Solution to urn models of pairwise interaction with application to social, physical, and biological sciences.

We investigate a family of urn models that correspond to one-dimensional random walks with quadratic transition probabilities that have highly diverse applications. Well-known instances of these two-urn models are the Ehrenfest model of molecular diffusion, the voter model of social influence, and the Moran model of population genetics. We also provide a generating function method for diagonalizing the corresponding transition matrix that is valid if and only if the underlying mean density satisfies a linear differential equation and express the eigenvector components as terms of ordinary hypergeometric functions.

Analysis of the high-dimensional naming game with committed minorities.

The naming game has become an archetype for linguistic evolution and mathematical social behavioral analysis. In the model presented here, there are N individuals and K words. Our contribution is developing a robust method that handles the case when K=O(N).

Solution of the multistate voter model and application to strong neutrals in the naming game.

We consider the voter model with M states initially in the system. Using generating functions, we pose the spectral problem for the Markov transition matrix and solve for all eigenvalues and eigenvectors exactly. With this solution, we can find all future probability probability distributions, the expected time for the system to condense from M states to M-1 states, the moments of consensus time, the expected local times, and the expected number of states over time.

Solution of the voter model by spectral analysis.

An exact spectral analysis of the Markov propagator for the voter model is presented for the complete graph and extended to the complete bipartite graph and uncorrelated random networks. Using a well-defined Martingale approximation in diffusion-dominated regions of phase space, which is almost everywhere for the voter model, this method is applied to compute analytically several key quantities such as exact expressions for the m time-step propagator of the voter model, all moments of consensus times, and the local times for each macrostate. This spectral method is motivated by a related method for solving the Ehrenfest urn problem and by formulating the voter model on the complete graph as an urn model.

Simultaneous tissue factor expression and phosphatidylserine exposure account for the highly procoagulant pattern of melanoma cell lines.

A correlation between cancer and hypercoagulability has been described for more than a century. Patients with cancer are at increased risk for thrombotic complications, and the clotting initiator protein, tissue factor (TF), is possibly involved in this process. In addition to TF, the presence of negatively charged phospholipids, particularly phosphatidylserine (PS), is necessary to support some of the blood-clotting reactions.