University of Missouri - Saint Louis

The Graduate School

Announcement

An oral examination in defense of the dissertation for the degree

Doctor of Philosophy in Applied Mathematics

Robert A. Murphy
M.S. in Systems Science and Mathematics, May, 1996, Washington University in St. Louis
B.S. in Mathematics, December, 1991, University of Missouri – Columbia

Partial Connectivity of Wireless Sensor Networks with Applications

 

Abstract

Given a bounded subset of the 2-dimensional plane, a discrete set of nodes is uniformly distributed throughout and each node is generated at a discrete time instant according to the Poisson distribution. Given some fixed, finite, real number, two nodes are said to connect and form an edge if their mutual distance is less than this number. Let G be the graph of all such edges over the set of generated nodes and let C be any set of mutually connected nodes. It is shown that there is a critical mutual distance such that the at least half of all generated nodes are mutually connected to form a connected cluster.

Now, suppose that the 2-dimensional plane is partitioned into hexagons chosen such that each can have inscribed, a circle of radius which is half the size of the mutual distance. Define another notion of connectivity on the generated nodes by saying that two nodes connect if each lies in the same hexagon or each lies in a hexagon which shares a common face with the hexagon containing the other node. It is shown that the original graph of edges contains the new graph of edges and that there is an inequality relationship among the critical mutual distances.

Finally, using results mentioned above, upper and lower bounds on the probability of connecting slightly more or slight less than half of all generated nodes are obtained and used to estimate the length of the interval of radii such that the probability of connecting at least half of all generated nodes will increase from some small positive value to a value near 1.

 

Date: September 2, 2011

Time: 9:00 a.m. to 11:00 a.m.

Place: 302 Express Scripts Hall

 

Defense of Dissertation Committee

 

Haiyan Cai, Ph.D. (Advisor)

Nevena Maric, Ph.D.
 

Adrian Clingher, Ph.D.

Wenjie He, Ph.D.
     

 


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