Connectomics, graph theory, and complexity

Graph theory is a branch of mathematics that can be used to understand connectomes. This note provides references on the basic principles of connectomics as well as graph theory.

Graph theory and network science

Online courses

• Introduction to Complexity (Santa Fe Institute 2015)
• Interactive complexity explainer (2019)
• Probabilistic graph models (Stanford 2023)
• Chaos and nonlinear dynamics lectures (Steven Strogatz 2014)
  • Video series summary (2021)
• Principles of complex systems (2023)
• History of complexity and network science video
• Dynamical systems in neuroscience video series (2021)
• Differential equations and dynamical systems video series (2022)
• Computational neuroscience animation channel
• Theoretical physics lecture notes and problems (Cambridge 2013)
• Graphs and networks resource set (2017)
• Brain networks (Russ Poldrack 2018)
• Logistic maps, fractals, and chaos video (Veritasium 2020)
• Criticality in the brain video (Quanta magazine 2023)
• Collection of explorable explainers
  • Complexity explorables
    • Kuramoto oscillators (2018)
      • Medium article about the history, applications, and limitations of Kuramoto oscillators
  • Biological neural networks (2018)
  • Attractor landscapes (2018)
  • Thinking in systems (2017)
  • Watts-Strogatz small-world networks (2011)
  • Brain mapping (2019)
  • Graph theory (2017)
• Network neuroscience
• Information theory and self-organisation lectures (2020)
• Chaos lectures (2017)
• Network diffusion (via the graph Laplacian) explainer (2021)
• MATLAB-based tutorial for analysing self-organising criticality (Ben Fulcher 2019)
• List of network analysis resources
• List of complexity resources

Textbooks

• Albert-László Barabási - Network Science (2015)
• Mark Newman - Networks (2018)
• Michele Coscia - The Atlas for the Aspiring Network Scientist (2025)
• Parr, Pezzulo, Friston - Active Inference (2022)
  • Walkthrough videos (2023)
• Porter and Gleeson - Dynamical Systems on Networks (2016)
• Newman and Barabási - The Structure and Dynamics of Networks (2006)
• Easley and Kleinberg - Networks, Crowds, and Markets (2010)
• Wasserman and Faust - Social Network Analysis (1994)
• Béla Bollobás - Random Graphs (2011)
• Eugene Izhikevich - Dynamical Systems in Neuroscience (2006)

Review articles

• Mark Newman - The Structure and Function of Complex Networks (2003)
• Albert and Barabási - Statistical mechanics of complex networks (2002)
• Boccaletti et al. - Complex networks: Structure and dynamics (2006)
• Carter Butts - Revisiting the Foundations of Network Analysis (2009)
• Barabási and Oltvai - Network biology: Understanding the cell's functional organization (2004)

Modularity

• Mark Newman - Communities, modules and large-scale structure in networks (2012)
• Santo Fortunato - Community detection in graphs (2009)
• Santo Fortunato - Community detection in networks: A user guide (2016)

Centrality

• Stephen Borgatti - Centrality and network flow (2005)
• Borgatti and Everett - A Graph-theoretic perspective on centrality (2006)

David Schoch (PhD thesis) - A Positional Approach for Network Centrality(2015)

Network database

Connectomics

Lay article on network neuroscience (WIRED 2020)

Panel discussion video about brain mapping via connectomics (World Science Festival 2018)

Andrew Zalesky's short videos on thresholding, comparing, and null modelling connectomes (2020)

Textbooks

• Fornito, Zalesky, Bullmore - Fundamentals of Brain Network Analysis (2016)
  • There are several copies in the lab, ask Alex
• Kennedy, van Essen, Christen - Micro-, Meso- and Macro- Connectomics of the Brain (2016)
• Olaf Sporns - Networks of the Brain (2010)
• Olaf Sporns - Discovering the Human Connectome (2012)
• Robert Rosenbaum - Modelling neural circuits made simple with Python (2024)
  • Supporting code

Articles

• Sporns et al. - The Human Connectome: A Structural Description of the Human Brain (2005)
• Bullmore and Sporns - Complex brain networks: graph theoretical analysis of structural and functional systems (2009)
• Rubinov and Sporns - Complex network measures of brain connectivity: Uses and interpretations (2010)
• Bullmore and Bassett - Brain Graphs: Graphical Models of the Human Brain Connectome (2011)
• Fornito et al. - Graph analysis of the human connectome: Promise, progress, and pitfalls (2013)
• Bullmore and Sporns - The economy of brain network organization (2012)
• van den Heuvel and Sporns - Network hubs in the human brain (2013)
• van den Heuvel et al. - Multimodal Analysis of Cortical Chemoarchitecture and Macroscale fMRI Resting-State Functional Connectivity (2016)
• Fornito et al. - The connectomics of brain disorders (2015)
• Centeno et al. - A hands‑on tutorial on network and topological neuroscience (2022)

Guide for commonly used parcellations

Software

Analysis

• Brain Connectivity Toolbox (MATLAB)
• Network-Based Statistic (MATLAB)
• 39 community detection algorithms (Python)
• Graph-tool (Python)
• Brain Dynamics Toolbox (MATLAB)
  • Tutorial (Ben Fulcher 2019)
• MATLAB tutorial for analysing neuronal connectivity in C. elegans (Ben Fulcher 2019)
• TAPAS: Translational Algorithms for Psychiatry-Advancing Science (MATLAB, Python)
• PyNets - machine learning with connectomes (Python)
• Bajada et al. - A tutorial and tool for exploring feature similarity gradients with MRI data (Matlab, Python; 2020)
• Bahrami et al. - A MATLAB toolbox for multivariate analysis of brain networks (2018)

Visualisation

• Chopra et al. - A Practical Guide for Generating Reproducible and Programmatic Neuroimaging Visualizations (2023)
  • Table of code-based brain visualisation tools
  • Code templates for R and Python
• NeuroMarvl - web-based GUI for brain networks, developed by our lab
• BrainNet Viewer - downloadable GUI for brain networks
• R tutorial on static and dynamic network visualisation
• Gephi - downloadable and web-based GUIs for network analysis and visualisation
• Cytoscape - downloadable and web-based GUIs for network analysis and visualisation
• BioImage Suite - downloadable and web-based GUIs for brain image processing and visualisation of maps and networks
• pyCirclize - Circos plots, chord diagrams, radar charts (Python)
• Tutorial for brain connectivity analyses (Python; 2019)
• tikz-network - Network visualisation in LaTeX