Stochastic Spatial Models: on Complex Networks, Coevolution, and Modeling Cancer

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Research will be carried out on various aspects of stochastic spatial models taking place on regular lattices and on complex networks. (i) With Ted Cox and Ed Perkins, the author has studied voter model perturbations. This work will be the starting point for studying evolutionary games with weak selection in order to try to resolve some recent conjectures of Martin Nowak. (ii) The proposer has studied various processes on random regular graphs in joint work with Shirhsendu Chatterjee. The main thrust of the new work will consider process on graphs with more general degree distributions, but we will also consider models in which a threshold of more than one neighbor with a different opinion is needed for an individual to change. (iii) A third theme is to study networks in which the state of the system and the connections in the network coevolve. This work leads to difficult questions about stochastic evolutions on the space of all labeled graphs.The world around us is spatially structured. In biological systems, organisms and infections only spread over short distances. Social systems have a network structure that is described by a graph with vertices being individuals and edges between those that interact. Connections between individuals may extend over large physical distances, but diseases, opinions, and new technologies spreading due to the interactions between an individual and their neighbors in the graph. Most mathematical analyses ignore spatial or network structure because its inclusion greatly complicates the analysis. However, as our past work with Simon Levin and others has shown, incorporating spatial structure changes the outcome of ecological competition. Our new work will focus on how spatial structure changes the impact of frequency dependent selection, and how the network structure of a social system changes the behavior of processes that occur in it. The models are idealized but the results give valuable insights about real systems.