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

Abstract

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.

Explaining the heterogeneity of functional connectivity findings in multiple sclerosis: An empirically informed modeling study

“…The difference between functional connectivity and functional network organization is that functional connectivity refers to the strength of connections between brain regions, whereas functional network organization (topology) corresponds to the pattern of connections between regions in the brain. Specifically, we assess network organization by computing two network topology measures based on the minimum spanning tree (MST) (Stam et al, 2014; Tewarie et al, 2015b): diameter and leaf fraction. (c) Diameter is defined as the longest shortest path within the MST and (d) leaf fraction refers to the fraction of regions in the MST with only one connection.…”

Minimum spanning tree analysis of the human connectome

“…One of these issues is the lack of a reference network, which describes the topology of a “standard” human brain; such a reference network is lacking even if scanning and preprocessing conditions are fixed (Fornito, Zalesky, & Breakspear, 2013; Stam et al, 2014). A reference could be useful to compare results of different studies and of different populations, provided that the preprocessing pipeline, node and edge definitions are similar.…”

“…However, if we were to normalize or correct for the greater number of connections in network A, the efficiency of both networks might be comparable. Therefore, an absolute difference in efficiency does not necessarily imply that the topological organization of the connections is also more efficient in one of the networks (Stam et al, 2014; van Wijk et al, 2010). Proportional thresholding is frequently applied to weighted connectivity matrices to overcome this problem and yield networks that are matched in terms of total number of connections; network characteristics are then based on this thresholded network containing a subset of strong connections.…”

“…Proportional thresholding is frequently applied to weighted connectivity matrices to overcome this problem and yield networks that are matched in terms of total number of connections; network characteristics are then based on this thresholded network containing a subset of strong connections. However, this approach affects the graph properties in an unpredictable manner, because the impact of setting a threshold varies, depending on the underlying network topology (Stam et al, 2014; van den Heuvel et al, 2017; van Wijk et al, 2010). Handling of the thresholding problem is likely to be one of the reasons that empirical studies have reported contradictory findings, for example of both increased and decreased characteristic shortest path length in patients with Alzheimer's Disease (Fornito et al, 2013; Tijms et al, 2013).…”

“…It connects all the nodes in the network without forming cycles or loops. The MST is unaffected by the thresholding problem provided that the ranking of edge weights remains unaltered, while its characteristics can be interpreted along the lines of conventional graph measures (Stam et al, 2014; Tewarie, van Dellen, Hillebrand, & Stam, 2015). Importantly, the MST is a unique subset of connections provided that all connection weights are unique, which makes it possible to create an empirical reference network of the connections in a healthy human brain.…”