The main focus of this research will be on spatial random processes. Motivated by applications in ecology we will consider processes that take place in two and three dimensional space. We will also study processes that take place on random graphs, which provide models for social networks. We will consider the spread of information and of diseases through these structures. In addition we will study of the behavior of evolutionary games, which were introduced many years ago to help understand animal behavior and have recently been used to understand the interactions of different cell types in cancer modeling. Despite the fact that it has been clearly established that spatial structure changes the outcome of evolutionary games, most applications assume homogeneous mixing and use the replicator equation to determine the dynamics. Having a well developed theory that predicts the behavior of spatial games will be useful for applications. Much of this theory has been developed under the assumption of weak selection, so it is important to understand whether the results are valid when selection is not weak.
Work will be carried out on evolutionary games, variants of the voter model, coalescing random walk, and epidemics. These processes will in most cases take place on static random graphs generated by the configuration model in which vertices are assigned i.i.d. degrees, but in one situation we will let the states of an SIS epidemic and the graph co-evolve. The degree heterogeneity of random graphs forces the development of new techniques in order to extend results known on the d-dimensional lattice. In addition, new phenomena occur on these structures, such as rigorously provable discontinuous phase transitions. Specific research goals include (i) Study the rate of decay of the density of coalescing random walks on a random graph. (ii) Disprove physicists? claim that the critical infection probability on a finite graph is 1 over the largest eigenvalue of the adjacency matrix. (iii) Show that the critical threshold of the SIS is positive if the degree distribution has an exponential tail. (iv) Study SIR and SIS models on evolving graphs where susceptible individuals cut their ties to infected neighbors and rewire to a randomly chosen individual. In the SIR version we want to determine the critical value. In the SIS case we want to show that when the rewiring rate is fixed and the infection rate is varied there is a discontinuous phase transition.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
