My thesis focuses on models for complex networks. This work was performed in collaboration with Ed Scheinerman at Johns Hopkins University and with my advisor, Milena Mihail (College of Computing) on generalizing the random dot product graph model. In partiuclar, we develop a robust and general model for complex networks based on the dot product. We are able to show that there is a general reduction of the diameter of the random dot product graph model to the diameter of a suitable class of Erdos-Renyi graphs. Further, we show that the model is clustered in a natural way and there is an explicit relationship between the underlying geometry of the random space to the degree distribution of the resulting model. This results generalize to the natural directed graph model. We also explore other modeling and algorithmic aspects of the random dot product graph model.

I am also currently doing research with Tom Trotter and his graduate students (Csaba, Dave, and Mitch) on the combinatorics of partially ordered sets and related questions. This collaboration has resulted in partial result on the structure of forbidden subposets for linear discrepancy 3 (see paper below).

On the side I am also working with Joel Sokol from the School of Industrial and Systems Engineering on a project in choice aggregation as it relates to the professional baseball draft. In essence, since there are so many players eligble for the draft (approximately 2000), there is no way for one scout to see them all, so a team must have some way of aggregating all of its scout's limited and potentially contradictory information.

Papers

• Stanley depth of squarefree monomial ideals (with M.T. Keller) accepted (2009) Journal of Algebra
• A Brooks-type Theorem for the Bandwidth of Interval Graphs (with M.T. Keller) submitted
• Interval partitions and Stanley Depth (with Cs. Biro, D.M. Howard, M.T. Keller, and W.T. Trotter) submitted
• Directed Random Dot Product Graphs(with E.R. Scheinerman) accepted (2008) Internet Mathematics
• A characterization of partially ordered sets with linear discrepancy equal to 2
  (with D.M. Howard and M.T. Keller), ORDER 24 no. 3 (2007), 139-153.
  The original publication is available at www.springerlink.com.
• Random Dot Product Graph Models for Social Networks (with E.R. Scheinerman) Lecture Notes in Computer Science 4863, Proceedings of WAW 2007, 138-149.
  The original publication is available at www.springerlink.com.