text: 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.
  @cite_5: 
  @cite_1: We consider using traceroute-like end-to-end measurement to infer the underlay topology for a group of hosts. One major issue is the measurement cost. Given N hosts in an asymmetric network without anonymous routers, traditionally full N(N-1) traceroutes are needed to determine the underlay topology. We investigate how to efficiently infer an underlay topology with low measurement cost, and propose a heuristic called Max-Delta. In the heuristic, a server selects appropriate host-pairs to measure in each iteration so as to reveal the most undiscovered information on the underlay. We further observe that the presence of anonymous routers significantly distorts and inflates the inferred topology. Previous research has shown that obtaining both exact and approximate topology in the presence of anonymous routers under certain consistency constraints is intractable. We hence propose fast algorithms on how to practically construct an approximate topology by relaxing some constraints. We investigate and compare two algorithms to merge anonymous routers. The first one uses Isomap to map routers into a multidimensional space and merges anonymous routers according to their interdistances. The second algorithm is based on neighbor router information, which trades off some accuracy with speed. We evaluate our inference algorithms on Internet-like and real Internet topologies. Our results show that almost full measurement is needed to fully discover the underlay topology. However, substantial reduction in measurements can be achieved if a little accuracy, say 5 , can be compromised. Moreover, our merging algorithms in the presence of anonymous routers can efficiently infer an underlay topology with good accuracy
  @cite_2: Many topology discovery systems rely on traceroute to discover path information in public networks. However, for some routers, traceroute detects their existence but not their address; we term such routers anonymous routers. This paper considers the problem of inferring the network topology in the presence of anonymous routers. We illustrate how obvious approaches to handle anonymous routers lead to incomplete, inflated, or inaccurate topologies. We formalize the topology inference problem and show that producing both exact and approximate solutions is intractable. Two heuristics are proposed and evaluated through simulation. These heuristics have been used to infer the topology of the 6Bone, and could be incorporated into existing tools to infer more comprehensive and accurate topologies.
summary: This paper attends to the problem of anonymous nodes and assumes a conservative, worst-case'' perspective that does not rely on any assumptions on the underlying network. There are already several works on the subject. @cite_2 initiated the study of possible candidate topologies for a given trace set and suggested computing the , that is, the topology with the minimal number of anonymous nodes, which turns out to be NP-hard. Consequently, different heuristics have been proposed @cite_5 @cite_1 .