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.

The Hidden Influence of Network Structures Many phenomena in our world spread through complex networks. Social networks provide a particularly vivid example of this. Every person can be described as a network node, and our friendships correspond to the connections between these nodes, known as network edges. An exemplary and largely unresolved question in this context is: "How does a collective opinion form from individual voices?" In more abstract terms, we investigated complex dynamic effects on networks based on simple edge interactions. To this end, we developed mathematical models that capture fundamental mechanisms found in many systems with network structures. How Quickly Does Order Emerge? Whether it involves opinion formation, energy distribution, or machine learning, we are interested in the long-term behavior of these systems. Inspired by thermodynamics, where countless interacting particles eventually transition into a state of rest, we investigated how quickly networks reach their equilibrium. Using newly developed mathematical tools based on the concept of entropy as a measure of disorder, we were able to prove universal laws. Our result in this regard is estimating the rate at which this equilibrium is reached. Our formulas provide predictions about how quickly a stable state develops, depending on specific network structures. Why Artificial Intelligence Sometimes Fails: Another success of our project are some mathematical insight into the "black box" of machine learning. Here, too, understanding the complex network structures of neural networks is essential. These are extremely powerful, but also prone to errors. In specific settings, we succeeded in mathematically proving the exact fundamental limits these systems encounter. This helps us better understand why AIs fail or make incorrect decisions in certain situations. At the same time, we have developed new approaches that make certain neural networks more robust against perturbations and manipulations. The most important findings of our work can therefore be summarized as follows: Mathematics helps us decode the hidden structures of complex networks and better understand their influences.