ATiNS: Stochastic Networks

Summary

Networks matter. They underpin communication and transportation systems, structure the flow of information on the Web and social media, and shape interactions in social, economic, and biological systems. But what can we learn from data capturing these complex interaction topologies? How do network structures influence dynamical processes such as diffusion or epidemic spreading? What roles do individual nodes play? How can we identify significant structural patterns or influential actors? And how can we analyze time series on systems with evolving network topologies?

This course introduces the foundations of network science and equips students with statistical and computational tools to analyze, model, and interpret network data across disciplines. Participants will learn how to quantify structural features, evaluate the influence of topology on dynamics, assess system robustness, and understand the emergence of collective behavior from local interactions.

The course combines lectures introducing core theoretical concepts with a research-led project, allowing students to apply methods to real-world network problems. Course materials include annotated lecture slides and a suite of interactive Jupyter notebooks. Students are expected to complete a final research paper to successfully pass the course.

Lectures

Chapter I: Introduction to Network Science and Graph Theory

The first chapter of our course motivates the growing need for network analysis techniques in science, industry, and society. We introduce graph-theoretic foundations of network analysis and explain how we can mathematically analyse patterns in network topologies.

  • Motivation
  • Foundations of Graph Theory
  • Communities and Node Centrality
  • Spatially Embedded Networks
Network Science Icon
Teaching CEE520 Power Grid

Chapter II: Statistical Ensembles of Networks

The second chapter introduces the ensemble perspective on complex networks and explains analytical techniques that enable us to make strong statements about macroscopic system qualities like connectedness, diameter, or robustness based on simple aggregate statistics.

  • Random Graph Ensembles
  • Small-world Networks
  • Degree-based ensembles
  • Generating functions
  • Percolation Transition
  • Scale-Free Networks

Chapter III: Dynamical Processes in Networks

The third chapter addresses the stochastic modelling of linear dynamical processes in networks. We show how we can use Markov chains to model random walks and diffusion and how spectral properties allow us to predict the evolution of processes.

  • Modelling Network Growth
  • Random walks and Markov Chains
  • Feedback Centralities
  • Spectral analysis of networks
  • Temporal Networks
Teaching CEE520 Temporal Graph