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 the field of , topologies are explored using pairwise end-to-end measurements, without the cooperation of nodes along these paths. This approach is quite flexible and applicable in various contexts, e.g., in social networks @cite_6 . For a good discussion of this approach as well as results for a routing model along shortest and second shortest paths see @cite_6 . For example, @cite_6 shows that for sparse random graphs, a relatively small number of cooperating participants is sufficient to discover a network fairly well.
@cite_6
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We consider the task of topology discovery of sparse random graphs using end-to-end random measurements (e.g., delay) between a subset of nodes, referred to as the participants. The rest of the nodes are hidden, and do not provide any information for topology discovery. We consider topology discovery under two routing models: (a) the participants exchange messages along the shortest paths and obtain end-to-end measurements, and (b) additionally, the participants exchange messages along the second shortest path. For scenario (a), our proposed algorithm results in a sub-linear edit-distance guarantee using a sub-linear number of uniformly selected participants. For scenario (b), we obtain a much stronger result, and show that we can achieve consistent reconstruction when a sub-linear number of uniformly selected nodes participate. This implies that accurate discovery of sparse random graphs is tractable using an extremely small number of participants. We finally obtain a lower bound on the number of participants required by any algorithm to reconstruct the original random graph up to a given edit distance. We also demonstrate that while consistent discovery is tractable for sparse random graphs using a small number of participants, in general, there are graphs which cannot be discovered by any algorithm even with a significant number of participants, and with the availability of end-to-end information along all the paths between the participants.  2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013