Abstract
Given a dense countable set in a metric space, the infinite random geometric graph is the random graph with the given vertex set and where any two points at distance less than 1 are connected, independently, with some fixed probability. It has been observed by Bonato and Janssen that in some, but not all, such settings, the resulting graph does not depend on the random choices, in the sense that it is almost surely isomorphic to a fixed graph. While this notion makes sense in the general context of metric spaces, previous work has been restricted to sets in Banach spaces. We study the case when the underlying metric space is a circle of circumference L, and find a surprising dependency of behaviour on the rationality of L.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Balister, P., Bollobás, B., Gunderson, K., Leader, I., Walters, M.: Random geometric graphs and isometries of normed spaces. Trans. Amer. Math. Soc. 370(10), 7361–7389 (2018). MR 3841851
Bonato, A., Janssen, J.: Infinite random geometric graphs. Ann. Comb. 15(4), 597–617 (2011). MR 2854782
Bonato, A., Janssen, J.: Infinite random geometric graphs from the hexagonal metric. In: Combinatorial Algorithms. Lecture Notes in Computer Science, vol. 7643, pp. 6–19. Springer, Heidelberg (2012). MR 3056370
Bonato, A., Janssen, J.: Infinite random graphs and properties of metrics. In: Recent Trends in Combinatorics, IMA Vol. Math. Appl., vol. 159, pp. 257–273. Springer, Cham (2016). MR 3526412
Bonato, A., Janssen, J., Quas, A.: Geometric random graphs and Rado sets in sequence spaces. Eur. J. Combin. 79, 1–14 (2019). MR 3895055
Bonato, A., Janssen, J., Quas, A.: Geometric random graphs and Rado sets of continuous functions. Discrete Anal. (2021). Paper No. 3, 21. MR 4244349
Cameron, P.J.: The random graph. In: The Mathematics of Paul Erdős, II. Algorithms Combin., vol. 14, pp. 333–351. Springer, Berlin (1997). MR 1425227
Erdős, P., Rényi, A.: Asymmetric graphs. Acta Math. Acad. Sci. Hungar. 14, 295–315 (1963). MR 156334
Rado, R.: Universal graphs and universal functions. Acta Arith. 9, 331–340 (1964). MR 172268
Acknowledgement
This work was supported in part by NSERC of Canada.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Angel, O., Spinka, Y. (2021). Geometric Random Graphs on Circles. In: Hernández‐Hernández, D., Leonardi, F., Mena, R.H., Pardo Millán, J.C. (eds) Advances in Probability and Mathematical Statistics. Progress in Probability, vol 79. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-85325-9_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-85325-9_2
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-85324-2
Online ISBN: 978-3-030-85325-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)