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.
Most mathematical models that describe the dynamics of interacting populations consist of systems of ordinary differential equations. These models, however, leave out any spatial structure, while it is known from past research that spatial models can result in predictions that differ from their nonspatial counterpart. In contrast, the framework of interacting particle systems is ideally suited to understand the role of space. In these models, particles are located on the vertices of a connected graph and can only interact with their neighbors, as defined by the edge set of the graph. The mathematical analysis of interacting particle systems aims to deduce the macroscopic behavior of the system from microscopic rules indicating the rate at which a vertex changes its state as a function of the configuration in its neighborhood. The development and analysis of such models have had a tremendous impact in the advance of physical sciences. The analysis of spatial stochastic models in life and social sciences is equally important but has been largely unexplored by researchers in the field of probability theory, in particular models of social dynamics. Outcomes related to the intellectual merit. The leitmotiv of the research resulting from this award has been to initiate the rigorous analysis of a wide variety of spatial stochastic models of interest in life and social sciences based on the framework of interacting particle systems. This includes some of the most popular agent-based models introduced by applied scientists but unexplored so far by mathematicians - the Axelrod model for the dynamics of cultures, the naming game for the dynamics of languages, spatial versions of the Galam's majority rule and evolutionary games, etc. - as well as new spatial stochastic models. For the latter, the PI's research includes in addition a modeling aspect in an effort to design processes that suitably mimics biological and social systems, by notably using hypergraphs to represent the network of interactions in order to model the inclusion of intermediate mesoscopic scales. For single type models and multitype non-symmetric models, the main objective is to determine whether types - species, opinions, languages, strategies, etc. - survive and/or coexist. The research on this class of models has revealed a number of examples for which the long-term behavior strongly depends on the topology of the network of interactions, and thus a number of disagreements between the spatial stochastic model and its nonspatial deterministic counterpart, showing the importance of local interactions in such systems. For symmetric multitype models, the main objective is to determine whether the system fluctuates or fixates, i.e., whether particles change their type infinitely often or only a finite number of times. Spatial simulations for this class of models are difficult to interpret, and the research of the PI in this area has disproved some important results conjectured by applied scientists, notably for the Axelrod model and the constrained voter model. Overall, this research program has resulted in 19 research papers published in or submitted to the main journals in probability theory and stochastic processes. Outcomes related to broader impacts. Because the research resulting from this award gives rigorous answers to important questions raised by applied scientists, its impact is obviously not limited to the field of stochastic processes. In particular, the work on popular models of social dynamics, and specifically the Axelrod model for the dissemination of cultures, already has an impact that the PI expects to see growing in the field of social sciences. The interdisciplinarity of this research has also been an important ingredient in terms of educational perspectives by promoting the involvement of students at different levels and with different backgrounds and interests. In particular, during the time window of this award, the PI has supervised six Honors students, one Master student and one Ph.D. student who, all together, worked on three different areas: spatial versions of the Galam's majority rule, spatial evolutionary games, and the Axelrod model for the dynamics of cultures. This work as a supervisor has resulted so far in four research papers written with four of these students. The graduate level stochastic modeling class designed by the PI has also been recently completed to include lectures on cultural dynamics and evolutionary game dynamics, which shows the potential of this research in terms of curriculum development. The PI has in addition presented some of his results to underrepresented minorities at the summer program of the Mathematical and Theoretical Biology Institute each summer during the four consecutive years of this award.
