A comparison between the topological properties of the measured Internet
topology, at the autonomous system level (AS graph), and the equivalent graphs
generated by two different power law topology generators is presented. Only one
of the synthetic generators reproduces the tier connectivity of the AS graph

Faloutsos

Labovitz

R.J. Mondragón

Shi Zhou

Zhou

English

arXiv

Crossref

Redundancy and robustness of AS-level Internet topology and its models

A comparison between the topological properties of the measured Internet topology, at the autonomous system level (AS graph), and the equivalent graphs generated by two different power law topology generators are presented. Only one of the synthetic generators reproduces the tier connectivity of the AS graph

Zhou, S

Mondragón, RJ

UCL Discovery

arXiv:cs/0402026v1  [cs.NI]  12 Feb 2004IEE ELECTRONIC LETTERS 1Redundancy and Robustness of the AS–levelInternet topology and its modelsShi Zhou and Rau´l J. Mondrago´nAbstract: A comparison between the topological propertiesof the measured Internet topology, at the autonomous systemlevel (AS graph), and the equivalent graphs generated bytwo different power law topology generators is presented.Only one of the synthetic generators reproduces the tierconnectivity of the AS graph.Introduction: Simulation plays an important role in thedevelopment of the Internet, as it can be used to compareand analyze new network protocols. In these simulations it iscrucial that the topology generators capture the key topologicalproperties of the Internet. For example, Labovitz et al [1]showed that the topology of the Internet has a major impacton the delayed BGP routing convergence.If each autonomous system (AS) of the Internet isrepresented by a node in a graph, Faloutsos et al [2]discovered that the link connectivity between these nodesfollows the power law P (k) ∝ k−γ , γ ≈ 2.2, where k is thenumber of links a node has. A good model of the AS-levelInternet topology not only has to reproduce the power lawlink connectivity but also the connectivity of the core of thenetwork. Tier 1 of the AS graph is the core of the networkwhich consists of a set of nodes which are very rich inlinks and are densely interconnected with each other, wecalled this set of nodes the rich–club [3]. This letter showsthat a power law topology generator without a rich–clubcould under–estimate network redundancy and over–estimatenetwork robustness of the Internet.TABLE INETWORK PROPERTIESAS graph IG graph FBA graphNumber of nodes, N 11122 11122 11122Number of links, L 30054 33349 33349Power–law exponent, γ 2.2 2.22 2.255Maximum k 2839 842 1793Maximum Kt 7482 4962 1191Average Kt 12.7 10.0 0.6Manuscript received on 19 August 2003, published on 22 January 2004in IEE Electronic Letters vol. 40, no. 2, pp. 151-15. This work is supportedby the U.K. Engineering and Physical Sciences Research Council (EPSRC)under Grant GR-R30136-01.S. Zhou and R. J. Mondrago´n are with the Department of Elec-tronic Engineering, Queen Mary, University of London, Mile End Road,London, E1 4NS, United Kingdom (e-mail: shi.zhou@elec.qmul.ac.uk;r.j.mondragon@elec.qmul.ac.uk).Methodology: We compared the traceroute AS graph [4]measured on the 1st of April 2002 against the syntheticnetworks generated by the Fitness Baraba´si–Albert (FBA)model [5] and Interactive Growth (IG) model [6]. The twomodels create networks using a node growth mechanism,where a new node attaches to other nodes and prefers to attachitself to nodes that have large numbers of links. The FBAmodel is an example of a generator where the preferentialattachment is controlled by a fitness parameter. This parameteradjust the node’s ability of acquiring connections with othernodes. The IG model is an example of a generator wherea new node attaches itself to other nodes and also createsnew links between nodes that already exist on the network.As shown in table 1, these two topology models generatenetworks that have similar sizes and power law degreedistributions as the AS graph. 0.1% 1% 10% 100% 1%  10%  100%Rich-club connectivityDecending node rank0.1%FBA graphAS graphIG graphFig. 1. Rich–club connectivity.Each node in a network is sorted in decreasing numberof the links that the node contains. The node rank r is theposition that a node has on this ordered list. The positionis normalized by the total number of nodes N . The rich–club are the nodes with rank less than r (e.g. 1%). The coreconnectivity of the network (tier 1) is measured using the rich–club connectivity, which is the ratio of the actual number oflinks between the members of the rich–club to the maximumpossible number of links (fully connected). Figure 1 showsthat the IG graph closely matches the AS graph’s rich–clubconnectivity. For example, the 1% best connected nodes have35% of the maximum possible number of links, whereas inthe FBA graph they only have 4% of the maximum possiblenumber of links.IEE ELECTRONIC LETTERS 2In a network a circuit of length three is called a triangle.The number of alternative routes in a network increases withthe number of triangles. The triangle coefficient Kt of a nodeis defined as the number of triangles that share the node.Figure 2 and table 1 show that the AS graph and the IG graphhave significantly more triangles than the FBA graph. Hencethe FBA model produces networks that are less flexible totraffic routing and have a lower degree of network redundancy. 0.01% 0.1% 1% 10% 100% 1  10  100  1000  10000Cumulative distributionTriangle coefficientFBA graphAS graphIG graphFig. 2. Cumulative distribution of triangle coefficient. 0% 20% 40% 60% 80% 100% 0%  5%  10%  15%  20%Normalized size of largest clusterRatio of nodes under attack  BA graphAS graphIG graphFig. 3. Network robustness under node attack.The removal of the best connected nodes from a networkis called node attack. This resembles the Internet scenariowhere a node, AS or router, is out of service due to severetraffic congestion or infection by a malicious virus. Figure3 shows that the AS graph and the IG graph are extremelyfragile under node attack. The removal of only a few ofthe AS graph’s richest nodes can break down the networkintegrity. By comparison, the FBA graph is not so vulnerableand shows higher degree of resilience to node attack.Conclusion: The AS graph have a densely interconnectedcore structure, which plays a dominant role in the network.The rich–club acts as a super traffic hub providing a largeselection of shortcuts. Realistic models of the Internet shouldreproduce this core structure to avoid under–estimatingthe network redundancy and over–estimating the networkrobustness.REFERENCES[1] C. Labovitz, A. Ahuja, R. Wattenhofer, S. Venkatachary. “The Impact ofInternet Policy and Topology on Delayed Routing Convergence.” Proc.of INFOCOMM 2001.[2] M. Faloutsos, P. Faloutsos, and C. Faloutsos, “On Power–Law Relation-ships of the Internet Topology,” Proc. of ACM/SIGCOMM 1999.[3] S. Zhou and R. J. Mondragon, “Analyzing and modelling the AS–level Internet topology,” Proc. of HET-NETs’03, (arXive:cs.NI/0303030),2003.[4] The ITDK0204 data used in this research was collected as part ofCAIDA’s Skitter initiative, http://www.caida.org. Support for skitter isprovided by DARPA, NSF, and CAIDA membership.[5] G. Bianconi and A. L. Baraba´si, “Competition and multiscaling inevolving networks,” Europhysics Letters, vol. 54, no. 4, pp. 436–442,2001.[6] S. Zhou and R. J. Mondragon, “Towards modelling theInternet topology – the Interactive Growth model,” Proc. ofITC18, (arXive:cs.NI/0303029), 2003.

Redundancy and Robustness of the AS-level Internet topology and its models

Redundancy and Robustness of the AS-level Internet topology and its
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Redundancy and Robustness of the AS-level Internet topology and its models

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