomenon is a simple qualitative way to differentiate betweenpower law topologies and provides a criterion for new networkmodels. To show this, we compared the measured rich-club of theAS graph with networks obtained using the Barabási–Albert (BA)scale-free network model, the Fitness BA model and the Inet–3.0model.Index Terms—Internet, modeling, networks, topology.I. INTRODUCTIONTHE analysis of the Internet topology at the autonomoussystem (AS) level by Faloutsos et al. [1] showed that theprobability that a node has links has a power-law tail for large, following , . Subramanian et al. [2],using a heuristic argument based on the commercial relation-ship between ASes, found that the Internet has a tier structure.Tier 1 consists of a “core” of ASs which are well connected toeach other. There have been some attempts to model the net-work using Transit-Stub and Tiers generators. Tangmunarunkitet al. [3] showed that the degree distributions produced by thesestructure-based generators are not power-laws. Their results arebased on qualitative metrics and they recognized that there isa need for further studies to characterize network topologies.There exist network models that produce power-law networks,e.g., the Barabási and Albert (BA) scale-free model [4] and theInet–3.0 model [5] to mention just two of them.We address in this letter the following questions: Can wecharacterizethecoretieroftheASwithoutmakinganyheuristicassumptions? Does the power-law network generators producea tier structure similar to the one measured in the Internet? Toanswer these questions we introduce the rich-club phenomenonas a quantitative way to characterize a core tier without makingany heuristic assumption on the network elements interaction.One of the main properties of power-law networks is that asmall number of nodes have large numbers of links, we callthese nodes, rich nodes. In this letter we show that the AS graphshows a rich-club, i.e., a core tier. The members of the club tendtobeverywellconnectedbetweeneachother, theycreateatightManuscript received November 29, 2002. The associate editor coordinatingthe review of this letter and approving it for publication was Prof. J. Choe. Thiswork is supported by the U.K. Engineering and Physical Sciences ResearchCouncil (EPSRC) under Grant GR-R30136-01.The authors are with the Department of Electronic Engineering, Queen MaryCollege, University of London, Mile End Road, London E1 4NS, U.K. (e-mail:shi.zhou@elec.qmul.ac.uk; r.j.mondragon@elec.qmul.ac.uk).Digital Object Identifier 10.1109/LCOMM.2004.823426group where if two members of the club do not share a link, itis very likely that they share a common node that can link them,thatistheaveragehopdistanceisbetweenoneandtwo.Also,wehave compared the rich-club measured in the AS graph with theone produced by the BA model, the Fitness BA model [6] andthe Inet–3.0 model, where the synthetic networks are created tomodel the AS graph. Our results show that the BA and FitnessBA model do not create a rich-club. The Inet–3.0 model createsa rich-club but with a deficit in the number of core-links. Noticethatinthisletter,wearenottryingtocharacterizealltheexistingpower-law network generators, but to show that it is possibleto distinguish between them by studying the properties of therich-club.II. THE AS GRAPH AND ITS MODELSA. AS GraphA number of studies [1], [5], [7] on the AS-level Internettopology used the so-called original AS connectivity maps. Theoriginal maps are based on BGP routing tables collected bythe University of Oregon Route Views Project [8]. Chen et al.[9] constructed the extended maps [10] using additional datasources,suchastheInternetRoutingRegistry(IRR)dataandtheLooking Glass (LG) data. The extended maps have 20%–50%more links than the original maps and provide a more completepicture of the AS graph. The AS connectivity data used in thisletter is an extended map measured on May 26, 2001. Table Ishows some properties of the data. For comparison, Table I alsoshows the synthetic graphs generated using the following net-work models.B. BA ModelThe model [4] generates networks with a power-law degreedistribution by using two generic mechanisms: growth, wherethe network “grows” by attaching a new node with links todifferent nodes present in the network; and preferential at-tachment, where new nodes are attached preferentially to nodesthat are already wellconnected. The probability thata new nodewill be connected to node with degree is(1)The BA model has generated great interest in various re-search areas and has been used as a starting-point in researchinto the error and attack tolerance of the Internet [7], [11], [12].The properties shown in Table I are of a network where eachnew node has three new links ( ) which are preferen-tially connected to three already existing nodes. Notice that thismodel generates networks with a power-law link distribution of[13].1089-7798/04$20.00 © 2004 IEEEZHOU AND MONDRAGON: THE RICH-CLUB PHENOMENON IN INTERNET TOPOLOGY 181TABLE IPARAMETERS OF THE FOUR NETWORKS.C. Fitness BA ModelThe model [6] is a modification of the BA model. It uses gen-eralized preferential attachment which assures that, even a rel-atively young node with a small number of links, can acquirenew links at a high rate if it has a large fitness parameter. Thereason we study this model is that, for the uniform fitness pa-rameter distribution, the network generated by this model has apower-law exponent similar to that of the AS graph.D. Inet–3.0 ModelThe model [5] generates networks in three steps: 1) build aspanning tree with all nodes that have degrees greater than one;2) connect all nodes with degree one to nodes in the spanningtree with a linear preference; and 3) connect the remaining freelinks in the spanning tree. The model was designed to matchthe measurements of the original maps of the AS graph. Thenumber of links generated by the model depend on the numberofnodesandtheexponentofthepower-law.ChoosingthesetwoparameterstomatchtheASgraphofTableI,themodeltypicallygenerates 26% less links than the AS graph.III. RICH-CLUB PHENOMENONThe rich-club is characterized by the rich-club connectivityand the node-node link distribution, which measure the inter-connection between rich nodes.A. Rich-Club ConnectivityNodes in the network are sorted by decreasing number oflinks that each node contains. There are instances where groupsof nodes contain identical numbers of links. Where this occurs,they are arbitrarily assigned a position within that group. Thenode rank denotes the position of a node on this ordered list.is normalized by the total number of nodes . The rich-clubis defined, for the purposes of this study, as nodes with rankless than (e.g., 1%). The rich-club connectivity isdefined as the ratio of the total actual number of links to themaximum possible number of links between members of therich-club. The maximum possible number of links betweennodes is .Fig. 1 shows the rich-club coefficient against node rankon a log-log scale. It shows that the rich nodes of the ASgraph are very well connected between each other. The top 1%rich nodes have 32% of the maximum possible number of links,Fig. 1. Rich-club connectivity ￿(r) against node rank r.comparedwith oftheInet-3.0andonlyof the BA and the Fitness BA graphs.B. Node–Node Link DistributionWe define as the number of links connecting nodeswith rank to nodes with rank , where . Fig. 2 showsthe node-node link distribution against correspondingnode rank and . The node ranks are divided into 5% bins.In the AS graph (Fig. 2(a)), rich nodes are connected prefer-entially to other rich nodes. The number of links between thetop 5% rich nodes (far corner) is significantly larger than thenumbers of links connecting the rich nodes to other nodes withsmaller degrees (see the column with ).The node-node link distribution of the Inet-3.0 (Fig. 2(b)) issimilar to that of the AS graph, however, the number of linksbetweenthetop5%richnodesoftheInet–3.0model(farcorner,3697 links) is significantly smaller than that of the AS graph(8919 links).The link distributions of the BA and the Fitness BA graphs(Fig. 2(c),(d)) are different from that of the AS graph. The top5% rich nodes of the BA and the Fitness BA graphs are con-nected to all nodes with similar probabilities regardless of thenode degree. The graphs generated by these two models do notcontain a rich-club.We end this section with the following observation: in the ASgraph, if we take the rich-club to comprise the nodes ranking1% or less, the probability distribution of node degrees betweenthe members of the club is not a power-law, instead is more likethe distribution obtained from a random network.182 IEEE COMMUNICATIONS LETTERS, VOL. 8, NO. 3, MARCH 2004Fig. 2. Node-node link distribution – number of links l(r ;r ) against noderanks r and r .IV. DISCUSSION AND CONCLUSIONThe AS graph has a core tier that we called the rich-club.Membership of the rich-club, as defined above, could be furtherlimited by imposing the requirement of a relationship betweenthe members of the club and their connectivity, for examplethat the average hop distance is less than 1.5. This relationshipcan be made more explicit if we approximate the rich-club toa random network. In this case the average hop distance, where isthe number of nodes contained in the rich-club and is theiraverage node degree which is approximated by .We noticed that the BA and Fitness BA model do not havea rich-club due to the growth dynamics of the models. All newlinks connect with new nodes. Due to the preferential attach-ment, the probability for a new node to become a rich node de-creasesasthenetworkgrows.Asaresult,richnodesarenotwellconnected between each other. This suggests a simple modifi-cation to these models to generate a rich-club. As the networkgrows, new links appear which are preferentially attached be-tween the existing nodes.We believe that modeling the rich-club phenomenon isimportant because the connectivity between rich nodes canbe crucial for network properties, such as network routingefficiency, redundancy and robustness. In the AS graph, thereis a large number of alternative routing paths between the clubmembers, their average path length is very small (1 to 2 hops).The rich-club acts as a super traffic hub and provides a largeselection of shortcuts. Hence scale-free models without therich-club phenomenon may under-estimate the efficiency andflexibility of the traffic routing in the AS graph. Also, networkswithout the rich-club may over-estimate the robustness of thenetwork to a node attack, where the removal of a few of itsrichest club members can break down the network integrity.REFERENCES[1] M. Faloutsos, P. Faloutsos, and C. Faloutsos, “On power-law relation-ships of the internet topology,” in Proc. ACM/SIGCOMM, Comput.Commun. Rev., vol. 29, 1999, pp. 251–262.[2] L. Subramanian, S. Agarwal, J. Rexford, and R. H. Katz, “Character-izing the internet hierarchy from multiple vantage points,” in Proc. IN-FOCOM 2002, June 2002, pp. 618–627.[3] H. Tangmunarunkit, R. Govindan, S. Jamin, S. Shenker, and W. Will-inger, “Network topology generators: Degree-based vs. structural,” inProc. ACM/SIGCOMM 2002, 2002, pp. 147–159.[4] A. L. Barabási and R. Albert, “Emergence of scaling in random net-works,” Science, vol. 286, pp. 509–512, 1999.[5] J. Winick and S. Jamin, “Inet–3.0 Internet Topology Generator,” Univ.Michigan, Ann Arbor, Tech. Rep. UM-CSE-TR-456-02, 2002.[6] G. Bianconi and A. L. Barabási, “Competition and multiscaling inevolving networks,” Europhys. Lett., vol. 54, no. 4, p. 436, 2001.[7] R. Albert, H. Jeong, and A. L. Barabási, “Error and attack tolerance ofcomplex networks,” Nature, vol. 406, pp. 378–381, 2000.[8] Route Views Project. Univ. Oregon, Eugene. [Online]. Available: http://antc.uoregon.edu/route-views[9] Q. Chen, H. Chang, R. Govindan, S. Jamin, S. J. Shenker, and W. Will-inger, “The origin of power laws in internet topologies (Revisited),” inProc. INFOCOM 2002, 2002, pp. 608–617.[10] Topology Project. Univ. Michigan, Ann Arbor. [Online]. Available:http://topology.eecs.umich.edu[11] R. Albert and A. L. Barabási, “Statistical mechanics of complex net-works,”, arXiv: cond-mat/0106096 v1, 2000.[12] P. Holme, B. J. Kim, C. N. Yoon, and S. K. Han, “Attack vulnerabilityof complex networks,” Phys. Rev. E, vol. 65, 2002.[13] A. L. Barabási, R. Albert, and H. Jeong, “Mean-field theory forscale-free random networks,” Physica A, vol. 272, pp. 173–187, 1999.

Rich-club phenomenon of the Internet topology

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