Stochastic processes on hypergraphs and dynamic graphs

Arizona State University, Scottsdale AZ

Investigators

Abstract

The theoretical framework of interacting particle systems has been successful in modeling physical, biological and sociological systems that are spatial by nature. In such systems, members under consideration are traditionally located on the vertices of a static graph, with the state of each member being updated according to transition rates that depend upon the configuration in its local neighborhood. Motivated by a variety of problems that arise from ecology, sociology and neuroscience, the primary objective of this research project is to introduce and analyze simple stochastic models that extend the framework of interacting particle systems as the components under consideration evolve either on hypergraphs or on dynamic graphs rather than traditional static graphs. Hypergraph structures will be employed to model systems including an intermediate mesoscopic scale: contact process with destruction of cubes and hyperplanes as a model of populations subject to catastrophic events, and voter type model on hypergraphs as a model of the majority rule in explicit space. This results in systems in which all the vertices connected by the same hyperedge are simultaneously updated, either spontaneously or at a rate that depends upon the global configuration of the hyperedge. Dynamic graph structures will be employed to model systems of components that have the ability to shape their spatial environment: coupling of contact processes with varying parameters as a model of host-symbiont interactions, and random walks on weighted graphs as a model of electrochemical signals in the brain. This results in systems involving dynamics on the graph similar to that of traditional interacting particle systems, but also dynamics of the graph, with a feedback between dynamics on and of the graph. Beyond the mathematical analysis of some specific models, the aim of this research program is also to initiate the development of a theoretical framework, inspired from interacting particle systems, in order to model more realistically systems that are ubiquitous in nature. In the early seventies, Frank Spitzer in the United States and Roland Dobrushin in Soviet Union independently introduced a new theoretical framework, known as interacting particle systems, in order to understand the dynamics of systems that are spatial by nature. This framework not only had a major impact in a wide variety of fields such as physics, biology and sociology, but it also gave rise to a number of challenging and exciting mathematical problems. The main objective of research in this area is to understand the macroscopic behavior and the spatial patterns that emerge from the microscopic interactions that dictate the local dynamics of large systems of components such as atoms, cells, plants of different species or individuals with different opinions. Motivated by a variety of problems that arise from ecology, sociology and neuroscience, the primary objective of this research program is to extend, starting from simple examples, the traditional framework of interacting particle systems following two directions. The first extension consists of systems that include an intermediate mesoscopic scale: populations that undergo catastrophic events modeled by the removal of all the individuals in large blocks, and opinion dynamics including the emergence of large discussion groups. The second extension, also known as adaptive networks in the physics literature, consists of systems of components that have the ability to shape their spatial environment: host-pathogen and host-mutualist systems in which symbionts affect the mortality of their host, and electrochemical signals that can strengthen or weaken the connections between neurons. This research project is also the first step towards the development of more realistic models of biological and sociological systems that cannot be captured by traditional interacting particle systems.