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75
A FirstPrinciples Approach to Understanding the Internet's Routerlevel Topology
, 2004
"... A detailed understanding of the many facets of the Internet's topological structure is critical for evaluating the performance of networking protocols, for assessing the effectiveness of proposed techniques to protect the network from nefarious intrusions and attacks, or for developing improved ..."
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Cited by 214 (19 self)
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A detailed understanding of the many facets of the Internet's topological structure is critical for evaluating the performance of networking protocols, for assessing the effectiveness of proposed techniques to protect the network from nefarious intrusions and attacks, or for developing improved designs for resource provisioning. Previous studies of topology have focused on interpreting measurements or on phenomenological descriptions and evaluation of graphtheoretic properties of topology generators. We propose a complementary approach of combining a more subtle use of statistics and graph theory with a firstprinciples theory of routerlevel topology that reflects practical constraints and tradeoffs. While there is an inevitable tradeoff between model complexity and fidelity, a challenge is to distill from the seemingly endless list of potentially relevant technological and economic issues the features that are most essential to a solid understanding of the intrinsic fundamentals of network topology. We claim that very simple models that incorporate hard technological constraints on router and link bandwidth and connectivity, together with abstract models of user demand and network performance, can successfully address this challenge and further resolve much of the confusion and controversy that has surrounded topology generation and evaluation.
Towards Capturing Representative ASLevel Internet Topologies
 Computer Networks Journal
, 2002
"... Recent studies concerning the Internet connectivity at the AS level have attracted considerable attention. These studies have exclusively relied on the BGP data from Oregon routeviews [1] to derive some unexpected and intriguing results. The Oregon routeviews data sets reflect AS peering relations ..."
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Cited by 182 (25 self)
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Recent studies concerning the Internet connectivity at the AS level have attracted considerable attention. These studies have exclusively relied on the BGP data from Oregon routeviews [1] to derive some unexpected and intriguing results. The Oregon routeviews data sets reflect AS peering relationships, as reported by BGP, seen from a handful of vantage points in the global Internet. The possibility that these data sets from Oregon routeviews may provide only a very sketchy picture of the complete interAS connections that exist in the actual Internet has received surprisingly little scrutiny. In this paper, we will use the term "AS peering relationship" to mean that there is "at least one direct routerlevel connection" between two existing ASs, and that these two ASs agree to exchange traffic by enabling BGP between them. By augmenting the Oregon routeviews data sets with BGP summary information from a large number of Internet Looking Glass sites and with routing policy information from Internet Routing Registry (IRR) databases, we find that (1) a significant number of existing AS connections remain hidden from most BGP routing tables, (2) the AS connections to tier1 ASs are in general more easily observed than those to non tier1 ASs, and (3) there are at least about 2550% more AS connections in the Internet than commonlyused BGPderived AS maps reveal (but only about 2% more ASs). These findings point out the need for an increased awareness of and a more critical attitude toward the applicability and completeness of given data sets at hand when establishing the generality of any particular observations about the Internet.
The Origin of Power Laws in Internet Topologies Revisited
 In IEEE INFOCOM 2002
, 2002
"... In a recent paper, Faloutsos et al. [1] found that the inter Autonomous System (AS) topology exhibits a powerlaw vertex degree distribution. This result was quite unexpected in the networking community and stirred significant interest in exploring the possible causes of this phenomenon. The work of ..."
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Cited by 110 (3 self)
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In a recent paper, Faloutsos et al. [1] found that the inter Autonomous System (AS) topology exhibits a powerlaw vertex degree distribution. This result was quite unexpected in the networking community and stirred significant interest in exploring the possible causes of this phenomenon. The work of Barabasi and Albert [2] and its application to network topology generation in the work of Medina et al. [3] have explored a promising class of models that yield strict powerlaw vertex degree distributions. In this paper, we reexamine the BGP measurements that form the basis for the results reported in [1]. We find that by their very nature (i.e., being strictly BGPbased), the data provides a very incomplete picture of Internet connectivity at the AS level. The AS connectivity maps constructed from this data (the original maps) typically miss 2050% or even more of the physical links in AS maps constructed using additional sources (the extended maps). Subsequently, we find that while the vertex degree distributions resulting from the extended maps are heavytailed, they deviate significantly from a strict power law. Finally, we show that available historical data does not support the connectivitybased dynamics assumed in [2]. Together, our results suggest that the Internet topology at the AS level may well have developed over time following a very different set of growth processes than those proposed in [2].
On Certain Connectivity Properties of the Internet Topology
 IN PROC. 35TH ACM SYMP. ON THEORY OF COMPUTING
, 2003
"... We show that random graphs in the preferential connectivity model have constant conductance, and hence have worstcase routing congestion that scales logarithmically with the number of nodes. Another immediate implication is constant spectral gap between the first and second eigenvalues of the rando ..."
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Cited by 89 (3 self)
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We show that random graphs in the preferential connectivity model have constant conductance, and hence have worstcase routing congestion that scales logarithmically with the number of nodes. Another immediate implication is constant spectral gap between the first and second eigenvalues of the random walk matrix associated with these graphs. We also show that the expected frugality (overpayment in the VickreyClarkeGroves mechanism for shortest paths) of a random graph is bounded by a small constant.
Compact routing on Internetlike graphs
 In Proc. IEEE INFOCOM
, 2004
"... Abstract — The ThorupZwick (TZ) compact routing scheme is the first generic stretch3 routing scheme delivering a nearly optimal pernode memory upper bound. Using both direct analysis and simulation, we derive the stretch distribution of this routing scheme on Internetlike interdomain topologies. ..."
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Cited by 64 (7 self)
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Abstract — The ThorupZwick (TZ) compact routing scheme is the first generic stretch3 routing scheme delivering a nearly optimal pernode memory upper bound. Using both direct analysis and simulation, we derive the stretch distribution of this routing scheme on Internetlike interdomain topologies. By investigating the TZ scheme on random graphs with powerlaw node degree distributions, Pk � k −γ, we find that the average TZ stretch is quite low and virtually independent of γ. In particular, for the Internet interdomain graph with γ � 2.1, the average TZ stretch is around 1.1, with up to 70 % of all pairwise paths being stretch1 (shortest possible). As the network grows, the average stretch slowly decreases. The routing table is very small, too. It is well below its upper bounds, and its size is around 50 records for 10 4node networks. Furthermore, we find that both the average shortest path length (i.e. distance) d and width of the distance distribution σ observed in the real Internet interAS graph have values that are very close to the minimums of the average stretch in the d and σdirections. This leads us to the discovery of a unique critical point of the average TZ stretch as a function of d and σ. The Internet distance distribution is located in a close neighborhood of this point. This is remarkable given the fact that the Internet interdomain topology has evolved without any direct attention paid to properties of the stretch distribution. It suggests the average stretch function may be an indirect indicator of the optimization criteria influencing the Internet’s interdomain topology evolution.
Understanding internet topology: principles, models, and validation
 IEEE/ACM TRANSACTIONS ON NETWORKING
, 2005
"... Building on a recent effort that combines a firstprinciples approach to modeling routerlevel connectivity with a more pragmatic use of statistics and graph theory, we show in this paper that for the Internet, an improved understanding of its physical infrastructure is possible by viewing the phys ..."
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Cited by 51 (8 self)
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Building on a recent effort that combines a firstprinciples approach to modeling routerlevel connectivity with a more pragmatic use of statistics and graph theory, we show in this paper that for the Internet, an improved understanding of its physical infrastructure is possible by viewing the physical connectivity as an annotated graph that delivers raw connectivity and bandwidth to the upper layers in the TCP/IP protocol stack, subject to practical constraints (e.g., router technology) and economic considerations (e.g., link costs). More importantly, by relying on data from Abilene, a Tier1 ISP, and the Rocketfuel project, we provide empirical evidence in support of the proposed approach and its consistency with networking reality. To illustrate its utility, we: 1) show that our approach provides insight into the origin of high variability in measured or inferred routerlevel maps; 2) demonstrate that it easily accommodates the incorporation of additional objectives of network design (e.g., robustness to router failure); and 3) discuss how it complements ongoing community efforts to reverseengineer the Internet.
Mathematics and the Internet: A Source of Enormous Confusion and Great Potential
"... For many mathematicians and physicists, the Internet has become a popular realworld domain for the application and/or development of new theories related to the organization and behavior of largescale, complex, and dynamic systems. In some cases, the Internet has served both as inspiration and just ..."
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Cited by 46 (7 self)
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For many mathematicians and physicists, the Internet has become a popular realworld domain for the application and/or development of new theories related to the organization and behavior of largescale, complex, and dynamic systems. In some cases, the Internet has served both as inspiration and justification for the popularization of new models and mathematics within the scientific enterprise. For example, scalefree network models of the preferential attachment type [8] have been claimed to describe the Internet’s connectivity structure, resulting in surprisingly general and strong claims about the network’s resilience to random failures of its components and its vulnerability to targeted attacks against its infrastructure [2]. These models have, as their trademark, powerlaw type node degree distributions that drastically distinguish them from the classical ErdősRényi type random graph models [13]. These “scalefree ” network models have attracted significant attention within the scientific community and have been partly responsible for launching and fueling the new field of network science [42, 4]. To date, the main role that mathematics has played in network science has been to put the physicists’ largely empirical findings on solid grounds Walter Willinger is at AT&T LabsResearch in Florham Park, NJ. His email address is walter@research.att. com.
Internet Connectivity at the ASlevel: An OptimizationDriven Modeling Approach
, 2003
"... Two ASs are connected in the Internet AS graph only if they have a business "peering relationship." By focusing on the AS subgraph ASPC whose links represent providercustomer relationships, we develop a new optimizationdriven model for Internet growth at the ASPC level. The model's ..."
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Cited by 37 (7 self)
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Two ASs are connected in the Internet AS graph only if they have a business "peering relationship." By focusing on the AS subgraph ASPC whose links represent providercustomer relationships, we develop a new optimizationdriven model for Internet growth at the ASPC level. The model's defining feature is an explicit construction of a novel class of intuitive, multiobjective, local optimizations by which the different customer ASs determine in a fully distributed and decentralized fashion their "best" upstream provider AS. Key criteria that are explicitly accounted for in the formulation of these multiobjective optimization problems are (i) ASgeography, i.e., locality and number of PoPs within individual ASs; (ii) ASspecific business models, abstract toy models that describe how individual ASs choose their "best" provider; and (iii) AS evolution, a historic account of the "lives" of individual ASs in a dynamic ISP market. We show that the resulting model is broadly robust, perforce yields graphs that match inferred AS connectivity with respect to a number of different metrics, and is ideal for exploring the impact of new peering incentives or policies on ASlevel connectivity.