Reconstruction of geometric random graphs with the Simple algorithm

TL;DR

This paper extends the Simple graph reconstruction algorithm to geometric random graphs, achieving near-optimal query complexity proportional to the number of edges, and demonstrates effective partial reconstruction with fewer queries.

Contribution

It adapts the Simple algorithm for geometric random graphs with diverging degrees, providing near-edge-optimal query complexity and partial reconstruction guarantees.

Findings

Query complexity is $ ilde{O}(n^{2k+1})$ for geometric graphs with radius $r o n^k$.

At least 75% of non-edges can be reconstructed with $n^{1+o(1)}$ queries.

Simulations confirm the query complexity aligns with the number of edges.

Abstract

Graph reconstruction can efficiently detect the underlying topology of massive networks such as the Internet. Given a query oracle and a set of nodes, the goal is to obtain the edge set by performing as few queries as possible. An algorithm for graph reconstruction is the Simple algorithm (Mathieu & Zhou, 2023), which reconstructs bounded-degree graphs in $\tilde{O}(n^{3/2})$ queries. We extend the use of this algorithm to the class of geometric random graphs with connection radius $r \sim n^k$, with diverging average degree. We show that for this class of graphs, the query complexity is $\tilde{O}(n^{2k+1})$ when k > 3/20. This query complexity is up to a polylog(n) term equal to the number of edges in the graph, which means that the reconstruction algorithm is almost edge-optimal. We also show that with only $n^{1+o(1)}$ queries it is already possible to reconstruct at least 75% of the non-edges of a geometric random graph, in both the sparse and dense setting. Finally, we show that the number of queries is indeed of the same order as the number of edges on the basis of simulations.