Elsevier

NeuroImage

Volume 60, Issue 4, 1 May 2012, Pages 2096-2106
NeuroImage

On the use of correlation as a measure of network connectivity

https://doi.org/10.1016/j.neuroimage.2012.02.001Get rights and content

Abstract

Numerous studies have demonstrated that brain networks derived from neuroimaging data have nontrivial topological features, such as small-world organization, modular structure and highly connected hubs. In these studies, the extent of connectivity between pairs of brain regions has often been measured using some form of statistical correlation. This article demonstrates that correlation as a measure of connectivity in and of itself gives rise to networks with non-random topological features. In particular, networks in which connectivity is measured using correlation are inherently more clustered than random networks, and as such are more likely to be small-world networks. Partial correlation as a measure of connectivity also gives rise to networks with non-random topological features. Partial correlation networks are inherently less clustered than random networks. Network measures in correlation networks should be benchmarked against null networks that respect the topological structure induced by correlation measurements. Prevalently used random rewiring algorithms do not yield appropriate null networks for some network measures. Null networks are proposed to explicitly normalize for the inherent topological structure found in correlation networks, resulting in more conservative estimates of small-world organization. A number of steps may be needed to normalize each network measure individually and control for distinct features (e.g. degree distribution). The main conclusion of this article is that correlation can and should be used to measure connectivity, however appropriate null networks should be used to benchmark network measures in correlation networks.

Highlights

► Correlation networks more clustered than random networks. ► Correlation as a measure of connectivity induces nontrivial topological structure. ► Null networks proposed to normalize for structure induced by correlation.

Introduction

Recent years have witnessed a surge of interest in mapping and modeling the complicated web of connectivity that comprises the brain, termed the connectome. Large-scale brain networks have been central to this work (Bullmore and Sporns, 2009, Habeck and Moeller, 2011, He and Evans, 2010, Kaiser, 2011, Wig et al., 2011). They consist of a network of brain regions together with a measure of connectivity between every possible pair of these regions (Rubinov and Sporns, 2011).

In functional brain networks, connectivity is often measured using some form of statistical correlation. The brain activity at one region is correlated with the activity at another to quantify the strength and sign of any statistical dependency. When repeated for every possible pair of regions, the result is a network characterization of the brain's connectivity, where brain regions represent network nodes and correlation strengths correspond to connection weights.

Numerous studies have demonstrated functional brain networks have various nontrivial topological features, such as small-world organization, modular structure and highly connected hubs (Achard et al., 2006, Bassett and Bullmore, 2006, Hayasaka and Laurienti, 2010, Telesford et al., 2010, van den Heuvel and Hulshoff Pol, 2010, van den Heuvel et al., 2008). However, the interpretation of the small-world concept as well as other topological measures derived from path length is not straightforward in functional networks. This is because functional networks are intrinsically fully connected, and thus the “path length” between a pair of regions is already explicitly captured by the strength of correlation in brain activity; namely, the weight of the direct connection (Rubinov and Sporns, 2010).

The purpose of this article is to draw attention to the pitfalls of probing brain networks for small-world properties. Networks are said to be small-world if they are substantially more clustered than random networks, yet have approximately the same characteristic path length as random networks (Watts and Strogatz, 1998). This article begins by showing that networks in which connectivity is measured using correlation can inherently satisfy the two criteria required of small-world networks (Section 2). In particular, the use of correlation as a measure of connectivity in and of itself gives rise to networks that are inherently more clustered than random networks, and as such are more likely to be small-world networks.

A similar observation was recently reported (Bialonski et al., 2011), where the length and spectral content of multivariate electroencephalographic recordings were found to influence topological structure. The main contributions of this article are to provide an explanation for this observation, namely why correlation networks have a propensity to be small-world networks (Section 3), as well as to present strategies for generating null networks to normalize for the inherent topological structure induced by correlation measurements (Section 4). Related null networks have been generated using Markov chain (Bansal et al., 2009) and generative models (Volz, 2004).

In Section 3, evidence is presented suggesting that although strong positive correlation between X and Y as well as Y and Z theoretically does not imply strong positive correlation between X and Z (Langford et al., 2001), in functional brain networks, this is the case. In particular, strong positive correlation on the indirect path (X-Y-Z) implies correlation on the direct connection (X-Z) that is significantly greater than expected in random networks (although this does not mean that indirect paths necessarily predict direct connections in all instances). This effect is referred to as transitivity and is used to explain why correlation networks are inherently more clustered than random networks.

In Section 4, null networks are presented to normalize for the inherent topological structure found in correlation networks, resulting in more conservative estimates of small-world organization. Traditionally, normalization has been performed with respect to randomly rewired networks matched for degree distribution (Maslov and Sneppen, 2002). This kind of topology randomization is shown to overestimate the degree to which brain networks are small-world networks.

Whenever we refer to correlation networks in this article, we mean any kind of network in which connectivity is measured using the correlation coefficient. While we focus on functional brain networks, our conclusions and methods also apply to metabolite, protein and gene correlation networks (Gillis and Pavlidis, 2011, Junker and Schreiber, 2008), financial portfolio networks (Hirschberger et al., 2004) and anatomical brain networks in which connectivity is measured using correlation in cortical thickness or volume (Bassett et al., 2008, He et al., 2007).

Section snippets

Correlation induces small-world organization

A set of N random vectors were generated. Vectors comprised n elements, each independently sampled from a normal distribution of zero mean and unity variance. The correlation between every possible pair of vectors was calculated, yielding an N × N random correlation matrix with values ranging between -1 and 1 (Holmes, 1991). Vectors were generated independent of each other to ensure no interdependence. Non-random topological structure evident in the correlation matrix was therefore a matter of

Indirect paths predict direct connections

It has been shown that correlation networks are inherently more clustered than random networks, and as such are more likely to be small-world networks. In this section, an attempt is made to elucidate why this is the case.

The clustering coefficient can be defined in terms of triangles and triples. A triangle of node i is a subgraph comprising three nodes, one of which is i, and three connections. A triple of node i is a subgraph comprising three nodes, one of which is i, and two connections,

Normalization

Networks are customarily benchmarked against an ensemble of null networks to quantify the extent to which topological properties such as small-world organization are expressed. Topology randomization is the most widely used strategy for generating null networks, and is typically performed by randomly rewiring the observed network (Maslov and Sneppen, 2002).

This kind of random rewiring not only annihilates intrinsic topological structure in the empirical data, it also annihilates topological

Conclusion

Networks express pairwise relationships between sets of objects. Implicit to network analysis is that the mere measurement of these pairwise relations is topologically neutral. That is, measurement performed between one pair of objects does not affect measurements performed on other pairs in such a way that produces non-random topological structure.

This is not the case for networks measured using correlation or partial correlation. The use of correlation as a measure of connectivity gives rise

Acknowledgments

A.Z. is grateful for the support provided by Professor Trevor Kilpatrick as part of the inaugural Melbourne Neuroscience Institute Fellowship. This work was also supported by the Australian Research Council [DP0986320 to A.Z.]; the Melbourne Neuroscience Institute; and the National Health and Medical Research Council of Australia [C.J. Martin Fellowship to A.F. ID: 454797]. We are grateful to Dr Mikail Rubinov for reviewing the first draft of this manuscript and for providing many fruitful

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