Entropy Methods for Interacting Particle Models on Networks

Many phenomena in our world emerge from network structures. An illustrative example are social networks. Every person represents a network node and the connections between the people correspond to the edges. Social Media is changing the fabric of our society, with deep consequences that we only begin to understand. A fundamental and not well understood question in this context is How does a collective opinion arise out of many diverse voices? The basic network dynamics considered for this project are relevant in many areas, such as statistical physics, where charged or oscillating particles are aligned in grids or crystals, in biology or in the global energy grid. Furthermore, a big part of the success of machine learning can be attributed to neural network structures. We are specifically interested in mathematically describing the fundamental qualities of the long-time behavior of the dynamics. This is strongly dependent on the precise structure of the network: How many friends (or edges) does a typical user have? How many edges are needed in average to connect two random nodes? etc. To better understand the impact of specific structures on the dynamics, it is necessary to extend the mathematical theory of graphs. A successful approach of recent years idealizes a large network as a grey scale image on the unit square. Mathematically, this corresponds to graph density functions so called Graphons. To allow for an even wider range of structures (including typical social media graphs), we further describe graphs as operators. These are abstract objects that describe modifications of functions. Operators are well understood and fit naturally into the framework of network dynamics. We describe behavioral patterns such as opinion formations mathematically via differential equations that are inspired by the concepts of Thermodynamics. There, the changes over time of a very large amount of interacting particles are described in a statistically average manner as an evolution equation. To analyze the long-time behavior of the solutions, entropy functionals are a central tool. They are a measure of disorder (or entropy) that characterize the particle interactions macroscopically. As in our project the interactions take place on very general networks, we develop entropy methods that incorporate the mentioned network operators. In conclusion, the aim of the project is developing mathematical models that shed light on the long-time behavior of particle interactions on large networks of very diverse structures. To better understand the emerging dynamics, this requires us to design new entropy methods.