Comparing Brain Networks of Different Size and Connectivity Density Using Graph Theory

By B. van Wijk, Cornelis J. Stam, Andreas Daffertshofer. PLoS ONE, 2010

1,181 citations recorded by the source corpus.

Graph theory is a valuable framework to study the organization of functional and anatomical connections in the brain. Its use for comparing network topologies, however, is not without difficulties. Graph measures may be influenced by the number of nodes (N) and the average degree (k) of the network. The explicit form of that influence depends on the type of network topology, which is usually unknown for experimental data. Direct comparisons of graph measures between empirical networks with different N and/or k can therefore yield spurious results. We list benefits and pitfalls of various approaches that intend to overcome these difficulties. We discuss the initial graph definition of unweighted graphs via fixed thresholds, average degrees or edge densities, and the use of weighted graphs. For instance, choosing a threshold to fix N and k does eliminate size and density effects but may lead to modifications of the network by enforcing (ignoring) non-significant (significant) connections. Opposed to fixing N and k, graph measures are often normalized via random surrogates but, in fact, this may even increase the sensitivity to differences in N and k for the commonly used clustering coefficient and small-world index. To avoid such a bias we tried to estimate the N,k-dependence for empirical networks, which can serve to correct for size effects, if successful. We also add a number of methods used in social sciences that build on statistics of local network structures including exponential random graph models and motif counting. We show that none of the here-investigated methods allows for a reliable and fully unbiased comparison, but some perform better than others.

Subfields: Cognitive Neuroscience, Experimental and Cognitive Psychology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/fbe0aeb6d1aaca06c84c985ce63b7c061569dd99

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Citations in the corpus

The trees and the forest: Characterization of complex brain networks with minimum spanning trees. (2014, positive): “For instance in the case of comparing networks with different N, a bias can be expected that cannot be completely solved with normalized measures (van Wijk et al., 2010; Zalesky et al., 2010).”
On the use of correlation as a measure of network connectivity (2012, background): “Received 26 November 2011 Revised 31 January 2012 Accepted 3 February 2012 Available online 11 February 2012”
Learning Graph Representations with Recurrent Neural Network Autoencoders (2018, background): “For instance, in neuroscience, comparing brain networks represented by graphs helps to identify brains with neurological disorders [43].”
Graph analysis of the human connectome: Promise, progress, and pitfalls (2013, positive): “------Figure 4 about here------ In a systematic analysis, van Wijk et al. (2010) examined the biases associated with a range of thresholding techniques, including τ- and κ-based thresholding, normalization by matched surrogate data, the use of exponential random graph models (Robins, et al., 2007;…”
The minimum spanning tree: An unbiased method for brain network analysis (2015, positive): “However, this may result in either inclusion of spurious or noisy connections in networks (for too high density values or too high average degree) or the exclusion of relevant connection in networks (for too low density values or too low average degree) (van Wijk et al., 2010).”
Investigating the structure of semantic networks in low and high creative persons (2014, methodology): “Network parameters calculated, with the MatLab Brain Connectivity Toolbox (Rubinov and Sporns, 2010) were: CC, ASPL, the average mean amount of edges per node [ < k > , van Wijk et al., 2010 and the network’s diameter (D)].”
Nine quick tips for analyzing network data (2019, background): “In this case, special care should be taken when comparing values because metrics differences can be a side effect of differences in simple network characteristics such as the number of nodes or edges (see common pitfalls mentioned in [26] for brain networks and a discussion on covariation of metrics”
Proportional thresholding in resting‐state fMRI functional connectivity networks and consequences for patient‐control connectome studies: Issues and recommendations (2017, positive): “However, as discussed in the graph theoretical studies of Van Wijk et al. (2010) and others (e.g. (Alexander-Bloch et al., 2010; Fornito et al., 2013; van den Heuvel and Fornito, 2014) the inclusion of lower and thus potentially less reliable correlations as functional network edges can have an e ﬀ…”
GRETNA: a graph theoretical network analysis toolbox for imaging connectomics (2015, background): “Note that the same connectivity strength threshold usually leads to a different number of edges in the resultant networks, which could confound between-group comparisons in network topology (van Wijk et al., 2010).”
The Segregation and Integration of Distinct Brain Networks and Their Relationship to Cognition (2016, background): “To be able to compare graphs most directly across conditions and participants, we equated the number of edges across all graphs (van Wijk et al., 2010; Garrison et al., 2015).”