## The Annals of Applied Probability

### A geometric Achlioptas process

#### Abstract

The random geometric graph is obtained by sampling $n$ points from the unit square (uniformly at random and independently), and connecting two points whenever their distance is at most $r$, for some given $r=r(n)$. We consider the following variation on the random geometric graph: in each of $n$ rounds in total, a player is offered two random points from the unit square, and has to select exactly one of these two points for inclusion in the evolving geometric graph.

We study the problem of avoiding a linear-sized (or “giant”) component in this setting. Specifically, we show that for any $r\ll(n\log\log n)^{-1/3}$ there is a strategy that succeeds in keeping all component sizes sublinear, with probability tending to one as $n\to\infty$. We also show that this is tight in the following sense: for any $r\gg(n\log\log n)^{-1/3}$, the player will be forced to create a component of size $(1-o(1))n$, no matter how he plays, again with probability tending to one as $n\to\infty$. We also prove that the corresponding offline problem exhibits a similar threshold behaviour at $r(n)=\Theta(n^{-1/3})$.

These findings should be compared to the existing results for the (ordinary) random geometric graph: there a giant component arises with high probability once $r$ is of order $n^{-1/2}$. Thus, our results show, in particular, that in the geometric setting the power of choices can be exploited to a much larger extent than in the classical Erdős–Rényi random graph, where the appearance of a giant component can only be delayed by a constant factor.

#### Article information

Source
Ann. Appl. Probab., Volume 25, Number 6 (2015), 3295-3337.

Dates
Revised: May 2014
First available in Project Euclid: 1 October 2015

https://projecteuclid.org/euclid.aoap/1443703775

Digital Object Identifier
doi:10.1214/14-AAP1074

Mathematical Reviews number (MathSciNet)
MR3404637

Zentralblatt MATH identifier
1326.05143

#### Citation

Müller, Tobias; Spöhel, Reto. A geometric Achlioptas process. Ann. Appl. Probab. 25 (2015), no. 6, 3295--3337. doi:10.1214/14-AAP1074. https://projecteuclid.org/euclid.aoap/1443703775

#### References

• [1] Achlioptas, D., D’Souza, R. M. and Spencer, J. (2009). Explosive percolation in random networks. Science 323 1453–1455.
• [2] Azar, Y., Broder, A. Z., Karlin, A. R. and Upfal, E. (1994). Balanced allocations. In Proceedings of the Twenty-Sixth Annual ACM Symposium on Theory of Computing 593–602. ACM, New York.
• [3] Azar, Y., Broder, A. Z., Karlin, A. R. and Upfal, E. (1999). Balanced allocations. SIAM J. Comput. 29 180–200.
• [4] Balogh, J., Bollobás, B., Krivelevich, M., Müller, T. and Walters, M. (2011). Hamilton cycles in random geometric graphs. Ann. Appl. Probab. 21 1053–1072.
• [5] Bohman, T. and Frieze, A. (2001). Avoiding a giant component. Random Structures Algorithms 19 75–85.
• [6] Bohman, T., Frieze, A. and Wormald, N. C. (2004). Avoidance of a giant component in half the edge set of a random graph. Random Structures Algorithms 25 432–449.
• [7] Bohman, T. and Kim, J. H. (2006). A phase transition for avoiding a giant component. Random Structures Algorithms 28 195–214.
• [8] Bohman, T. and Kravitz, D. (2006). Creating a giant component. Combin. Probab. Comput. 15 489–511.
• [9] Cooper, C. and Frieze, A. (2009). The cover time of random geometric graphs. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms 48–57. SIAM, Philadelphia, PA.
• [10] Flaxman, A. D., Gamarnik, D. and Sorkin, G. B. (2005). Embracing the giant component. Random Structures Algorithms 27 277–289.
• [11] Gilbert, E. N. (1961). Random plane networks. J. Soc. Indust. Appl. Math. 9 533–543.
• [12] Gonnet, G. H. (1981). Expected length of the longest probe sequence in hash code searching. J. ACM 28 289–304.
• [13] Haenggi, M., Andrews, J., Baccelli, F., Dousse, O. and Franceschetti, M. (2009). Stochastic geometry and random graphs for the analysis and design of wireless networks. IEEE Journal on Selected Areas in Communications 27 1029–1046.
• [14] Higham, D., Rasajski, M. and Przulj, N. (2008). Fitting a geometric graph to a protein–protein interaction network. Bioinformatics 24 1093–1099.
• [15] Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs. Wiley, New York.
• [16] Kingman, J. F. C. (1993). Poisson Processes. Oxford Studies in Probability 3. Oxford Univ. Press, New York.
• [17] Krivelevich, M., Loh, P.-S. and Sudakov, B. (2009). Avoiding small subgraphs in Achlioptas processes. Random Structures Algorithms 34 165–195.
• [18] Krivelevich, M., Lubetzky, E. and Sudakov, B. (2010). Hamiltonicity thresholds in Achlioptas processes. Random Structures Algorithms 37 1–24.
• [19] Krivelevich, M. and Spöhel, R. (2012). Creating small subgraphs in Achlioptas processes with growing parameter. SIAM J. Discrete Math. 26 670–686.
• [20] McDiarmid, C. and Müller, T. (2011). On the chromatic number of random geometric graphs. Combinatorica 31 423–488.
• [21] Mitzenmacher, M., Richa, A. W. and Sitaraman, R. (2001). The power of two random choices: A survey of techniques and results. In Handbook of Randomized Computing, Vol. I, II. Comb. Optim. 9 255–312. Kluwer Academic, Dordrecht.
• [22] Müller, T. (2008). Two-point concentration in random geometric graphs. Combinatorica 28 529–545.
• [23] Mütze, T., Spöhel, R. and Thomas, H. (2011). Small subgraphs in random graphs and the power of multiple choices. J. Combin. Theory Ser. B 101 237–268.
• [24] Penrose, M. (2003). Random Geometric Graphs. Oxford Studies in Probability 5. Oxford Univ. Press, Oxford.
• [25] Penrose, M. D. (1999). On $k$-connectivity for a geometric random graph. Random Structures Algorithms 15 145–164.
• [26] Riordan, O. and Warnke, L. (2012). Achlioptas process phase transitions are continuous. Ann. Appl. Probab. 22 1450–1464.
• [27] Spencer, J. and Wormald, N. (2007). Birth control for giants. Combinatorica 27 587–628.
• [28] Thiedmann, R., Manke, I., Lehnert, W. and Schmidt, V. (2011). Random geometric graphs for modelling the pore space of fibre-based materials. Journal of Materials Science 46 7745–7759.
• [29] Thliveris, A., Halberg, R., Clipson, L., Dove, W., Sullivan, R., Washington, M., Stanhope, S. and Newton, M. (2005). Polyclonality of familial murine adenomas: Analyses of mouse chimeras with low tumor multiplicity suggest short-range interactions. Proc. Natl. Acad. Sci. USA 102 6960–6965.