Large scale behavior of graph dynamics
Open Access
- Author:
- Zhou, Datong
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- May 09, 2024
- Committee Members:
- Anna Mazzucato, Major Field Member
Pierre-Emmanuel Jabin, Chair & Dissertation Advisor
Bruce Gluckman, Outside Unit & Field Member
Leonid Berlyand, Major Field Member
Pierre-Emmanuel Jabin, Program Head/Chair - Keywords:
- many-particle systems
multi-agent systems
biological neural networks
numerical analysis - Abstract:
- This dissertation establishes novel quantitative approaches for understanding the complexities of various dynamics on graphs. The first set of topics includes many-particle and multi-agent systems on graphs that exhibit generalized mean-field scaling, with a specific focus on biological neuronal networks governed by integrate-and-fire (IF) dynamics. The second set of topics investigates advection equations on finite-degree graphs, the transport plan of Wasserstein distance (which could be considered as bipartite graphs), and their applications in numerical analysis. Previously, networks of IF neurons had only been researched in depth in the case of identical connections (complete graph), to which various classical tools in many-particle systems apply. For a given graph with potentially richer structures, the well-posedness theory can still be extended. However, it becomes drastically challenging to study the dynamics across multiple graphs and establish any convergence property, as the influence of graph structure difference is little understood. We introduce two approaches to capture the large-scale dynamics, each involving several novel statistical notions. Through novel commutator estimates, these approaches enable us, for the first time, to derive mean-field limits for IF neuron networks without assuming a priori graph structure. Similar circumstances also appear in the numerical analysis problems we study, despite the difference between these problems. While the well-posedness of the equations and their numerical schemes are established, quantitative estimates across the two are long-standing problems. We approach these problems through an in-depth investigation of the particular graph structure that causes the fluctuations in the particular problem. We achieve the first proof of numerical convergence for non-linear coupled systems involving advection PDE on non-Cartesian meshes by studying such meshes as finite degree graphs with complex topology and a non-Euclidean distance structure. Also, by using a novel commutator estimate for the Wasserstein distance transport map, we establish the first quantitative convergence rate for a deterministic particle method solving the porous media equation.