The Alan Turing Institute

# Geometry and topology of complex interconnected systems

## Introduction

Geometry plays an important part in understanding the behaviour of systems. For example, periodic or recurrent systems (systems which repeat every so often) are naturally described by a circle – this includes the yearly cycle of the seasons and daily commuting traffic patterns. This may not be a geometric circle. For example, the seasons vary from year to year, so are characterised by an ‘angle’ which shows where in the year we are and, e.g., that it’s more likely that a hot day occurs in summer.

Understanding this shape is better done in terms of topology, or ‘rubber sheet’ geometry, which maintains the qualitative features of the shape but also includes bending and stretching. Understanding these types of structures is critical for performing statistical analyses based on collected data as well as making and testing predictions.

## Explaining the science

While understanding the overall shape is important in analysing data, geometry still plays a critical role. A typical example is the scale at which we analyse a system. Local behaviour may be geometrically intricate, but at larger scales emergent phenomena may appear giving more insight into the system, e.g. the circle in the case of a yearly cycle. In complex systems, structures can interact at several scales, resulting in different phenomena at a number of different scales. This is perhaps best understood in terms of temporal scales, where we can understand and interpret events in the short term, mid-term, or long-term. Constructing a quantitative measure is critical, e.g. is the climate changing on a scale of 10 years, 100 years, or 1000 years?

This issue of scale is ubiquitous and requires an understanding of the interplay between geometry (quantitative measures) and topology (overall shape). One way of understanding this relationship is through algebraic invariants, which are descriptors of the global structure which are also directly related to the local geometry. To further complicate matters, complex systems are most often random. This requires a further understand of how randomness affects both the local geometry and global topology. The Lorenz system – a simple chaotic system which has two attractors (the two holes which the system circles). Understanding the system away from the middle is straightforward (it will continue around the hole) while at the center, it is nearly impossible to predict which hole it will go around next. Knowing this can be used to plot a complex trajectory in 3 dimensions and analyse it with 2 mutually exclusive impulses describing which hole it is circling (far right). These appear at a given scale and so we must understand the ‘scale’ of the holes (far left).

## Project aims

The project is at the intersection of geometry, topology, algebra, and probability. The goal is to answer the following questions:

• How probabilistic effects appear at different scales in complex interconnected systems?
• How can an understanding be used to verify hypotheses about different complicated datasets?
• How can prediction and statistical analysis take these structures into account?

This project is part of the Data-centric engineering programme's 'Mathematical foundations' challenge.

## Applications

As we collect more and more data, tools to understand non-linear and high dimensional structures are becoming  increasingly important. How these structures appear and their role in understanding systems and data has been largely unexplored, however as we consider the complex interaction of many different factors, it is likely that they will play an important role in future developments of analyses and interpretation. The applications include understanding and predicting random systems arising from any number of real-world systems

Turing Fellow

## Researchers and collaborators

### Dr Wajid Mannan

Postdoctoral Research Assistant, QMUL