Solution of urn models by generating functions with applications to social, physical, biological, and network sciences

Abstract

Examples of such an urn model date back to the Ehrenfest "dog-flea" model of molecular diffusion, created in 1907. In this model, one ball is chosen randomly to move to the opposite urn. The model was invented to describe the diffusion of particles within a closed container. This model was later diagonalized by Mark Kac in 1947 by using an innovative generating function method to calculate all eigenvalues and eigenvectors of the Markov transition matrix. We not only improve upon this method, but the models that we solve have a wider range of applications. In addition to the physical application of the Ehrenfest model, we have related such models to social interactions on networks, as well as genetic drift.

Through the process of diagonalization, we have uncovered solutions that would otherwise be untenable by other methods. For instance, dynamics for small N, higher moments of consensus time, and the long time behavior of these urn models can be easily determined by these methods. Furthermore, an exact solution for all future probability distributions of some popular urn models had not been found prior to the results given in this thesis. Due to the breadth of models that can be studied in this manner, this thesis represents a significant step forward in the subject of complex systems theory.

We improve upon the generating function method of Kac significantly to handle the more complex features of such urn models. The method of Kac reduces the eigenvalue problem to an ordinary differential equation that is solved exactly. Since our urn models draw two balls instead of one ball in the Ehrenfest model, we solve a partial differential equation instead. Interestingly, the method of solution given in this thesis easily solves the Ehrenfest model, where the original generating function method of Kac does not solve the general urn models easily.

The primary subject of this thesis is the notion of an urn model and their applications to complex systems. It is demonstrated that several models of social, physical, and biological science are special cases of a large class of urn models that are then exactly solved. These models prescribe 2 or more urns with N balls distributed among them. Two balls are then drawn randomly and then redistributed among these urns. This redistribution is also stochastic, with distributions only depending on which urns the balls came from and the order in which they were drawn.