Network Topologies, Power Laws, and Hierarchy Hongsuda Tangmunarunkit USC-ISI Rarnesh Govindan USC-ISI Sugih Jamin Univ. of Michigan Scott Shenker ACIRI Walter Willinger AT&T It has long been thought that the Internet, and its constituent networks, are hierarchies! in nature. Consequently, the network topology generators most widely used by the Internet research community, G T - I T M and Tiers, create networks with a deliberately hierarchical structure. However, recent work by Faloutsos etal. revealed that the Internet's degree distribution - - the distribution of the number of connections touters or Autonomous Systems (ASs) have - - is a power-law. The degree distributions produced by the GTITM and Tiers generators are not power-laws. To rectify this problem, several new network generators have recently been proposed that produce more realistic degree distributions; these new generators do not attempt to create a hierarchical structure but instead focus solely on the degree distribution. There are thus two families of network generators, structural generators that treat hierarchy as fundsmental and degree-based generators that treat the degree distribution as fundamental. The ~uiding principles of the two f~-milies of generators are very different; the degree distribution is a very local property, having to do with the numbers of connections at individual nodes, while hierarchy is very nonlocal and describes the overall structure of the network. In this research 1 we attempt to gain more understanding about the topological structure of the Interuet. Underlying our investigation is the issue of whether the hierarchical structure or the degree distribution is more fundamental in modeling the Interuet. More specifically, we address the following questions: Q u e s t i o n # 1 Which generated networks moat closely resemble the Interact.~ To mmswer this question, we use several topology metrics to compare networks produced by these two fmmilies of generators to current measurements of the l.uternet g r a p h - - t h e router-level (RL) sad Autonomous System (AS) connectivity maps. These metrics are intended to capture some basic larye-acale or overall properties of network structures, as opposed to purely local properties; we make a judgment that the large-scale properties are more fundamental than the local properties. The degree distribution is a rather local property. Even though the structural generators do not match the degree distribution of the In*This work was supported in part by the Defense Advaaced Research Projects Agency under grant F30602-00-2-055. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the Defense Advanced Research Projects Agency. lhttp;//www.isi.edu/~hongsuda/publicatiou contains full papers. ternet topologies, they could nonetheless create networks with large-scale properties that were similar to the Internet graphs. Our results suggest that the AS and RL graphs have similar properties and the degree-based generators are significantly better at representing the Internet t h a n the structural generators. This leaves us with the seeming paradox that while the Internet certainly has hierarchy, it appears that the Internet graphs are better modeled by network generators that completely ignore hierarchy! Resolving this paradox leads us to our second question. Q u e s t i o n ~ 2 Is ~hsre any relationship beb~een hierarchical structure and power-law degree distributions.~ We introduce a novel measure of hierarchy, and use this to investigate the nature of hierarchy in the generated and measured networks. We find that while the degree-based generators do not explicitly inject hierarchy into the network, the powerlaw nature of the degree distribution results in a substautial leve] of hierarchy--not as strict as the hierarchy present in the structural generators, but significantly more hierarchical than random graphs. This relatively loose form of hierarchy, produced merely by the presence of the power-law degree distribution, more accurately reflects the nature of hierarchy in the Iuternet than the strict hierarchy produced by the structural generators. Q u e s t i o n ~ 3 Where does the power.law degree distribu. tion in the A S topology come from.° 2 From an Internet router-level topology, an AS-overlay can be computed. This computation allows us to estimate the AS size distribution, where size is defined by the number of routers in the AS. We find that the distribution of AS sizes exhibits a powerlaw. Moreover, there is a strong correlation between AS size and degree. Based on the ubiquity of highly-variable size distributions in real-world entities such as cities by population size, file size, web document, etc., we conjecture that the power-law degree distribution in the AS topology may simply follow from its power-law Size distribution. The work presented here is preliminary, and is only a first step towards an understanding of the issues we've addressed. Thus far, our metrics and analysis rely on shortest paths on the graph. We plan to incorporate policy constraints into our work especially in the context of measured networks. 2I.u collaboration with John Doyle, Caltech. ACM SIGCOMM Computer Communication Review

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