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.
Internet mapping by traceroute sampling was pioneered by Pansiot and Grad in @cite_10 , and the scale-free nature of the degree distribution was observed by Faloutsos, Faloutsos, and Faloutsos in @cite_21 . Since 1998, the Cooperative Association for Internet Data Analysis (CAIDA) project has archived traceroute data that is collected daily @cite_17 . The bias introduced by traceroute sampling was identified in computer experiments by Lakhina, Byers, Crovella, and Xie in @cite_4 and Petermann and De Los Rios @cite_16 , and formally proven to hold in a model of one-monitor, all-target traceroute sample by Clauset and Moore @cite_2 and, in further generality, by Achlioptas, Clauset, Kempe, and Moore @cite_1 . Computer experiments by to Guillaume, Latapy, and Magoni @cite_14 and an analysis using the mean field approximation of statistical physics due to Dall'Asta, Alvarez-Hamelin, Barrat, V ' a zquez, and Vespignani @cite_11 argue that, despite the bias introduced by traceroute sampling, some sort of scale-free behavior can be inferred from the union of traceroute-sampled paths.
@cite_14 @cite_4 @cite_11 @cite_21 @cite_1 @cite_2 @cite_16 @cite_10 @cite_17
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Internet maps are generally constructed using the traceroute tool from a few sources to many destinations. It appeared recently that this exploration process gives a partial and biased view of the real topology, which leads to the idea of increasing the number of sources to improve the quality of the maps. In this paper, we present a set of experiments we have conducted to evaluate the relevance of this approach. It appears that the statistical properties of the underlying network have a strong influence on the quality of the obtained maps, which can be improved using massively distributed explorations. Conversely, some statistical properties are very robust, and so the known values for the Internet may be considered as reliable. We validate our analysis using real-world data and experiments, and we discuss its implications. 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. 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. Understanding the structure of the Internet graph is a crucial step for building accurate network models and designing efficient algorithms for Internet applications. Yet, obtaining its graph structure is a surprisingly difficult task, as edges cannot be explicitly queried. Instead, empirical studies rely on traceroutes to build what are essentially single-source, all-destinations, shortest-path trees. These trees only sample a fraction of the network's edges, and a recent paper by found empirically that the resuting sample is intrinsically biased. For instance, the observed degree distribution under traceroute sampling exhibits a power law even when the underlying degree distribution is Poisson.In this paper, we study the bias of traceroute sampling systematically, and, for a very general class of underlying degree distributions, calculate the likely observed distributions explicitly. To do this, we use a continuous-time realization of the process of exposing the BFS tree of a random graph with a given degree distribution, calculate the expected degree distribution of the tree, and show that it is sharply concentrated. As example applications of our machinery, we show how traceroute sampling finds power-law degree distributions in both δ-regular and Poisson-distributed random graphs. Thus, our work puts the observations of on a rigorous footing, and extends them to nearly arbitrary degree distributions. Department of Physics and Astronomy,University of New Mexico, Albuquerque NM 87131(aaron,moore)@cs.unm.edu(Dated: February 2, 2008)A great deal of eﬀort has been spent measuring topological features of the Internet. However, itwas recently argued that sampling based on taking paths or traceroutes through the network froma small number of sources introduces a fundamental bias in the observed degree distribution. Weexamine this bias analytically and experimentally. For Erd˝os-R´enyirandom graphs with mean degreec, we show analytically that traceroute sampling gives an observed degree distribution P(k) ∼ k The increased availability of data on real networks has favoured an explosion of activity in the elaboration of models able to reproduce both qualitatively and quantitatively the measured properties. What has been less explored is the reliability of the data, and whether the measurement technique biases them. Here we show that tree-like explorations (similar in principle to traceroute) can indeed change the measured exponents of a scale-free network. Multicasting has an increasing importance for network applications such as groupware or videoconferencing. Several multicast routing protocols have been defined. However they cannot be used directly in the Internet since most inter-domain routers do no implement multicasting. Thus these protocols are mainly tested either on a small scale inside a domain, or through the Mbone, whose topology is not really the same as Internet topology. The purpose of this paper is to construct a graph using actual routes of the Internet, and then to use this graph to compare some parameters - delays, scaling in term of state or traffic concentration - of multicast routing trees constructed by different algorithms - source shortest path trees and shared trees.