Large Connectivity for Dynamic Random Geometric Graphs

We provide the first rigorous analytical results for the connectivity of dynamic random geometric graphs - a model for mobile wireless networks in which vertices move in random directions in the unit torus. The model presented here follows the one described. We provide precise asymptotic results for the expected length of the connectivity and disconnectivity periods of the network. We believe that the formal tools developed in this work could be extended to be used in more concrete settings and in more realistic models, in the same manner as the development of the connectivity threshold for static random geometric graphs has affected a lot of research done on ad hoc networks.

[1]  C. Avin,et al.  How to Explore a Fast-Changing World , 2008 .

[2]  Distribution of path durations in mobile ad-hoc networks - Palm's Theorem to the rescue , 2006, Comput. Networks.

[3]  J.,et al.  A guide to concentration bounds , 2001 .

[4]  Maria J. Serna,et al.  Walkers on the Cycle and the Grid , 2008, SIAM J. Discret. Math..

[5]  Josep Díaz,et al.  On the probability of the existence of fixed-size components in random geometric graphs , 2009, Advances in Applied Probability.

[6]  Ramin Hekmat,et al.  Ad-hoc networks - fundamental properties and network topologies , 2006 .

[7]  Mathew D. Penrose,et al.  On k-connectivity for a geometric random graph , 1999, Random Struct. Algorithms.

[8]  M. Penrose.  The longest edge of the random minimal spanning tree , 1997 .

[9]  Donald F. Towsley,et al.  Properties of random direction models , 2005, Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies..

[10]  Mathew D. Penrose,et al.  Random Geometric Graphs , 2003 .

[11]  Piyush Gupta,et al.  Critical Power for Asymptotic Connectivity in Wireless Networks , 1999 .

[12]  M. J. Appel,et al.  The connectivity of a graph on uniform points on [0,1]d , 2002 .

[13]  Tracy Camp,et al.  A survey of mobility models for ad hoc network research , 2002, Wirel. Commun. Mob. Comput..

[14]  Bhaskar Krishnamachari,et al.  Monotone properties of random geometric graphs have sharp thresholds , 2003, math/0310232.

[15]  Ivan Stojmenovic,et al.  Simulation and Modeling of Wireless, Mobile, and AD HOC Networks , 2004 .

[16]  Chen Avin,et al.  How to Explore a Fast-Changing World (Cover Time of a Simple Random Walk on Evolving Graphs) , 2008, ICALP.

[17]  R.A. Guerin,et al.  Channel occupancy time distribution in a cellular radio system , 1987, IEEE Transactions on Vehicular Technology.

[18]  P. R. Kumar,et al.  Critical power for asymptotic connectivity , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[19]  Béla Bollobás,et al.  Random Graphs: Preface to the Second Edition , 2001 .

[20]  Béla Bollobás,et al.  Random Graphs: Notation , 2001 .

[21]  Leonidas J. Guibas,et al.  Kinetic Connectivity for Unit Disks , 2001, Discret. Comput. Geom..

[22]  Kevin C. Almeroth,et al.  Towards realistic mobility models for mobile ad hoc networks , 2003, MobiCom '03.

[23]  Jordi Petit,et al.  A Guide to Concentration Bounds , 2001 .

[24]  Paolo Santi,et al.  An evaluation of connectivity in mobile wireless ad hoc networks , 2002, Proceedings International Conference on Dependable Systems and Networks.