NSF Grant DMS-10-05282 (08/2010 - 07/2014)
Stochastic processes on hypergraphs and dynamic graphs
Most mathematical models that describe dynamics of interacting populations consist of systems of ordinary differential equations.
Stochastic models can be naturally derived from deterministic models by assuming that the population size is finite and that all pairs of individuals have equal probability to interact, which is referred to as global interactions.
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 nonspatial models.
In contrast, the framework of interacting particle systems is ideally suited to understand the role of space in a wide variety of applied sciences.
Members of the population are located on the vertices of a connected graph and can only interact with their adjacent vertices, which is referred to as local interactions.
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.
We propose to introduce and analyze spatial stochastic processes of interest in a wide variety of fields, such as ecology, sociology and neuroscience.
Even through these processes are based on the contact process and the voter model, they do not fall into the traditional framework of interacting particle systems.
In particular, one of the main objectives of this project is to develop, starting from simple examples, a new theoretical framework that extends interacting particle systems following two directions motivated by various areas of applied sciences.
First extension: Hypergraphs.
We will introduce models in which arbitrarily large subsets of vertices may flip simultaneously, whereas traditional interacting particle systems exclude simultaneous jumps.
These models will be obtained by replacing the graph that represents the underlying spatial structure with a hypergraph, i.e., the edge set is replaced with a collection of subsets of the vertex set with arbitrary cardinality that are called hyperedges.
This will allow us to model the inclusion of intermediate mesoscopic scales and different levels of interactions (see figure).
This framework will be applied to an ecological model including catastrophic events and an opinion model including the presence of arbitrarily large discussion groups.
Second extension: Dynamic graphs.
We will also introduce models in which the underlying spatial structure is a dynamic graph whose vertex and edge sets evolve stochastically.
This involves dynamics on the graph similar to that of a traditional interacting particle system, but also dynamics of the graph at rates which are functions of the spatial configuration.
In particular, dynamics on the graph and dynamics of the graph are coupled (see figure).
This framework will be employed to model the evolution of a population of symbionts that shape the structure of their habitat of hosts, as well as the evolution of a neural network where an electrochemical signal moving randomly may strengthen or weaken neural connections.
Research papers resulting from this grant
NSA Grant MPS-14-040958 (03/2015 - 09/2017)
Stochastic spatial models of social dynamics
The development and analysis of mathematical models in general and stochastic processes in particular have had a tremendous impact in the advance of physical sciences and more recently life sciences, often leading to predictions that were unexpected based on experimental studies.
The use of stochastic models in the rapidly growing field of social sciences is equally important, which has led applied scientists to develop a number of stochastic models of social dynamics with an emphasis on agent-based models, referred to as interacting particle systems in mathematics, to understand the dynamics of complex systems at the population level.
The main objective of research in this field is to deduce the macroscopic behavior that emerges from microscopic rules and the topological structure of an underlying network of interactions.
The recent development of these models has resulted in the formulation of conjectures still unexplored by researchers in the field of probability and stochastic processes.
Motivated by the need for analytical results in social sciences, the main objective of this proposal is to prove (or disprove) important conjectures for some of the most popular models of social dynamics and new results for natural variants of these models.
These stochastic spatial models are classified into four branches of social sciences:
||Opinion dynamics with an emphasis on the Deffuant model, an averaging process that includes a confidence threshold preventing agents who strongly disagree to interact.
||Cultural dynamics with variants of the very popular Axelrod model including a confidence threshold as in the Deffuant model and a variable number of states.
||Language dynamics where our proposed work focuses on a spatial version of the naming game introduced by Steels to describe the emergence of conventions and shared lexicons in a population of individuals interacting through successive conversations.
||Evolutionary game dynamics of interest in social sciences with a focus on the birth-death and the death-birth updating processes and the best-response dynamics.
Even though there has been a number of attempts to study these models using computer simulations, a rigorous mathematical analysis remains necessary mainly because spatial simulations are difficult to interpret and might be misleading.
The first work of the PI on the Axelrod model gives for instance rigorous proofs of fluctuation and clustering results that contradict conjectures formulated by statistical physicists based on simulations.
Similarly, our proposed works on variants of the Axelrod model and the best-response dynamics show results that cannot be observed for systems whose size is small enough to be handled by a computer.
In terms of impact, our research program has important contributions in both social sciences and mathematics.
||Impact in social sciences -
By giving rigorous answers to some of the most important questions asked by the leading social scientists, this proposal is obviously a key step for the advance of social sciences and our understanding of social dynamics of populations.
||Impact in mathematics -
Even though the framework of interacting particle systems has been used to model physical and biological systems, the mathematical properties of these models differ significantly from the characteristics of models of social dynamics.
In particular, the analysis of stochastic spatial models of social dynamics requires the development of new techniques, some of which are introduced in this project description.
Therefore, from a mathematical perspective, our research program represents an important contribution to the theory of stochastic processes in general and interacting particle systems in particular.
Research papers resulting from this grant