The trees and the forest: Characterization of complex brain networks with minimum spanning trees

• Comparing brain networks is a challenge for modern network science.
• The minimum spanning tree (MST) is a unique representation of weighted brain networks.
• The MST reflects traffic flow and hierarchy in the underlying system.
• MST sensitivity is comparable to classical graph theoretical brain network analysis.

In recent years there has been a shift in focus from the study of local, mostly task-related activation to the exploration of the organization and functioning of large-scale structural and functional complex brain networks. Progress in the interdisciplinary field of modern network science has introduced many new concepts, analytical tools and models which allow a systematic interpretation of multivariate data obtained from structural and functional MRI, EEG and MEG. However, progress in this field has been hampered by the absence of a simple, unbiased method to represent the essential features of brain networks, and to compare these across different conditions, behavioural states and neuropsychiatric/neurological diseases. One promising solution to this problem is to represent brain networks by a minimum spanning tree (MST), a unique acyclic subgraph that connects all nodes and maximizes a property of interest such as synchronization between brain areas. We explain how the global and local properties of an MST can be characterized. We then review early and more recent applications of the MST to EEG and MEG in epilepsy, development, schizophrenia, brain tumours, multiple sclerosis and Parkinson's disease, and show how MST characterization performs compared to more conventional graph analysis. Finally, we illustrate how MST characterization allows representation of observed brain networks in a space of all possible tree configurations and discuss how this may simplify the construction of simple generative models of normal and abnormal brain network organization.