Abstract
We consider the following dynamic Boolean model introduced by van den Berg et al. (Stoch. Process. Appl. 69:247–257, 1997). At time 0, let the nodes of the graph be a Poisson point process in \({\mathbb{R}^d}\) with constant intensity and let each node move independently according to Brownian motion. At any time t, we put an edge between every pair of nodes whose distance is at most r. We study three fundamental problems in this model: detection (the time until a target point—fixed or moving—is within distance r of some node of the graph); coverage (the time until all points inside a finite box are detected by the graph); and percolation (the time until a given node belongs to the infinite connected component of the graph). We obtain precise asymptotics for these quantities by combining ideas from stochastic geometry, coupling and multi-scale analysis.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alon N., Spencer J.H.: The Probabilistic Method, 3rd edn. Wiley, New York (2008)
Berezhovskii A.M., Makhovskii Yu.A., Suris R.A.: Wiener sausage volume moments. J. Stat. Phys. 57, 333–346 (1989)
van den Berg J., Meester R., White D.G.: Dynamic boolean model. Stoch. Process. Appl. 69, 247–257 (1997)
Ciesielski Z., Taylor S.J.: First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path. Trans. Am. Math. Soc. 103, 434–450 (1962)
Clementi, A., Pasquale, F., Silvestri, R.: MANETS: high mobility can make up for low transmission power. In: Proceedings of the 36th International Colloquium on Automata, Languages and Programming (ICALP) (2009)
Drewitz, A., Gärtner, J., Ramírez, A.F., Sun, R.: Survival probability of a random walk among a poisson system of moving traps (2010). ar**v:1010.3958v1
Gupta, P., Kumar, P.R.: Critical power for asymptotic connectivity in wireless networks. In: McEneany, W.M., Yin, G., Zhang, Q. (eds.) Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W.H. Fleming, pp. 547–566. Birkhäuser, Boston (1998)
Gupta, P., Kumar, P.R.: The capacity of wireless networks. IEEE Transactions on Information Theory, 46, 388–404 (2000) [correction in IEEE Transactions on Information Theory 49, p. 3117 (2000)]
Kesidis, G., Konstantopoulos, T., Phoha, S.: Surveillance coverage of sensor networks under a random mobility strategy. In: Proceedings of the 2nd IEEE International Conference on Sensors (2003)
Kesten H., Sidoravicius V.: The spread of a rumor or infection in a moving population. Ann. Probab. 33, 2402–2462 (2005)
Konstantopoulos, T.: Response to Prof. Baccelli’s lecture on modelling of wireless communication networks by stochastic geometry. Comput. J. Adv. Access (2009)
Lam, H., Liu, Z., Mitzenmacher, M., Sun, X., Wang, Y.: Information dissemination via random walks in d-dimensional space (2011). ar**v:1104.5268
Liu, B., Brass, P., Dousse, O., Nain, P., Towsley, D.: Mobility improves coverage of sensor networks. In: Proceedings of the 6th ACM International Conference on Mobile Computing and Networking (MobiCom) (2005)
Mattila P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge (1995)
Meester R., Roy R.: Continuum Percolation. Cambridge University Press, Cambridge (1996)
Moreau, M., Oshanin, G., Bénichou, O., Coppey, M.: Lattice theory of trap** reactions with mobile species. Phys. Rev. E 69 (2004)
Mörters P., Peres Y.: Brownian Motion. Cambridge University Press, Cambridge (2010)
Penrose M.: The longest edge of the random minimal spanning tree. Ann. Appl. Probab. 7, 340–361 (1997)
Penrose M.: On k-connectivity for a geometric random graph. Random Struct. Algorithms 15, 145–164 (1999)
Penrose M.: Random Geometric Graphs. Oxford University Press, Oxford (2003)
Penrose M., Pisztora A.: Large deviations for discrete and continuous percolation. Adv. Appl. Probab. 28, 29–52 (1996)
Pettarin, A., Pietracaprina, A., Pucci, G., Upfal, E.: Infectious random walks (2010). ar**v:1007.1604
Sinclair, A., Stauffer, A.: Mobile geometric graphs, and detection and communication problems in mobile wireless networks (2010). ar**v:1005.1117v1
Spitzer F.: Electrostatic capacity, heat flow, and Brownian motion. Z. Wahrscheinlichkeitstheorie verw. Geb. 3, 110–121 (1964)
Stoyan D., Kendall W.S., Mecke J.: Stochastic Geometry and its Applications, 2nd edn. Wiley, New York (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
A. Sinclair research supported in part by NSF grant CCF-0635153 and by a UC Berkeley Chancellor’s Professorship.
A. Stauffer research supported by a Fulbright/CAPES scholarship and NSF grants CCF-0635153 and DMS-0528488. Part of this work was done while the author was doing a summer internship at Microsoft Research, Redmond, WA.
Rights and permissions
About this article
Cite this article
Peres, Y., Sinclair, A., Sousi, P. et al. Mobile geometric graphs: detection, coverage and percolation. Probab. Theory Relat. Fields 156, 273–305 (2013). https://doi.org/10.1007/s00440-012-0428-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-012-0428-1