Network Inference from TraceRoute Measurements: Internet Topology   `Species'
  Internet mapping projects generally consist in sampling the network from a limited set of sources by using traceroute probes. This methodology, akin to the merging of spanning trees from the different sources to a set of destinations, leads necessarily to a partial, incomplete map of the Internet. Accordingly, determination of Internet topology characteristics from such sampled maps is in part a problem of statistical inference. Our contribution begins with the observation that the inference of many of the most basic topological quantities -- including network size and degree characteristics -- from traceroute measurements is in fact a version of the so-called `species problem' in statistics. This observation has important implications, since species problems are often quite challenging. We focus here on the most fundamental example of a traceroute internet species: the number of nodes in a network. Specifically, we characterize the difficulty of estimating this quantity through a set of analytical arguments, we use statistical subsampling principles to derive two proposed estimators, and we illustrate the performance of these estimators on networks with various topological characteristics. 
Fabien Viger, Alain Barrat, Luca Dall'Asta, Cun-Hui Zhang, and Eric D.   Kolaczyk