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.
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 .
@cite_5 @cite_1 @cite_2
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Many systems require information about the topology of networks on the Internet, for purposes like management, efficiency, testing of new protocols and so on. However, ISPs usually do not share the actual topology maps with outsiders; thus, in order to obtain the topology of a network on the Internet, a system must reconstruct it from publicly observable data. The standard method employs traceroute to obtain paths between nodes; next, a topology is generated such that the observed paths occur in the graph. However, traceroute has the problem that some routers refuse to reveal their addresses, and appear as anonymous nodes in traces. Previous research on the problem of topology inference with anonymous nodes has demonstrated that it is at best NP-complete. In this paper, we improve upon this result. In our previous research, we showed that in the special case where nodes may be anonymous in some traces but not in all traces (so all node identifiers are known), there exist trace sets that are generable from multiple topologies. This paper extends our theory of network tracing to the general case (with strictly anonymous nodes), and shows that the problem of computing the network that generated a trace set, given the trace set, has no general solution. The weak version of the problem, which allows an algorithm to output a "small" set of networks- any one of which is the correct one- is also not solvable. Any algorithm guaranteed to output the correct topology outputs at least an exponential number of networks. Our results are surprisingly robust: they hold even when the network is known to have exactly two anonymous nodes, and every node as well as every edge in the network is guaranteed to occur in some trace. On the basis of this result, we suggest that exact reconstruction of network topology requires more powerful tools than traceroute. Computing the topology of a network in the Internet is a problem that has attracted considerable research interest. The usual method is to employ Traceroute, which produces sequences of nodes that occur along the routes from one node (source) to another (destination). In every trace thus produced, a node occurs by either its unique identifier, or by the anonymous identifier "*". We have earlier proved that there exists no algorithm that can take a set of traces produced by running Traceroute on network N and compute one topology which is guaranteed to be the topology of N. This paper proves a much stronger result: no algorithm can produce a small set of topologies that is guaranteed to contain the topology of N, as the size of the solution set is exponentially large. This result holds even when every edge occurs in a trace, all the unique identifiers of all the nodes are known, and the number of nodes that are irregular (anonymous in some traces) is given. On the basis of this strong result, we suggest that efforts to exactly reconstruct network topology should focus on special cases where the solution set is small. In topology recognition, each node of an anonymous network has to deterministically produce an isomorphic copy of the underlying graph, with all ports correctly marked. This task is usually unfeasible without any a priori information. Such information can be provided to nodes as advice. An oracle knowing the network can give a (possibly different) string of bits to each node, and all nodes must reconstruct the network using this advice, after a given number of rounds of communication. During each round each node can exchange arbitrary messages with all its neighbors and perform arbitrary local computations. The time of completing topology recognition is the number of rounds it takes, and the size of advice is the maximum length of a string given to nodes.We investigate tradeoffs between the time in which topology recognition is accomplished and the minimum size of advice that has to be given to nodes. We provide upper and lower bounds on the minimum size of advice that is sufficient to perform topology recognition in a given time, in the class of all graphs of size n and diameter D ? α n , for any constant α < 1 . In most cases, our bounds are asymptotically tight. Topology recognition is one of the fundamental distributed tasks in networks. Each node of an anonymous network has to deterministically produce an isomorphic copy of the underlying graph, with all ports correctly marked. This task is usually unfeasible without any a priori information. Such information can be provided to nodes as advice. An oracle knowing the network can give a possibly different string of bits to each node, and all nodes must reconstruct the network using this advice, after a given number of rounds of communication. During each round each node can exchange arbitrary messages with all its neighbors and perform arbitrary local computations. The time of completing topology recognition is the number of rounds it takes, and the size of advice is the maximum length of a string given to nodes. We investigate tradeoffs between the time in which topology recognition is accomplished and the minimum size of advice that has to be given to nodes. We provide upper and lower bounds on the minimum size of advice that is sufficient to perform topology recognition in a given time, in the class of all graphs of size n and diameter D≤αn, for any constant α<1. In most cases, our bounds are asymptotically tight. More precisely, if the allotted time is D-k, where 0<k≤D, then the optimal size of advice is i¾?n 2 logn D-k+1. If the allotted time is D, then this optimal size is i¾?n logn. If the allotted time is D+k, where 0<k≤D 2, then the optimal size of advice is i¾?1+logn k. The only remaining gap between our bounds is for time D+k, where D 2<k≤D. In this time interval our upper bound remains O1+logn k, while the lower bound that holds for any time is 1. This leaves a gap if D∈ologn. Finally, we show that for time 2D+1, one bit of advice is both necessary and sufficient. Our results show how sensitive is the minimum size of advice to the time allowed for topology recognition: allowing just one round more, from D to D+1, decreases exponentially the advice needed to accomplish this task.