Topology Discovery of Sparse Random Graphs With Few Participants

TL;DR

This paper investigates the problem of reconstructing sparse random graph topologies using limited end-to-end measurements from a small subset of nodes, providing algorithms with guarantees and establishing fundamental lower bounds.

Contribution

It introduces algorithms for topology discovery with sub-linear participant requirements and proves bounds on the minimal number of participants needed for accurate reconstruction.

Findings

Sub-linear participants suffice for accurate topology recovery in certain models.

The proposed algorithms achieve a sub-linear edit-distance guarantee.

There are graph classes that cannot be reconstructed regardless of participant number.

Abstract

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.