Improved Reconstruction of Random Geometric Graphs

Authors Varsha Dani, Josep Díaz, Thomas P. Hayes, Cristopher Moore

Thumbnail PDF


  • Filesize: 1.28 MB
  • 17 pages

Document Identifiers

Author Details

Varsha Dani
  • Department of Computer Science, Rochester Institute of Technology, Rochester, NY, USA
Josep Díaz
  • Department of Computer Science, Polytechnic University of Catalonia, Barcelona, Spain
Thomas P. Hayes
  • Department of Computer Science, University of New Mexico, Albuquerque, NM, USA
Cristopher Moore
  • Santa Fe Institute, NM, USA

Cite AsGet BibTex

Varsha Dani, Josep Díaz, Thomas P. Hayes, and Cristopher Moore. Improved Reconstruction of Random Geometric Graphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 48:1-48:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Embedding graphs in a geographical or latent space, i.e. inferring locations for vertices in Euclidean space or on a smooth manifold or submanifold, is a common task in network analysis, statistical inference, and graph visualization. We consider the classic model of random geometric graphs where n points are scattered uniformly in a square of area n, and two points have an edge between them if and only if their Euclidean distance is less than r. The reconstruction problem then consists of inferring the vertex positions, up to the symmetries of the square, given only the adjacency matrix of the resulting graph. We give an algorithm that, if r = n^α for α > 0, with high probability reconstructs the vertex positions with a maximum error of O(n^β) where β = 1/2-(4/3)α, until α ≥ 3/8 where β = 0 and the error becomes O(√{log n}). This improves over earlier results, which were unable to reconstruct with error less than r. Our method estimates Euclidean distances using a hybrid of graph distances and short-range estimates based on the number of common neighbors. We extend our results to the surface of the sphere in ℝ³ and to hypercubes in any constant dimension.

Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
  • Reconstruction algorithm
  • distances in RGG
  • d-dimensional hypercube
  • 3 dimensional sphere


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Emmanuel Abbe. Community detection and stochastic block models: Recent developments. Journal of Machine Learning Research, 18(177):1-86, 2018. URL:
  2. Ittai Abraham, Shiri Chechik, David Kempe, and Aleksandrs Slivkins. Low-distortion inference of latent similarities from a multiplex social network. SIAM J. Comput., 44(3):617-668, 2015. Google Scholar
  3. Ernesto Araya Valdivia and De Castro Yohann. Latent distance estimation for random geometric graphs. In H. Wallach, H. Larochelle, A. Beygelzimer, F. dquotesingle Alché-Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 32, 2019. URL:
  4. Ery Arias-Castro, Antoine Channarond, Bruno Pelletier, and Nicolas Verzelen. On the estimation of latent distances using graph distances. Electronic Journal of Statistics, 15(1):722-747, 2021. Google Scholar
  5. J. Aspnes, D.K. Goldenberg, and Y.R. Yang. On the computational complexity of sensor network localization. In Proc. ALGOSENSORS-04, pages 32-44, 2004. Google Scholar
  6. Thomas Bläsius, Tobias Friedrich, Anton Krohmer, and Sören Laue. Efficient embedding of scale-free graphs in the hyperbolic plane. IEEE/ACM Transactions on Networking, 26(2):920-933, 2018. Google Scholar
  7. M. Bradonjic, T. Elsasser, T. Friedrich, T. Sauerwald, and A. Stauffer. Efficient broadcast on random geometric graphs. In 21st. ACM-SIAM SODA, pages 1412-1421, 2010. Google Scholar
  8. H. Breu and D.G. Kirkpatrick. Unit disk graph recognition in np-hard. Compt. Geometry Theory Appl., 9(1-2):3-24, 1998. Google Scholar
  9. J. Bruck, J. Gao, and J. Jiang. On the computational complexity of sensor network localization. In ACM MOBIHOC-05, pages 181-192, 2005. Google Scholar
  10. J. Bubeck, J. Ding, R. Eldan, and M. Racz. Testing for high-dimensional geometry in random graphs. Random Structures and Algorithms, 49(3):503-532, 2016. Google Scholar
  11. Josep Diaz, Colin McDiarmid, and Dieter Mitsche. Learning random points from geometric graphs or orderings. Random Structures & Algorithms, 2019. Google Scholar
  12. Josep Diaz, Dieter Mitsche, Guillem Perarnau, and Xavier Pérez-Giménez. On the relation between graph distance and Euclidean distance in random geometric graphs. Advances in Applied Probability, 48(3):848-864, 2016. Google Scholar
  13. R. Ellis, J.L. Martin, and C. Yan. Random geometric graph diameter in the unit ball. Algorithmica, 47(4):421-438, 2007. Google Scholar
  14. Edward. E. Gilbert. Random plane networks. J. Soc. Industrial Applied Mathematics, 9(5):533-543, 1961. Google Scholar
  15. Peter D. Hoff, Adrian E. Raftery, and Mark S. Handcock. Latent space approaches to social network analysis. Journal of the American Statistical Association, 97:1090+, December 2002. Google Scholar
  16. Maksim Kitsak, Ivan Voitalov, and Dmitri Krioukov. Link prediction with hyperbolic geometry. Phys. Rev. Research, 2:043113, 2020. Google Scholar
  17. Dmitri Krioukov, Fragkiskos Papadopoulos, Maksim Kitsak, Amin Vahdat, and Marián Boguñá. Hyperbolic geometry of complex networks. Phys. Rev. E, 82:036106, September 2010. URL:
  18. Shane Lubold, Arun G. Chandrasekhar, and Tyler H. McCormick. Identifying the latent space geometry of network models through analysis of curvature, 2021. URL:
  19. Cristopher Moore. The computer science and physics of community detection: Landscapes, phase transitions, and hardness. Bull. EATCS, 121, 2017. URL:
  20. Elchanan Mossel, Joe Neeman, and Allan Sly. Reconstruction and estimation in the planted partition model. Probability Theory and Related Fields, 162(3-4):431-461, 2015. Google Scholar
  21. M. Muthukrishnan and G. Pandurangan. The bin-covering technique for thresholding random geometric graph properties,. In 16th. ACM-SIAM SODA, pages 989-998, 2005. Google Scholar
  22. S. Parthasarathy, D. Sivakoff, M. Tian, and Y. Wang. A quest to unravel the metric structure behind perturbed networks. In Proc of the 33rd SoCG, pages 53:1-53:16, 2017. Google Scholar
  23. Mathew Penrose. Random Geometric Graphs. Oxford University Press, 2003. Google Scholar
  24. Purnamrita Sarkar, Deepayan Chakrabarti, and Andrew W Moore. Theoretical justification of popular link prediction heuristics. In Twenty-Second International Joint Conference on Artificial Intelligence, IJCAI'11, pages 2722-2727, 2011. Google Scholar
  25. Cosma Rohilla Shalizi and Dena Asta. Consistency of maximum likelihood for continuous-space network models, 2019. URL:
  26. M. Walters. Random geometric graphs. Surveys in Combinatorics, pages 365-402, 2011. Google Scholar
  27. Yi-Jiao Zhang, Kai-Cheng Yang, and F. Radicchi. Systematic comparison of graph embedding methods in practical tasks, 2021. URL: