More is less: Connectivity in fractal regions

Ad-hoc networks are often deployed in regions with complicated boundaries. We show that if the boundary is modeled as a fractal, a network requiring line of sight connections has the counterintuitive property that increasing the number of nodes decreases the full connection probability. We characterise this decay as a stretched exponential involving the fractal dimension of the boundary, and discuss mitigation strategies. Applications of this study include the analysis and design of sensor networks operating in rugged terrain (e.g. railway cuttings), mm-wave networks in industrial settings and vehicle-to-vehicle/vehicle-to-infrastructure networks in urban environments.

[1]  Jeffrey G. Andrews,et al.  Stochastic geometry and random graphs for the analysis and design of wireless networks , 2009, IEEE Journal on Selected Areas in Communications.

[2]  B. V. R. Reddy,et al.  Node centrality in wireless sensor networks: Importance, applications and advances , 2013, 2013 3rd IEEE International Advance Computing Conference (IACC).

[3]  S. Lalley,et al.  Statistically self-affine sets: Hausdorff and box dimensions , 1994 .

[4]  C. Dettmann,et al.  STRUCTURE FACTOR OF DETERMINISTIC FRACTALS WITH ROTATIONS , 1993 .

[5]  Brian D. O. Anderson,et al.  Towards a Better Understanding of Large-Scale Network Models , 2010, IEEE/ACM Transactions on Networking.

[6]  M. Penrose.  CONNECTIVITY OF SOFT RANDOM GEOMETRIC GRAPHS , 2013, 1311.3897.

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

[8]  Justin P. Coon,et al.  Network connectivity through small openings , 2013, ISWCS.

[9]  Dmitri Vassiliev,et al.  Spectral asymptotics, renewal theorem, and the Berry conjecture for a class of fractals , 1996 .

[10]  Junmei Tang,et al.  Fractal Dimension of a Transportation Network and its Relationship with Urban Growth: A Study of the Dallas-Fort Worth Area , 2004 .

[11]  Justin P. Coon,et al.  Connectivity in dense networks confined within right prisms , 2014, 2014 12th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt).

[12]  B. Mandelbrot.  How Long Is the Coast of Britain ? , 2002 .

[13]  Y. Ahmet Sekercioglu,et al.  Swarm robotics reviewed , 2012, Robotica.

[14]  Yunhao Liu,et al.  Beyond Trilateration: On the Localizability of Wireless Ad Hoc Networks , 2009, IEEE/ACM Transactions on Networking.

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

[16]  Carl P. Dettmann,et al.  Connectivity of Soft Random Geometric Graphs over Annuli , 2016 .

[17]  Justin P. Coon,et al.  Network Connectivity in Non-Convex Domains With Reflections , 2015, IEEE Communications Letters.

[18]  Umberto Spagnolini,et al.  Wireless Cloud Networks for Critical Industrial Quality Control , 2013, ISWCS.

[19]  François Ingelrest,et al.  SensorScope: Application-specific sensor network for environmental monitoring , 2010, TOSN.

[20]  Justin P. Coon,et al.  Full Connectivity: Corners, Edges and Faces , 2012, ArXiv.

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

[22]  Klaus Moessner,et al.  Survey of Experimental Evaluation Studies for Wireless Mesh Network Deployments in Urban Areas Towards Ubiquitous Internet , 2013, IEEE Communications Surveys & Tutorials.

[23]  Victoria J. Hodge,et al.  Wireless Sensor Networks for Condition Monitoring in the Railway Industry: A Survey , 2015, IEEE Transactions on Intelligent Transportation Systems.

[24]  E. Weibel,et al.  Fractals in Biology and Medicine , 1994 .

[25]  Hussein Zedan,et al.  A comprehensive survey on vehicular Ad Hoc network , 2014, J. Netw. Comput. Appl..

[26]  Dharma P. Agrawal,et al.  Ad Hoc and Sensor Networks: Theory and Applications , 2006 .

[27]  M. Walters.  Surveys in Combinatorics 2011: Random geometric graphs , 2011 .

[28]  R. Mantegna,et al.  Fractals in biology and medicine. , 1995, Chaos, solitons, and fractals.

[29]  B. Mandelbrot.  How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension , 1967, Science.

[30]  Potential theory and analytic properties of a Cantor set , 1993 .

[31]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[32]  Guoqiang Shen,et al.  Fractal dimension and fractal growth of urbanized areas , 2002, Int. J. Geogr. Inf. Sci..

[33]  J. Dall,et al.  Random geometric graphs. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.