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Reconstruction of geometric random graphs with the Simple algorithm

Clara Stegehuis, Lotte Weedage

TL;DR

This work extends the Simple graph reconstruction algorithm to geometric random graphs with growing average degree, showing that query complexity scales as $\tilde{O}(n^{2k+1})$ for $k>3/20$ and $\tilde{O}(n^{3/2-4k/3})$ for $0<k<3/20$ when $r\sim n^k$, with $r=o(\sqrt{n})$ yielding $\tilde{O}(n^{3/2})$; a seed set of size $|S|=\log(n)n^{\epsilon}$ suffices to identify at least $75\%$ of the non-edges, and simulations corroborate that the reconstruction cost is close to the edge count. The approach leverages the geometry to bound graph distances with Euclidean distances via a two-step argument: constructing nearby seeds and distinguishing distant node pairs, enabling near-optimal reconstruction in dense GRGs. These results provide the first query-complexity bounds for reconstructing dense graphs with unbounded maximum degree and highlight how geometric structure accelerates reconstruction. The findings have implications for topology discovery in large networks and motivate extensions to other geometric models, such as hyperbolic random graphs, and to approximate or verified reconstruction tasks.

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.

Reconstruction of geometric random graphs with the Simple algorithm

TL;DR

This work extends the Simple graph reconstruction algorithm to geometric random graphs with growing average degree, showing that query complexity scales as for and for when , with yielding ; a seed set of size suffices to identify at least of the non-edges, and simulations corroborate that the reconstruction cost is close to the edge count. The approach leverages the geometry to bound graph distances with Euclidean distances via a two-step argument: constructing nearby seeds and distinguishing distant node pairs, enabling near-optimal reconstruction in dense GRGs. These results provide the first query-complexity bounds for reconstructing dense graphs with unbounded maximum degree and highlight how geometric structure accelerates reconstruction. The findings have implications for topology discovery in large networks and motivate extensions to other geometric models, such as hyperbolic random graphs, and to approximate or verified reconstruction tasks.

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 queries. We extend the use of this algorithm to the class of geometric random graphs with connection radius , with diverging average degree. We show that for this class of graphs, the query complexity is 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 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.
Paper Structure (10 sections, 9 theorems, 41 equations, 8 figures, 1 algorithm)

This paper contains 10 sections, 9 theorems, 41 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Query oracles.
  4. Graph reconstruction algorithms.
  5. Graph embedding.
  6. Model and main results
  7. Proof of Theorem \ref{['thm:main']}
  8. Proof of Theorem \ref{['lem:connection_to_theorem']}
  9. Simulations
  10. Conclusion

Key Result

Theorem 1

There exist absolute constants $C_1, C_2$ and $c$ such that, for all $n \geq 1$, all $r \geq C_1\sqrt{\log n}$, with probability at least $1 - C_2/n^2$, all pairs of nodes $u,v$ satisfy where

Figures (8)

  • Figure 1: Regions in which the four seeds are located.
  • Figure 2: Worst case location of the furthest seed $s$ relative to $v$.
  • Figure 3: Optimal circle packing for 4 circles on a flat torus.
  • Figure 4: Simulated query complexity of the Simple algorithm for a GRG with $r \sim n^{k}$, $k = 0.1$ and $k = 0.3$. The upper dashed line represents the theoretical query complexity (up to a constant) from Theorem \ref{['thm:main']}.
  • Figure 5: Simulated query complexity of the Simple algorithm for a GRG with $r \sim n^{0.3}$ on a torus and on a square. The upper dashed line represents the theoretical query complexity $n^{2k+1}$ (up to a constant) from Theorem \ref{['thm:main']} and the lower dashed line is the number of edges in the toroidal GRG.
  • ...and 3 more figures

Theorems & Definitions (18)

  • Definition 1: distinguishable
  • Theorem 1: Theorem 3.9 from dani2023reconstruction
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Lemma 2
  • proof : Proof
  • ...and 8 more