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-v|| p ≤ 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

Theorem (E.,Martin,Yan) Given connectivity in the d-dim unit ball, graph path distance is well-approximated by l p -distance. u v u’ v’ With high probability, for any pair of nodes u,v, Moreover, the u,v-path is found by greedy hops plus local flooding. p=1, unit disk (dim d=2) An asymptotic diameter bound

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.)