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 destination, has been argued to introduce uncontrolled sampling biases that might produce statistical properties of the sampled graph which sharply differ from the original ones. In this paper, we explore these biases and provide a statistical analysis of their origin. We derive an 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 depends 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 broad distributions of connectivity. We complement the analytical discussion with a throughout numerical investigation of simulated mapping strategies in network models with different topologies. 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. Moreover, we characterize the level of redundancy and completeness of the exploration process as a function of the topological properties of the network. Finally, we study numerically how the fraction of vertices and edges discovered in the sampled graph depends on the particular deployements of probing sources. The results might hint the steps toward more efficient mapping strategies.
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 conduced 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, we show that the exploration process induces some properties on the maps. We validate our analysis using real-world data and experiments and we discuss its implications. 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.