Graph analysis of the human connectome: Promise, progress, and pitfalls
Introduction
The uptake of graph theoretical tools into brain connectivity research has proceeded at such a phenomenal rate that one could gain the impression that graph theory was only very recently developed. Yet graph theory, as a branch of mathematics, dates back over two hundred years, at least as far as Leonhard Euler's thesis on the ‘bridges of Königsberg’ (Euler, 1736). Euler introduced the notion of representing an interconnected system as the edges (bridges) and nodes (locations) of a graph, and showed how their organization could yield complex topological properties with specific implications (here, efficiently traversing bridges). Graph theory is generally considered to be a branch of combinatorics — that is, concerned with the study of discrete structures — and has been used to address a broad range of problems from “covering algorithms” (i.e. how to color maps) to the Traveling Salesman Problem (i.e. in which order should a salesman visit a set of multiple cities to minimize his distance traveled). Already we encounter two of the basic elements of graph theory: First, it was born from a real world problem and remains an extraordinarily pragmatic and empirically useful field. Second, it is dependent upon a discretization of the system under study; an important issue when it comes to imaging neuroscience. Importantly, graph theory is ideally suited to study complex systems of interacting elements, the brain being one example.
Several major advances in the 20th century preceded the rapid uptake of graph theory into neuroscience. Chief amongst these was the work on random graphs — pioneered by Erdos and Renyi (Erdos and Renyi, 1959) — and its later extension to random scale-free networks (Barabási and Albert, 1999) — that is, random networks with a power law degree distribution (degree referring to the number of connections possessed by each network node). Another seminal discovery concerned the so-called “small-world” phenomenon; the simultaneous presence of locally clustered connectivity and short path lengths between nodes in many real-world networks (Stephan et al., 2000, Watts and Strogatz, 1998). This organization provides a topological foundation for the dual properties of functional integration (short path lengths) and functional segregation (high clustering) — key organizational principles of the brain (Friston et al., 2004, Tononi et al., 1994). The realization that many complex systems found in nature, ranging from phylogenies, social interactions, electrical and telecommunication grids, transportation systems and metabolic networks, can be characterized by one or multiple non-trivial organizational properties, including power-law scaling, small-worldness and other features, such as modularity (Fortunato, 2010), hierarchy (Ravasz and Barabasi, 2003) and rich-club ordering (Colizza et al., 2006), pointed to a set of similarities (“universalities”) in very diverse systems (Newman, 2003, Strogatz, 2001) including the brain (Bullmore et al., 2009).
The connectome, representing the complete set of neural elements and inter-connections comprising the brain (Sporns et al., 2005), is a perfect candidate for graph theoretic analysis. Indeed, it was seminal work examining the organization of large-scale connectivity networks in the Caenorhabditis elegans (White et al., 1986), cat (Scannell and Young, 1993, Scannell et al., 1999), and non-human primate nervous systems (Felleman and Van Essen, 1991, Hilgetag et al., 1996, Sporns et al., 2000, Young, 1992), as well as early computational analyses of the principles of cortical organization (Stephan et al., 2000, Tononi et al., 1999), that illustrated the potential neuroscientific applications of graph theory. This research mandated a gold standard in connectivity data and provided an impetus for early connectivity databases in non-human primates, such as “CoCoMac” (ww.cocomac.org), which can be seen as a prelude to The Human Connectome Project (Stephan et al., 2000, Stephan et al., 2001, Kötter, 2004; see also Stephan, 2013 in this issue). The parsimonious nature, computational properties and intuitive appeal of small world networks made their uptake into brain connectivity research — via analyses of CoCoMac — almost immediate (Hilgetag et al., 2000b, Sporns and Zwi, 2004, Sporns et al., 2000, Stephan et al., 2000) and represents a compelling example of the confluence of computational and empirical neuroscience. Though much of this early work focused on structural connectivity data in non-human species, the rapidly emerging interest in functional connectivity led to the first demonstrations, obtained from MEG data (Stam, 2004b) then from fMRI data (Achard et al., 2006, Eguiluz et al., 2005, Salvador et al., 2005), of small-world and scale-free properties in human brain functional networks. These analyses where then extended to the first in vivo structural maps of the human connectome generated using diffusion imaging (Hagmann et al., 2007, Hagmann et al., 2008, Skudlarski et al., 2008, Zalesky and Fornito, 2009) and found rapid applications in the clinical neurosciences (Bassett et al., 2008, Fornito and Bullmore, 2010, Fornito and Bullmore, 2012, Fornito et al., 2012b, He et al., 2009, Liu et al., 2008, Micheloyannis et al., 2006, Rubinov et al., 2009a, Stam et al., 2007, van den Heuvel et al., 2010, Verstraete et al., 2011, Xie and He, 2011, Zalesky et al., 2011).
The exponential growth in brain connectivity research arguably constitutes something akin to a scientific revolution (Kuhn, 1962) either complementing or even subverting the prior prominence of functional specialization in the brain (Friston, 2011). In this context, graph theory provides a more compelling framework for the analysis of large-scale brain network architecture than traditional, mass univariate neuroimaging and carries the potential to revolutionize our understanding of brain organization (Bullmore and Sporns, 2009, Sporns, 2011b, Sporns, 2012, Stam and Reijneveld, 2007). However, the extent to which this promise can be realized is critically dependent upon the validity of the graph representation itself. As a branch of combinatorics, graph theory is reliant upon an unambiguous discretization of the brain into distinct nodes and their interconnecting edges, neither of which are trivial. Moreover, the application of graph theory to neuroscientific data poses several challenges with important implications for how results should be interpreted. Our goal in this article is to highlight these challenges, draw attention to potential pitfalls and discuss progress towards addressing them. Specifically, we do this in relation to the two major steps involved in connectomic analysis: (1) building an accurate map of the connectome; and (2) analyzing and making sense of the resulting data. We focus principally on in vivo macro-scale connectomics with MRI, a field that has rapidly adopted many graph theoretic concepts and techniques and which is central to large-scale initiatives such as The Human Connectome Project (Van Essen et al., 2012). We note however, that graph theory can be used to characterize data acquired using other modalities, such as EEG/MEG (Bassett et al., 2006, Brookes et al., 2011, Hipp et al., 2011, Kitzbichler et al., 2011, Rubinov et al., 2009a, Stam, 2004a, Stam et al., 2007, Zalesky et al., 2012a), and that many of the issues discussed here also apply to these analyses. Excellent introductions to the basic concepts of graph theory and their application to neuroscience have been provided elsewhere (Albert and Barabasi, 2002, Bullmore and Bassett, 2011, Bullmore and Sporns, 2009, Newman, 2003, Sporns, 2011b, Sporns, 2012, Sporns et al., 2004, Stam and Reijneveld, 2007).
Section snippets
Building a connectome
The validity of any graph-based model of a complex system depends on the extent to which its nodes and edges represent true subsystems and their interactions, respectively, of the system under investigation. In some applications, this is straightforward. For example, in social networks, each node represents a person and edges can represent Facebook links (Lewis et al., 2008), email traffic (Barabasi, 2005), co-authored publications (Newman and Girvan, 2004) or some other measure of social
Analyzing the connectome
Once nodes and edges are accurately defined, the tools of graph theory can be used to characterize a wide array of network properties. When used judiciously, these methods can provide novel insights into brain organization. When used carelessly, they may lead to misleading or incorrect conclusions. In the following, we consider several important issues: namely, addressing the multiple comparison problem in connectomics, graph thresholding, the interpretation of topological measures, reference
Discussion
The accumulation of large connectomic datasets across spatial scales, data modalities and clinical populations has ushered in an exciting era of neuroimaging science, while also challenging the field to find meaningful summary statistics of system organization. A graph theoretical approach provides the opportunity to address this challenge, providing a rich repertoire of summary metrics, new means of constructing and testing specific hypotheses, and a conceptual approach that positions brain
Acknowledgments
AF was supported by the Australian National Health and Medical Research Council (ID: 1050504), a University of Melbourne CR Roper Fellowship and a Monash University Larkins Fellowship. MB acknowledges the support of the Australian Research Council, the National Health and Medical Research Council and the Brain Network Recovery Group JSMF22002082 and thanks Anton Lord for the data presented in Fig. 6A. AZ was supported by the inaugural Melbourne Neuroscience Institute Fellowship.
Conflict of
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