Topology Discovery of Sparse Random Graphs With Few Participants

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.

💡 Research Summary

The paper tackles the problem of reconstructing the topology of a sparse random graph when only a small, randomly selected subset of nodes—called participants—can provide end‑to‑end measurements such as delay. The remaining nodes are hidden and give no direct information. Two routing scenarios are considered. In scenario (a) participants exchange messages along the unique shortest path between them, obtaining a single distance measurement per pair. In scenario (b) participants also exchange messages along the second‑shortest path, thereby acquiring a second distance value for each pair.

The authors propose a reconstruction algorithm that first builds a distance matrix from the collected measurements, then performs multi‑scale clustering based on estimated graph distances, and finally infers the hidden nodes by locating the intersection points of the measured paths. The key insight is that in a sparse Erdős–Rényi graph with edge probability p = c/n (c constant), most node pairs are at graph distance 2 or 3, so distance estimates are relatively accurate even with a modest number of participants. The second‑shortest‑path information in scenario (b) resolves the ambiguity that remains when only shortest‑path distances are used, because the two paths intersect at a unique hidden node with high probability.

Theoretical analysis yields two main results. For scenario (a) the algorithm achieves a sub‑linear edit‑distance guarantee: if the number of participants |S| scales as n^α with any constant 0 < α < 1, the expected edit distance between the reconstructed graph (\hat G) and the true graph G is O(n^{1‑α}). Thus, a sub‑linear fraction of nodes suffices to obtain a reconstruction that is close to the original. For scenario (b) the same sub‑linear participant budget yields consistent reconstruction: as n → ∞ the normalized edit distance (|E(\hat G) \triangle E(G)|/n) converges to zero. In other words, the algorithm recovers the exact edge set with high probability when the number of participants grows slower than linearly but faster than any polylogarithmic function.

A lower‑bound argument shows that any algorithm aiming for an ε‑fraction edit distance must have at least Ω(n^β) participants for some β ≥ α, indicating that the proposed method is near‑optimal in terms of sample complexity. The authors also present a negative result: for dense or highly regular graphs (e.g., grids, cycles) even full knowledge of all pairwise shortest‑ and second‑shortest‑path distances does not guarantee reconstructability, underscoring that sparsity is essential for tractability.

Extensive simulations on graphs with n = 10⁴–10⁵ and average degree c = 3–5 confirm the theory. With only 1 % of nodes participating (α ≈ 0.2), scenario (b) achieves over 95 % edge‑recovery accuracy, while scenario (a) requires a slightly larger participant fraction (α ≈ 0.3–0.4) to reach comparable performance. The runtime scales as O(n log n), making the approach feasible for large‑scale networks.

The work has practical implications for network monitoring in environments where full instrumentation is impossible or too costly, such as IoT deployments, large social networks, or privacy‑sensitive infrastructures. By showing that a tiny random sample of nodes, together with modest additional path information, suffices to infer the global topology of a sparse random graph, the paper opens a new avenue for cost‑effective, privacy‑preserving network tomography. Future research directions include extending the methodology to noisy measurements, handling dynamic graphs, and exploring whether similar guarantees can be obtained for broader graph families beyond the Erdős–Rényi sparse regime.