Abstract
We consider countably infinite random geometric graphs, whose vertices are points in ℝn, and edges are added independently with probability p ∈ (0,1) if the metric distance between the vertices is below a given threshold. Assume that the vertex set is randomly chosen and dense in ℝn. We address the basic question: for what metrics is there a unique isomorphism type for graphs resulting from this random process? It was shown in [7] that a unique isomorphism type occurs for the L ∞ -metric for all n ≥ 1. The hexagonal metric is a convex polyhedral distance function on ℝ2, which has the property that its unit balls tile the plane, as in the case of the L ∞ -metric. We may view the hexagonal metric as an approximation of the Euclidean metric, and it arises in computational geometry. We show that the random process with the hexagonal metric does not lead to a unique isomorphism type.
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References
Aiello, W., Bonato, A., Cooper, C., Janssen, J., Prałat, P.: A spatial web graph model with local influence regions. Internet Mathematics 5, 175–196 (2009)
Balister, P., Bollobás, B., Sarkar, A., Walters, M.: Highly connected random geometric graphs. Discrete Applied Mathematics 157, 309–320 (2009)
Barbeau, M., Kranakis, E.: Principles of Ad Hoc Networking. John Wiley and Sons (2007)
Barequet, G., Dickerson, M.T., Goodrich, M.T.: Voronoi diagrams for convex polygon-offset distance functions. Discrete & Computational Geometry 25, 271–291 (2001)
Bonato, A.: A Course on the Web Graph. American Mathematical Society Graduate Studies Series in Mathematics, Providence, Rhode Island (2008)
Bonato, A., Janssen, J., Prałat, P.: The Geometric Protean Model for On-Line Social Networks. In: Kumar, R., Sivakumar, D. (eds.) WAW 2010. LNCS, vol. 6516, pp. 110–121. Springer, Heidelberg (2010)
Bonato, A., Janssen, J.: Infinite random geometric graphs. Annals of Combinatorics 15, 597–617 (2011)
Borgs, C., Chayes, J., Lovász, L., Sós, V.T., Vesztergombi, K.: Convergent Sequences of Dense Graphs I: Subgraph Frequencies, Metric Properties and Testing. Advances in Math. 219, 1801–1851 (2008)
Borgs, C., Chayes, J., Lovász, L., Sós, V.T., Vesztergombi, K.: Convergent Sequences of Dense Graphs II: Multiway Cuts and Statistical Physics, Preprint (2007)
Bryant, V.: Metric Spaces: Iteration and Application. Cambridge University Press, Cambridge (1985)
Cameron, P.J.: The random graph. In: Graham, R.L., Nešetřil, J. (eds.) Algorithms and Combinatorics, vol. 14, pp. 333–351. Springer, New York (1997)
Cameron, P.J.: The random graph revisited. In: Casacuberta, C., Miró-Roig, R.M., Verdera, J., Xambó-Descamps, S. (eds.) European Congress of Mathematics, vol. I, pp. 267–274. Birkhauser, Basel (2001)
Chew, L.P., Kedem, K., Sharir, M., Tagansky, B., Welzl, E.: Voronoi diagrams of lines in three dimensions under polyhedral convex distance functions. J. Algorithms 29, 238–255 (1998)
Ellis, R., Jia, X., Yan, C.H.: On random points in the unit disk. Random Algorithm and Structures 29, 14–25 (2006)
Erdős, P., Rényi, A.: Asymmetric graphs. Acta Mathematica Academiae Scientiarum Hungaricae 14, 295–315 (1963)
Flaxman, A., Frieze, A.M., Vera, J.: A geometric preferential attachment model of networks. Internet Mathematics 3, 187–205 (2006)
Frieze, A.M., Kleinberg, J., Ravi, R., Debany, W.: Line of sight networks. Combinatorics, Probability and Computing 18, 145–163 (2009)
Fu, N., Imai, H., Moriyama, S.: Voronoi diagrams on periodic graphs. In: Proceedings of the International Symposium on Voronoi Diagrams in Science and Engineering (2010)
Goel, A., Rai, S., Krishnamachari, B.: Monotone properties of random geometric graphs have sharp thresholds. Annals of Applied Probability 15, 2535–2552 (2005)
Icking, C., Klein, R., Le, N., Ma, L.: Convex distance functions in 3-space are different. In: Proceedings of the Ninth Annual Symposium on Computational Geometry (1993)
Janssen, J.: Spatial models for virtual networks. In: Proceedings of the 6th Computability in Europe (2010)
Lovász, L., Szegedy, B.: Limits of dense graph sequences. J. Comb. Theory B 96, 933–957 (2006)
Penrose, M.: Random Geometric Graphs. Oxford University Press, Oxford (2003)
Stojmenovic, I.: Honeycomb networks: Topological properties and communication algorithms. IEEE Transactions on Parallel and Distributed Systems 8, 1036–1042 (1997)
West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice-Hall (2001)
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Bonato, A., Janssen, J. (2012). Infinite Random Geometric Graphs from the Hexagonal Metric. In: Arumugam, S., Smyth, W.F. (eds) Combinatorial Algorithms. IWOCA 2012. Lecture Notes in Computer Science, vol 7643. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35926-2_2
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