Satellite swarms, random walks and a cup of tea
He brings order to chaos and analyses satellite swarms. During his PhD research, mathematician Oliver Nagy delved into random networks and how they reach equilibrium. Along the way, he also developed a handy tool. This knowledge is valuable for calculations related to communication networks.
‘Imagine a cup of warm tea,’ Nagy begins. ‘At first, the drink is calm, but when we pour in some milk, chaos ensues. Eventually, the liquid becomes uniformly coloured. If we were to stir the cup, this would happen even faster.’ This process, where a dynamic system moves from disorder to order, is the central subject of Nagy’s thesis. ‘The colour of the tea as it mixes represents how far the system is from equilibrium,’ he explains. ‘As the milk spreads and the colour becomes uniform, the system approaches a stable state.’
Just as the colour of the tea can illustrate how well it’s mixed, there are abstract methods to measure how far a system is from equilibrium.
From tea to theory
Just as the colour of the tea can illustrate how well it’s mixed, there are abstract methods to measure how far a system is from equilibrium. One such method is the total variation distance. Nagy: ‘If you are as far away from a stable situation as possible, the total variation distance is one. A value of zero means you’ve reached equilibrium.’
In the first two chapters of his thesis, Nagy delves into the mathematical concept of a ‘random walk’, which describes how particles move through a system. He and his colleagues developed models to track this evolution, and measure how long it takes for systems to return to stability. ‘The second system we studied approached equilibrium through a sudden jump, or cut-off, at a random time, which had not been observed before.’
Studying satellite swarms: a network in motion
Even highly dynamic systems can be studied through their long-term average, which is often static. Nagy applied this principle to the communication network within a satellite swarm: a group of small, randomly moving satellites without propulsion systems. Their unpredictable movements make it difficult to predict network behaviour and maintain communication – an engineering challenge. Nagy: ‘Focusing on the static network that emerges over time helps us tackle practical engineering questions, such as how to design robust networks that stay in function, even when one of the satellites temporarily falls out of range, and or how to reduce energy consumption.’
An unexpected byproduct
The satellites project also led to an unexpected byproduct: a new tool. ‘We encountered a mathematical challenge. How do you determine which satellites are the most important? Crucial information for when, for example, conserving energy and needing to shut down all but the top twenty most important satellites. Which ones do you choose?’
How to define importance in networks: the challenge of centrality measures
Nagy explains: ‘First, you need to define what it means to be the most important. Imagine a friendship network in high school. How do you determine who’s the most popular? Is it the person with the most close friends, or the one who participates in all the different social groups, or is it the person who has a vague connection with everyone? These are different centrality measures, and you can calculate them for any network.’ The problem is, which centrality measure should you use, and what are the consequences of choosing one over another?
Nagy and his colleagues developed a new way to compare different centrality measures. ‘Essentially, each centrality measure ranks the nodes in the network, identifying the most important, second most important, and so on. The key question is whether these measures agree on the same top nodes.’ Using this idea, Nagy’ created the ‘centrality comparison curve’, which visualises the overlap or disagreement between measures. It helps engineers identify crucial nodes for maintaining the network’s functionality without complex calculations. Nagy: ‘We run into this unforeseen problem and solved it along the way. That too is mathematics.’
PhD thesis and defense
Nagy defended his thesis titled Approaching equilibrium in a dynamic network on 16 January at the Academiegebouw in Leiden. His promoters are professors Frank den Hollander and Remco van der Hofstad (Technical University Eindhoven), and Associate professor Luca Avena (University of Florence).