Traceroute measurements are one of our main instruments to shed light onto the structure and properties of today's complex networks such as the Internet. This paper studies the feasibility and infeasibility of inferring the network topology given traceroute data from a worst-case perspective, i.e., without any probabilistic assumptions on, e.g., the nodes' degree distribution. We attend to a scenario where some of the routers are anonymous, and propose two fundamental axioms that model two basic assumptions on the traceroute data: (1) each trace corresponds to a real path in the network, and (2) the routing paths are at most a factor 1 alpha off the shortest paths, for some parameter alpha in (0,1]. In contrast to existing literature that focuses on the cardinality of the set of (often only minimal) inferrable topologies, we argue that a large number of possible topologies alone is often unproblematic, as long as the networks have a similar structure. We hence seek to characterize the set of topologies inferred with our axioms. We introduce the notion of star graphs whose colorings capture the differences among inferred topologies; it also allows us to construct inferred topologies explicitly. We find that in general, inferrable topologies can differ significantly in many important aspects, such as the nodes' distances or the number of triangles. These negative results are complemented by a discussion of a scenario where the trace set is best possible, i.e., "complete". It turns out that while some properties such as the node degrees are still hard to measure, a complete trace set can help to determine global properties such as the connectivity.
In contrast to this line of research on cardinalities, we are interested in the . If the inferred topologies share the most important characteristics, the negative results in @cite_3 @cite_7 may be of little concern. Moreover, we believe that a study limited to minimal topologies only may miss important redundancy aspects of the Internet. Unlike @cite_3 @cite_7 , our work is constructive in the sense that algorithms can be derived to compute inferred topologies.
@cite_7 @cite_3
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We calculate an extensive set of characteristics for Internet AS topologies extracted from the three data sources most frequently used by the research community: traceroutes, BGP, and WHOIS. We discover that traceroute and BGP topologies are similar to one another but differ substantially from the WHOIS topology. Among the widely considered metrics, we find that the joint degree distribution appears to fundamentally characterize Internet AS topologies as well as narrowly define values for other important metrics. We discuss the interplay between the specifics of the three data collection mechanisms and the resulting topology views. In particular, we how how the data collection peculiarities explain differences in the resulting joint degree distributions of the respective topologies. Finally, we release to the community the input topology datasets, along with the scripts and output of our calculations. This supplement hould enable researchers to validate their models against real data and to make more informed election of topology data sources for their specific needs Comparing graphs to determine the level of underlying structural similarity between them is a widely encountered problem in computer science. It is particularly relevant to the study of Internet topologies, such as the generation of synthetic topologies to represent the Internet's AS topology. We derive a new metric that enables exactly such a structural comparison: the weighted spectral distribution. We then apply this metric to three aspects of the study of the Internet's AS topology. i) We use it to quantify the effect of changing the mixing properties of a simple synthetic network generator. ii) We use this quantitative understanding to examine the evolution of the Internet's AS topology over approximately seven years, finding that the distinction between the Internet core and periphery has blurred over time. iii) We use the metric to derive optimal parameterizations of several widely used AS topology generators with respect to a large-scale measurement of the real AS topology.