Brain Structure and Function

, Volume 221, Issue 4, pp 2361–2366 | Cite as

Is the brain really a small-world network?

  • Claus C. Hilgetag
  • Alexandros Goulas
Open Access
Brain Mythology


Brain connectivity Connectome Hierarchical modular networks Large-world networks 

A matter of network topology

It is commonly assumed that the brain is a small-world network (e.g., Sporns and Honey 2006). Indeed, one of the present authors claimed as much 15 years ago (Hilgetag et al. 2000). The small-worldness is believed to be a crucial aspect of efficient brain organization that confers significant advantages in signal processing (e.g., Lago-Fernández et al. 2000). Correspondingly, the small-world organization is deemed essential for healthy brain function, as alterations of small-world features are observed in patient groups with Alzheimer’s disease (Stam et al. 2007), autism (Barttfeld et al. 2011) or schizophrenia spectrum diseases (Liu et al. 2008; Wang et al. 2012; Zalesky et al. 2011).

While the colloquial idea of a small, interconnected world has a long tradition (e.g., Klemperer 1938), the present concept of small-world features of networks is frequently associated with the Milgram experiment (Milgram 1967) that demonstrated surprisingly short paths across social networks (‘six degrees of separation’). The concept was formalized by Watts and Strogatz (1998), who derived small-world networks from regular networks by including a small proportion of random network shortcuts. Such an organization results in short paths across the whole network—almost as small as in random networks—combined with local ‘cliquishness’ (or clustering) of neighboring nodes, due to dense local interconnections. These features can be mathematically summarized by the small-world coefficient (Humphries et al. 2006), which is defined as the clustering coefficient of a given network (normalized by the clustering coefficient of a same-size random network) divided by the network’s normalized average shortest pathlength. While any network that has a small-world coefficient larger than one is formally a small-world network, for many researchers, the term has become associated with the specific Watts and Strogatz model that is based on the partial random rewiring of a regular network (Fig. 1a). Indeed, the estimation of the rewiring probability has been used to directly associate real-world networks with the Watts and Strogatz model (Humphries and Gurney 2008). Incidentally, the small-world coefficient might not faithfully capture the small-world property as originally described by Watts and Strogatz (1998). Therefore, an alternative coefficient has been proposed that compares the clustering of the network to a lattice instead of a random network (Telesford et al. 2011).
Fig. 1

Classical small-world network (a) versus hierarchical modular network (b). Classical small-world networks can be derived by partial random rewiring of regular networks, which results in high clustering and relatively short path lengths. While hierarchical modular networks may also possess these features, they can also be large-world networks with a finite topological dimension. This aspect makes them an intriguing model for brain networks. Adapted from Watts and Strogatz (1998) and Kaiser et al. (2007)

A large number of empirical network data conform to the small-world features of short paths combined with high clustering, including many neural networks—but do these features capture the essence of the topological organization of brain networks? Real neural networks are very different from both regular and random networks; thus, a small-world organization, which by Watts and Strogatz’s (1998) model may be perceived as a blend of the two, does not appear as an intuitive blueprint of the brain. Moreover, it has been known for a while in network science that many different topological arrangements are possible within the confines of the small-world properties. For instance, small-world networks can show diversity with respect to how their topological features change across scales (Amaral et al. 2000) and possess or lack a modular structure (Meunier et al. 2010). Thus, the small-world structure can co-exist with, but does not necessarily entail, diverse topological properties and dynamic features.

What are characteristic topological features of brain networks? Ubiquitously observed features are a heterogeneous, non-random degree distribution, resulting in some nodes with more connections than others, so-called hubs (Sporns et al. 2007), as well as modules of connections, in which some nodes are more frequently linked with each other than with the rest of the network (Hilgetag et al. 2000; Sporns et al. 2004; Bullmore and Sporns 2009). These features may be combined, in so-called hub modules or rich clubs (Zamora-López et al. 2010; van den Heuvel and Sporns 2011), and repeated across several scales of organization, resulting in hierarchical network arrangements (Sporns 2006; Müller-Linow et al. 2008; Kaiser and Hilgetag 2010; Fig. 1b). Such topological aspects appear highly relevant for the organization of dynamic patterns observed in the networks (Hütt et al. 2014). In particular, activity patterns are strongly shaped by modular and hub features of cortical connections (Müller-Linow et al. 2008; Garcia et al. 2012; Gómez-Gardeñes et al. 2010).

Another global characterization of the organization of neural networks is provided by the so-called topological dimension which describes how quickly the whole network can be accessed from any of its nodes (Moretti and Muñoz 2013). This measure is not simply equivalent to path length, because the average path length does not specifically capture how quickly local node neighborhoods grow. Thus, networks with similar characteristic path length can have a different topological dimension, due to their distinct path length distributions. In classical small-world networks as described by Watts and Strogatz (1998; Fig. 1a), the networks are very well connected globally via the random network shortcuts. This means that the number of accessible nodes grows exponentially with the distance of steps from an initial node, formally corresponding to an infinite topological dimension (while ignoring finite-size effects, which mean that the dimension of finite networks is always finite). By contrast, the finite topological dimension that is the defining feature of so-called large-world networks (Moretti and Muñoz 2013) implies that some parts of the network are relatively inaccessible. The topological dimension, thus, offers an alternative perspective on the global organization of brain connectivity. Importantly, the characterization of large-world networks by their topological dimension and the traditional definition of small-world networks by features of clustering and short paths are not mutually exclusive. Nonetheless, they have different consequences for the global dynamics and ultimately the function of networks, as discussed further below. However, before entering this discussion, it is worth considering some practical aspects of gathering and analyzing brain network data that have implications for the inferred network organization.

Small-worldness depends on the aperture of empirical studies

Brain connectivity is traditionally established by invasive histochemical tracing of connections in the brains of animal models. Extensive collations of such data for the macaque monkey cortex (Felleman and Van Essen 1991; Stephan et al. 2001) or for the cat (Scannell et al. 1999) formally show a small-world organization, although based on an arrangement of multiple, interlinked network modules, rather than a classical regular-with-random-links small-world network (Hilgetag and Kaiser 2004). More recently, however, extended quantitative compilations of connectivity data for the macaque cortex (Markov et al. 2014) appear to suggest a different picture. In particular, these data imply a very high density of cortico-cortical connections, with more than 60 % of the possible connections actually existing (Markov et al. 2013). Such high density renders the networks featureless, also in terms of a small-world organization, as the clustering and path lengths of these densely connected networks do not differ from those of same-size comparison networks (matched in the number of nodes, edges and degree distribution).

The measured network density is influenced by a number of factors, such as the amount of injected tracer, the thoroughness of sampling of labeled sections, or by treating projections as existing pathways no matter if they involve many or few fibers. The density of neural paths varies considerably, over five or more orders of magnitude, that is, from fewer than 10 to more than 100,000 labeled projection neurons (Markov et al. 2011). While it is still unclear how much the physiological or functional impact of a projection changes with its fiber density (e.g., Vanduffel et al. 1997), such graded networks might be more suitably analyzed by approaches that account for the differential weight of pathways (Rubinov and Sporns 2011). The issue is underlined by the fact that examples can be found where brain networks resemble a large-world network when only the stronger connections are taken into account, while incorporating the weakest connections shrinks them to a small-world network (Gallos et al. 2012). Thus, an analysis that takes into account the weight of the connections, especially in very dense networks, and employs weighted versions of network metrics including the small-world index (e.g. Bolaños et al. 2013) can be instructive.

Connectivity data for the human brain can be inferred from in vivo imaging techniques such as diffusion weighted imaging (DWI). However, these approaches face problems of limited specificity and sensitivity (Thomas et al. 2014). Hence, some connections might be unresolved by such methods (e.g., Zalesky and Fornito 2009; Li et al. 2012). Generally, experimental approaches for discovering networks have apertures that are tuned to particular scales and data features. Therefore, not all networks at all scales are accessible with the same method (such as electron microscopy or DWI). An additional methodological caveat is the application of thresholds to such in vivo data to create networks that can be analyzed (Rubinov and Sporns 2010), which discards weak existing connections. Such empirical limitations may lead to inaccurate brain network representations, including evidence of small-worldness.

Another important factor determining the density of links is the parcellation of the network nodes. The coarser the parcellation, the denser the network will appear, while a finer parcellation results in sparser connectivity. For instance, assembling the human connectome from diffusion data results in a density of 26 % at a coarse parcellation, whereas a high resolution parcellation results in a density of just 3 % (Samu et al. 2014). Moreover, if a relatively coarse parcellation scheme, such as the Regional Map (Kötter and Wanke 2005), is employed for representing connections from the CoCoMac primate connectivity database (, a very high network density is obtained (i.e. 79 %; Goulas et al. 2014). This density is much higher than the one obtained for using the same database with a different, more fine-grained parcellation scheme (i.e. ~1 %; Modha and Singh 2010) and very close to the high density of the primate cortico-cortical network estimated in more recent studies (Markov et al. 2014). The parcellation coarseness also leads to systematic changes in the small-worldness of the brain, with more finely grained networks showing a higher small-world index (Zalesky et al. 2010).

That begs the question, what is the neural network density and topology of the brain at the finest, cellular, level of parcellation? On average, the density of human brain connectivity at the cellular level is very sparse. The average number of synapses of neurons (~104) (Braitenberg and Schüz 1998) divided by the number of neural elements (~1010) (Herculano-Houzel 2012) results in a very low average probability of any two neurons in the brain making contact (10−6), implying a highly dispersed network. The dispersion may be ameliorated via the local clustering of connectivity, for instance, in neural modules such as columns and layers so that within these compartments, the density is likely much higher (Markram 2006, but see Stepanyants et al. 2009). To be true, we do not really know the exact organization of brain networks at the cellular scale, since extensive empirical microconnectome data for the mammalian brain are still lacking. The one existing example of a complete neuronal microconnectome, of the nematode C. elegans, may be too small to be helpful here. While this network fits the small-world features of high clustering and short pathlengths (Watts and Strogatz 1998), it also appears to have a finite topological dimension (cf. Supplementary Figure). However, it is difficult to make a conclusive statement about this point, given the small number of just 302 neuronal network nodes that can be evaluated (White et al. 1986).

Nonetheless, it appears to be a reasonable guess that the organization of the mammalian brain follows a hierarchical organization (Meunier et al. 2009), with dense modules at the local level (cellular circuits, laminar compartments) that are encapsulated in increasingly larger modules (cortical columns, areas, whole lobes), but with very sparse overall connectivity (Hilgetag and Hütt 2014; Fig. 1b). This organization may produce different network topologies at different scales; for example, synaptic connectivity within local neuronal populations might form small-world or random networks. At the global scale, however, such a network may have a finite topological dimension (Moretti and Muñoz 2013), unlike classical small-world networks.

Dynamics of small-world versus large-world networks

What are implications for brain dynamics, if the brain is organized as a small-world or a large-world network? As pointed out already by Watts and Strogatz (1998), the coexistence of high clustering and short average distances facilitates the integration and spreading of signals. Such a small-world organization might enhance dynamic complexity (Sporns et al. 2000), due to increased reentry (Edelman and Gally 2013) and signal integration. Additionally, network shortcuts can insert incoherent, remote information (‘topological noise’) into local coherent neighborhoods (Marr and Hütt 2006). Such shortcuts may enhance the robustness of classification or local decision-making tasks across networks (Moreira et al. 2004).

On the other hand, a non-modular small-world arrangement of the Watts and Strogatz type may not be the optimal topology for supporting limited self-sustained activity (Kaiser et al. 2007). Such sustained yet constrained network activity is an essential ingredient of healthy brain dynamics, by maintaining the balance between activity dissipating too quickly or becoming pathologically large. Sustained activity in an excitable system is also an important precondition for the phenomenon of criticality, the positioning of the system precisely at the boundary between order and chaos (or disorder). The feature of criticality has been associated with several desirable functional properties, such as a large dynamic range, high adaptability or optimal information processing (e.g., Shew and Plenz 2013). It has been observed that sustained network activity is better supported in hierarchical modular networks (Kaiser et al. 2007; Kaiser and Hilgetag 2010; Wang et al. 2011) than classical small-world networks. More generally, hierarchical modular networks that are large-world networks with a finite topological dimension possess so-called Griffith phases (Muñoz et al. 2010) that expand the parameter range of criticality. In contrast to the precise fine-tuning that is required in other systems to reach a critical point, for instance by carefully balancing local excitation and inhibition, an expanded range of criticality arises directly from the topology of large-world networks. This makes such systems dynamically appealing and robust, in addition to further aspects of structural robustness conferred by hierarchical modularity, such as the potential to assemble a large network from smaller subnetworks of similar organization, or split a large network into smaller units while maintaining their dynamical features (Robinson et al. 2009).


To decide if the brain really adheres to a small-world organization and to understand the implications of this organization, one needs to take several theoretical and empirical aspects into account. The small-world property is influenced by practical aspects of analyzing brain connectivity; for example, whether connections are treated as weighted or as binary. Proper topological assessment requires the re-examination of weighted networks, an approach that entails a new definition of small-world properties (Bolaños et al. 2013). More generally, the small-world property depends on the aperture of experimental methods for studying brain connectivity, the coarseness of used parcellations and the resulting density of the studied brain networks. In that respect, recent compilations of brain connectivity at the macroscopic level do not appear to form small-world networks, due to their high density. If considered at the cellular level, brain networks are also unlikely to form classical small-world networks. While detailed empirical data are still lacking, a reasonable guess is that the large-scale neuronal networks of the brain are arranged as globally sparse hierarchical modular networks. Even if they fit the general features of local clustering and relatively short average paths, the small-world concept can miss the point of other essential topological properties of such brain networks, such as their finite topological dimension, which can also be used to characterize them as large-world networks.

This means that, at cellular resolution, the brain may be a large-world network, rather than a classical small-world network. Intriguingly, such a topology might fundamentally enhance the brain’s dynamic stability and information processing abilities. Thus, while most researchers have by now become accustomed to the small world of brain connectivity and might find it quite comforting, it may be just as exciting to step out and explore the large world of the brain.



We are grateful to Marc-Thorsten Hütt for critical discussion of this manuscript. The work of CCH has been supported by DFG grants SFB 936/A1 and HI 1286/5-1.

Supplementary material

429_2015_1035_MOESM1_ESM.pdf (32 kb)
Supplementary material 1 (PDF 32 kb)


  1. Amaral L, Scala A, Barthelemy M, Stanley HE (2000) Classes of small-world networks. Proc Natl Acad Sci USA 97:11149–11152CrossRefPubMedPubMedCentralGoogle Scholar
  2. Barttfeld P et al (2011) A big-world network in ASD: dynamical connectivity analysis reflects a deficit in long-range connections and an excess of short-range connections. Neuropsychologia 49:254–263CrossRefPubMedGoogle Scholar
  3. Bolaños M, Bernat EM, He B, Aviyente S (2013) A weighted small world network measure for assessing functional connectivity. J Neurosci Methods 212:133–142CrossRefPubMedGoogle Scholar
  4. Braitenberg V, Schüz A (1998) Cortex: statistics and geometry of neuronal connectivity. Springer, BerlinCrossRefGoogle Scholar
  5. Bullmore E, Sporns O (2009) Complex brain networks: graph theoretical analysis of structural and functional systems. Nat Rev Neurosci 10:186–198CrossRefPubMedGoogle Scholar
  6. Edelman GM, Gally JA (2013) Reentry: a key mechanism for integration of brain function. Front Integr Neurosci 7:63CrossRefPubMedPubMedCentralGoogle Scholar
  7. Felleman DJ, Van Essen DC (1991) Distributed hierarchical processing in the primate cerebral cortex. Cereb Cortex 1:1–47CrossRefPubMedGoogle Scholar
  8. Gallos LK, Makse H, Sigman M (2012) A small world of weak ties provides optimal global integration of self-similar modules in functional brain networks. Proc Natl Acad Sci USA 109:2825–2830CrossRefPubMedPubMedCentralGoogle Scholar
  9. Garcia GC et al (2012) Building blocks of self-sustained activity in a simple deterministic model of excitable neural networks. Front Comput Neurosci 6:50CrossRefPubMedPubMedCentralGoogle Scholar
  10. Gómez-Gardeñes J, Zamora-López G, Moreno Y, Arenas A (2010) From modular to centralized organization of synchronization in functional areas of the cat cerebral cortex. PLoS ONE 5:e12313CrossRefPubMedPubMedCentralGoogle Scholar
  11. Goulas A, Bastiani M, Bezgin G, Uylings HBM, Roebroeck A et al (2014) Comparative analysis of the macroscale structural connectivity in the macaque and human brain. PLoS Comput Biol 10:e1003529CrossRefPubMedPubMedCentralGoogle Scholar
  12. Herculano-Houzel S (2012) The remarkable, yet not extraordinary, human brain as a scaled-up primate brain and its associated cost. Proc Natl Acad Sci 109:10661–10668CrossRefPubMedPubMedCentralGoogle Scholar
  13. Hilgetag CC, Hütt M-T (2014) Hierarchical modular brain connectivity is a stretch for criticality. Trends Cogn Sci. doi: 10.1016/j.tics.2013.10.016 PubMedGoogle Scholar
  14. Hilgetag CC, Kaiser M (2004) Clustered organization of cortical connectivity. Neuroinformatics 2:353–360CrossRefPubMedGoogle Scholar
  15. Hilgetag CC et al (2000) Anatomical connectivity defines the organization of clusters of cortical areas in the macaque monkey and the cat. Philos Trans R Soc B 355:91–110CrossRefGoogle Scholar
  16. Humphries MD, Gurney K (2008) Network “Small-World-Ness”: a quantitative method for determining canonical network equivalence. PLoS ONE 3:e0002051CrossRefPubMedGoogle Scholar
  17. Humphries MD, Gurney K, Prescott TJ (2006) The brainstem reticular formation is a small-world, not scale-free, network. Proc Biol Sci 273:503–511CrossRefPubMedPubMedCentralGoogle Scholar
  18. Hütt M-T, Kaiser M, Hilgetag CC (2014) Network-guided pattern formation of neural dynamics. Philos Trans R Soc B 369:20130522CrossRefGoogle Scholar
  19. Kaiser M, Hilgetag CC (2010) Optimal hierarchical modular topologies for producing limited sustained activation of neural networks. Front Neuroinform 4:8CrossRefPubMedPubMedCentralGoogle Scholar
  20. Kaiser M et al (2007) Criticality of spreading dynamics in hierarchical cluster networks without inhibition. New J Phys 9:110CrossRefGoogle Scholar
  21. Klemperer V (1938) Tagebücher 1937–1939, p 122 (“piccolo mondo moderno”), Aufbau Taschenbuch Verlag 1999Google Scholar
  22. Kötter R, Wanke E (2005) Mapping brains without coordinates. Philos Trans R Soc Lond B Biol Sci 360:751–766CrossRefPubMedPubMedCentralGoogle Scholar
  23. Lago-Fernández LF, Huerta R, Corbacho F, Sigüenza JA (2000) Fast response and temporal coherent oscillations in small-world networks. Phys Rev Lett 84:2758–2761CrossRefPubMedGoogle Scholar
  24. Li L, Rilling JK, Preuss TM, Glasser MF, Damen FW et al (2012) Quantitative assessment of a framework for creating anatomical brain networks via global tractography. NeuroImage 61:1017–1030CrossRefPubMedPubMedCentralGoogle Scholar
  25. Liu Y et al (2008) Disrupted small-world networks in schizophrenia. Brain 131:945–961CrossRefPubMedGoogle Scholar
  26. Markov NT et al (2011) Weight consistency specifies regularities of macaque cortical networks. Cereb Cortex 21:1254–1272CrossRefPubMedPubMedCentralGoogle Scholar
  27. Markov NT et al (2013) Cortical high-density counterstream architectures. Science 342:1238406CrossRefPubMedPubMedCentralGoogle Scholar
  28. Markov NT et al (2014) A weighted and directed interareal connectivity matrix for macaque cerebral cortex. Cereb Cortex 24:17–36CrossRefPubMedPubMedCentralGoogle Scholar
  29. Markram H (2006) The blue brain project. Nat Rev Neurosci 7:153–160CrossRefPubMedGoogle Scholar
  30. Marr C, Hütt M (2006) Similar impact of topological and dynamic noise on complex patterns. Phys Lett A 349:302–305CrossRefGoogle Scholar
  31. Meunier D, Lambiotte R, Fornito A, Ersche KD, Bullmore ET (2009) Hierarchical modularity in human brain functional networks. Front Neuroinform 3:37. doi: 10.3389/neuro.11.037.2009 CrossRefPubMedPubMedCentralGoogle Scholar
  32. Meunier D, Lambiotte R, Bullmore ET (2010) Modular and hierarchically modular organization of brain networks. Front Neurosci 4:200CrossRefPubMedPubMedCentralGoogle Scholar
  33. Milgram S (1967) The small world problem. Psychol Today 2:60–67Google Scholar
  34. Modha DS, Singh R (2010) Network architecture of the long-distance pathways in the macaque brain. Proc Natl Acad Sci USA 107:13485–13490CrossRefPubMedPubMedCentralGoogle Scholar
  35. Moreira A, Mathur A, Diermeier D, Amaral L (2004) Efficient system-wide coordination in noisy environments. Proc Natl Acad Sci USA 101:12085CrossRefPubMedPubMedCentralGoogle Scholar
  36. Moretti P, Muñoz MA (2013) Griffiths phases and the stretching of criticality in brain networks. Nat Commun 4:2521CrossRefPubMedGoogle Scholar
  37. Müller-Linow M et al (2008) Organization of excitable dynamics in hierarchical biological networks. PLoS Comput Biol 4:e1000190CrossRefPubMedPubMedCentralGoogle Scholar
  38. Muñoz MA et al (2010) Griffiths phases on complex networks. Phys Rev Lett 105:128701CrossRefPubMedGoogle Scholar
  39. Robinson PA et al (2009) Dynamical reconnection and stability constraints on cortical network architecture. Phys Rev Lett 103:108104CrossRefPubMedGoogle Scholar
  40. Rubinov M, Sporns O (2010) Complex network measures of brain connectivity: uses and interpretations. NeuroImage 52:1059–1069CrossRefPubMedGoogle Scholar
  41. Rubinov M, Sporns O (2011) Weight-conserving characterization of complex functional brain networks. NeuroImage 56:2068–2079CrossRefPubMedGoogle Scholar
  42. Samu D, Seth AK, Nowotny T (2014) Influence of wiring cost on the large-scale architecture of human cortical connectivity. PLoS Comput Biol 10:e1003557CrossRefPubMedPubMedCentralGoogle Scholar
  43. Scannell JW et al (1999) The connectional organization of the cortico-thalamic system of the cat. Cereb Cortex 9:277–299CrossRefPubMedGoogle Scholar
  44. Shew WL, Plenz D (2013) The functional benefits of criticality in the cortex. Neuroscientist 19:88–100CrossRefPubMedGoogle Scholar
  45. Sporns O (2006) Small-world connectivity, motif composition, and complexity of fractal neuronal connections. Biosystems 85:55–64CrossRefPubMedGoogle Scholar
  46. Sporns O, Honey CJ (2006) Small worlds inside big brains. Proc Natl Acad Sci USA 103:19219–19220CrossRefPubMedPubMedCentralGoogle Scholar
  47. Sporns O et al (2000) Theoretical neuroanatomy: relating anatomical and functional connectivity in graphs and cortical connection matrices. Cereb Cortex 10:127–141CrossRefPubMedGoogle Scholar
  48. Sporns O et al (2004) Organization, development and function of complex brain networks. Trends Cogn Sci 8:418–425CrossRefPubMedGoogle Scholar
  49. Sporns O et al (2007) Identification and classification of hubs in brain networks. PLoS ONE 10:e1049CrossRefGoogle Scholar
  50. Stam CJ et al (2007) Small-world networks and functional connectivity in Alzheimer’s disease. Cereb Cortex 17:92–99CrossRefPubMedGoogle Scholar
  51. Stepanyants A, Martinez LM, Ferecskó AS, Kisvárday ZF (2009) The fractions of short- and long-range connections in the visual cortex. Proc Natl Acad Sci USA 106:3555–3560CrossRefPubMedPubMedCentralGoogle Scholar
  52. Stephan KE et al (2001) Advanced database methodology for the Collation of Connectivity data on the Macaque brain (CoCoMac). Philos Trans R Soc B 356:1159–1186CrossRefGoogle Scholar
  53. Telesford QK, Joyce KE, Hayasaka S, Burdette JH, Laurienti PJ (2011) The ubiquity of small-world networks. Brain Connect 1:367–375CrossRefPubMedPubMedCentralGoogle Scholar
  54. Thomas C, Ye FQ, Okan Irfanoglu M, Modi P, Saleem KS, Leopold DA, Pierpaoli C (2014) Anatomical accuracy of brain connections derived from diffusion MRI tractography is inherently limited. Proc Natl Acad Sci 111:16574–16579CrossRefPubMedPubMedCentralGoogle Scholar
  55. van den Heuvel MP, Sporns O (2011) Rich-club organization of the human connectome. J Neurosci 31:15775–15786CrossRefPubMedGoogle Scholar
  56. Vanduffel W, Payne BR, Lomber SG, Orban GA (1997) Functional impact of cerebral connections. Proc Natl Acad Sci USA 94:7617–7620CrossRefPubMedPubMedCentralGoogle Scholar
  57. Wang S-J et al (2011) Sustained activity in hierarchical modular neural networks: self-organized criticality and oscillations. Front Comput Neurosci 5:30PubMedPubMedCentralGoogle Scholar
  58. Wang Q et al (2012) Anatomical insights into disrupted small-world networks in schizophrenia. NeuroImage 59:1085–1093CrossRefPubMedGoogle Scholar
  59. Watts DJ, Strogatz SH (1998) Collective dynamics of “small-world” networks. Nature 393:440–442CrossRefPubMedGoogle Scholar
  60. White JG, Southgate E, Thomson JN, Brenner S (1986) The structure of the nervous system of the nematode Caenorhabditis elegans. Philos Trans R Soc Lond B Biol Sci 314:1–340CrossRefPubMedGoogle Scholar
  61. Zalesky A, Fornito A (2009) A DTI-derived measure of cortico-cortical connectivity. Trans Med Imaging 28(7):1023–1036CrossRefGoogle Scholar
  62. Zalesky A, Fornito A, Harding IH, Cocchi L, Yucel M, Pantelis C, Bullmore ET (2010) Whole-brain anatomical networks: does the choice of nodes matter? Neuroimage 50(3):970–983CrossRefPubMedGoogle Scholar
  63. Zalesky A, Fornito A, Seal ML, Cocchi L, Westin C-F, Bullmore ET, Egan GF, Pantelis C (2011) Disrupted axonal fiber connectivity in schizophrenia. Biol Psychiatry 69:80–89CrossRefPubMedGoogle Scholar
  64. Zamora-López G et al (2010) Cortical hubs form a module for multisensory integration on top of the hierarchy of cortical networks. Front Neuroinform 4:1PubMedPubMedCentralGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Computational NeuroscienceUniversity Medical Center Hamburg-EppendorfHamburgGermany
  2. 2.Department of Health SciencesBoston UniversityBostonUSA
  3. 3.Max Planck Research Group Neuroanatomy and ConnectivityMax Planck Institute for Human Cognitive and Brain SciencesLeipzigGermany

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