Soutenances

Random Dynamics in Collective Behavior - Consensus, Clustering & Extinction of Populations

par M. Jens Fischer (Institut de Mathématiques de Toulouse)

Europe/Paris
Salle Pellos, bâtiment 1R2 (Institut de Mathématiques de Toulouse)

Salle Pellos, bâtiment 1R2

Institut de Mathématiques de Toulouse

118 route de Narbonne 31062 Toulouse Cedex 9
Description

This thesis is dedicated to the contribution of new tools to the analysis of complex systems, which describe the interactive behavior of heterogeneous agents within a population modeled by an evolving network.
In my thesis I propose a new interpretation of the evolving networks as generalized exclusion processes on a higher dimensional graph. By defining a further abstraction of said graph on the set of all fixed size k subsets of its vertex set, which I call the “k particle graph”, we obtain a canonical understanding of the particle configurations on some graph and assign a natural topology to the space of configurations. This first part is purely graph theoretical and resides in the field of intersection graph theory, generalizing the Johnson graph family.
The results from the first part can then be applied to Markov chain interpretations of exclusion processes. I show that a specific Markov chain, arising from a specific collective behavior model, has various applications to other problems and yields new insights into vertex induced sub-graphs, which are of great interest in the computer siences. For this Markov chain, classical questions regarding the stationary distribution, convergence speed and lumpability will be answered and lead to a new understanding of the constituent parts of a particle configuration on a graph.
The talk will be closed by a link between opinions in heterogeneous populations and absorbing random environments for the corresponding Markov chain of an associated exclusion process as well as the aesthetically pleasing implications for the k particle graph.