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Exact methods to study dynamical processes on real-world networks

Date

Real-world networks are messy and complex. They typically have a large numbers of nodes (vertices) and links (edges), high connectivity and intricate structures. Some of these networks are also dynamic, with states that change over time.

Being able to directly evaluate real-world networks allows us to understand the systems they represent, such as the behaviour of ecosystems, the dynamics of social networks, or the spread of disease through a population.

This project is a visual and interactive implementation of a method developed by Dr. Jonathan Ward, described in “Dimension-reduction of dynamics on real-world networks with symmetry", Proceedings of the Royal Society A, 2021. The method is useful for studying complex dynamical processes on networks, where the full state-space would be too large to handle directly.

By translating complex concepts into visualizations and explanations, this work aims to make the original research accessible to practitioners and to engage with a wider audience, particularly researchers trying to gain additional insights into their own complex real-world data.

The implementation involved recreating a pipeline for processing network data. For a given network, the first step is to convert it into a format compatible with “Saucy”, a tool used to find network symmetries, known as automorphisms in mathematics. Next, the network automorphisms are processed using a software package called GAP, which facilitates the handling of symmetries in an efficient way.

After determining the automorphisms, information about the network and its reduced state space is collected and analysed. This data is then visualized to provide a clear representation of the network and the symmetries found.

To make the findings accessible, an interactive dashboard has been developed using Plotly Dash. This dashboard displays the visualizations along with detailed information about the network’s properties, allowing users to explore and interact with the data intuitively. By integrating these elements, the project offers a tool for visualizing and analyzing the complex dynamics of real-world networks, including their symmetries and structure.

The project was developed in Python and is available on Github:

https://github.com/preetscient/symmetry-based-dimension-reduction

Documentation for the interactive dashboard is available online:

https://preetscient.github.io/symmetry-based-dimension-reduction/

This project enables users to explore and interpret the results of applying the dimension reduction method to complex network dynamics in their own real-world data. Advanced theoretical concepts have been translated into an accessible tool, bridging a gap between theory and practice.

This easy-to-use tool will help to encourage researchers to adopt this method in their own work, resulting in better insights and further innovation.

Ward, Jonathan A. 2021, Dimension-reduction of dynamics on real-world networks with symmetry. Proc. R. Soc. A.47720210026

http://doi.org/10.1098/rspa.2021.0026

Insights

  • Theoretical concepts have been translated into an accessible, interactive tool.
  • By providing a visual and user-friendly interface, this project bridges the gap between graph theory and practical applications.
  • The dashboard encourages experimentation with network dynamics and has the potential to facilitate new insights across various fields.

Research theme

  • Health
  • Societies
  • Environment

Programme theme

  • Visualisation Extended Reality
  • Mathematical and Computational Foundations
  • Data Science Infrastructures

People

Preeti Sharma – Data Scientist, Leeds Institute for Data Analytics, University of Leeds

Dr. Jonathan A. Ward – Lecturer, School of Mathematics, University of Leeds

Funders

This work was funded by the Leverhulme Trust Research Project Grant RPG-2023-187