Random Geometric Graph Model

1
Random Geometric Graph Model

• Model for ad hoc/sensor networks
• n nodes placed in d-dim space
• Connectivity threshold r
• Two nodes u,v connected iff u-vp r

Approach Apply probabilistic methods in
combinatorics and graph theory to determine
asymptotic properties Example The asymptotic
connectivity threshold is when every vertex has
n log n neighbors
2
An asymptotic diameter bound
Theorem (E.,Martin,Yan) Given connectivity in
the d-dim unit ball, graph path distance is
well-approximated by lp-distance.
p1, unit disk (dim d2)
With high probability, for any pair of nodes
u,v, Moreover, the u,v-path is found by greedy
hops plus local flooding.
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Discrete Applied Mathematics Courses

• Math 553 DAM I/Graph Theory. Theory and
  applications, including coloring problems and
  network flows.
• Math 554 DAM II/Combinatorics. Enumerative
  combinatorics, design theory, and coding theory.
• Math 535 Optimization I. Theoretical and
  algorithmic aspects of linear optimization.
• Math 557 Probabilistic Methods in Combinatorics.
  Basic to intermediate probabilistic counting
  techniques with applications to Ramsey Theory,
  random graphs, coding theory, and number theory.
• Math 5 Complex Networks. Real-world massive
  graph models and techniques for their analysis.
  (Planned course in conjunction with IGERT
  proposal or potential student enrollment.)