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.

Propensity and stickiness in the naming game: tipping fractions of minorities.

Agent-based models of the binary naming game are generalized here to represent a family of models parameterized by the introduction of two continuous parameters. These parameters define varying listener-speaker interactions on the individual level with one parameter controlling the speaker and the other controlling the listener of each interaction. The major finding presented here is that the generalized naming game preserves the existence of critical thresholds for the size of committed minorities.

Opinion dynamics and influencing on random geometric graphs.

We investigate the two-word Naming Game on two-dimensional random geometric graphs. Studying this model advances our understanding of the spatial distribution and propagation of opinions in social dynamics. A main feature of this model is the spontaneous emergence of spatial structures called opinion domains which are geographic regions with clear boundaries within which all individuals share the same opinion.

Phase transition to super-rotating atmospheres in a simple planetary model for a nonrotating massive planet: exact solution.

An energy-enstrophy model for the equilibrium statistical mechanics of barotropic flow on a massive nonrotating sphere is introduced and solved exactly for phase transitions to rotating solid-body atmospheres when the kinetic energy level is high. Unlike the Kraichnan theory which is a Gaussian model, we substitute a microcanonical enstrophy constraint for the usual canonical one, a step which is based on sound physical principles. This yields a spherical model with zero total circulation, microcanonical enstrophy constraint, and canonical constraint on energy, leaving angular momentum free as is required for any model whose objective is to predict super-rotation in planetary atmospheres.