Traceroute sampling is an important technique in exploring the internet router graph and the autonomous system graph. Although it is one of the primary techniques used in calculating statistics about the internet, it can introduce bias that corrupts these estimates. This paper reports on a theoretical and experimental investigation of a new technique to reduce the bias of traceroute sampling when estimating the degree distribution. We develop a new estimator for the degree of a node in a traceroute-sampled graph; validate the estimator theoretically in Erdos-Renyi graphs and, through computer experiments, for a wider range of graphs; and apply it to produce a new picture of the degree distribution of the autonomous system graph.
In addition to traceroute sampling, maps of the AS graph have been generated in two different ways, using BGP tables and using the WHOIS database. A recent paper by Mahadevan, Krioukov, Fomenkov, Dimitropoulos, claffy, and Vahdat provides a detailed comparison of the graphs that result from each of these measurement techniques @cite_6 .
@cite_6
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Despite great effort spent measuring topological features of large networks like the Internet, it was recently argued that sampling based on taking paths through the network (e.g., traceroutes) introduces a fundamental bias in the observed degree distribution. We examine this bias analytically and experimentally. For classic random graphs with mean degree c, we show analytically that traceroute sampling gives an observed degree distribution P(k) 1 k for k < c, even though the underlying degree distribution is Poisson. For graphs whose degree distributions have power-law tails P(k) k^-alpha, the accuracy of traceroute sampling is highly sensitive to the population of low-degree vertices. In particular, when the graph has a large excess (i.e., many more edges than vertices), traceroute sampling can significantly misestimate alpha. Mapping the Internet generally consists in sampling the network from a limited set of sources by using "traceroute"-like probes. This methodology, akin to the merging of different spanning trees to a set of destinations, has been argued to introduce uncontrolled sampling biases that might produce statistical properties of the sampled graph which sharply differ from the original ones. Here we explore these biases and provide a statistical analysis of their origin. We derive a mean-field analytical approximation for the probability of edge and vertex detection that exploits the role of the number of sources and targets and allows us to relate the global topological properties of the underlying network with the statistical accuracy of the sampled graph. In particular we find that the edge and vertex detection probability is depending on the betweenness centrality of each element. This allows us to show that shortest path routed sampling provides a better characterization of underlying graphs with scale-free topology. We complement the analytical discussion with a throughout numerical investigation of simulated mapping strategies in different network models. We show that sampled graphs provide a fair qualitative characterization of the statistical properties of the original networks in a fair range of different strategies and exploration parameters. The numerical study also allows the identification of intervals of the exploration parameters that optimize the fraction of nodes and edges discovered in the sampled graph. This finding might hint the steps toward more efficient mapping strategies. Considerable attention has been focused on the properties of graphs derived from Internet measurements. Router-level topologies collected via traceroute-like methods have led some to conclude that the router graph of the Internet is well modeled as a power-law random graph. In such a graph, the degree distribution of nodes follows a distribution with a power-law tail. We argue that the evidence to date for this conclusion is at best insufficient We show that when graphs are sampled using traceroute-like methods, the resulting degree distribution can differ sharply from that of the underlying graph. For example, given a sparse Erdos-Renyi random graph, the subgraph formed by a collection of shortest paths from a small set of random sources to a larger set of random destinations can exhibit a degree distribution remarkably like a power-law. We explore the reasons for how this effect arises, and show that in such a setting, edges are sampled in a highly biased manner. This insight allows us to formulate tests for determining when sampling bias is present. When we apply these tests to a number of well-known datasets, we find strong evidence for sampling bias. Traceroute is widely used to detect routing problems, characterize end-to-end paths, and discover the Internet topology. Providing an accurate list of the Autonomous Systems (ASes) along the forwarding path would make traceroute even more valuable to researchers and network operators. However, conventional approaches to mapping traceroute hops to AS numbers are not accurate enough. Address registries are often incomplete and out-of-date. BGP routing tables provide a better IP-to-AS mapping, though this approach has significant limitations as well. Based on our extensive measurements, about 10 of the traceroute paths have one or more hops that do not map to a unique AS number, and around 15 of the traceroute AS paths have an AS loop. In addition, some traceroute AS paths have extra or missing AS hops due to Internet eXchange Points, sibling ASes managed by the same institution, and ASes that do not advertise routes to their infrastructure. Using the BGP tables as a starting point, we propose techniques for improving the IP-to-AS mapping as an important step toward an AS-level traceroute tool. Our algorithms draw on analysis of traceroute probes, reverse DNS lookups, BGP routing tables, and BGP update messages collected from multiple locations. We also discuss how the improved IP-to-AS mapping allows us to home in on cases where the BGP and traceroute AS paths differ for legitimate reasons.