Graph Theoretic Analysis of Human Brain Networks

  • Alex FornitoEmail author
Part of the Neuromethods book series (NM, volume 119)


The human brain is a highly interconnected network. It is thus suitable for investigation with graph theory, a branch of mathematics concerned with understanding systems of interacting elements. Graph theory has become a popular tool for analyzing human MRI data. In this work, brain networks are modeled as graphs of nodes connected by edges. The nodes represent distinct brain regions and the edges represent some measure of structural or functional interaction between regions. This representation enables the computation of a broad range of metrics that quantify diverse aspects of network organization, thus offering a powerful framework for understanding brain structure and function in both health and disease. This chapter overviews the principles and methods involved in building and analyzing graph theoretic models of the brain using MRI. It explains basic concepts, provides examples of how graph theory has shed new light on brain organization, and considers some limitations of current applications.

Key words

Connectome Connectivity Graph analysis Network Complexity MRI DTI fMRI 


  1. 1.
    Cherniak C (1990) The bounded brain: toward quantitative neuroanatomy. J Cogn Neurosci 2:58–68PubMedCrossRefGoogle Scholar
  2. 2.
    Sporns O, Tononi G, Kötter R (2005) The human connectome: a structural description of the human brain. PLoS Comput Biol 1:e42PubMedPubMedCentralCrossRefGoogle Scholar
  3. 3.
    Van Essen DC et al (2012) The Human Connectome Project: a data acquisition perspective. Neuroimage 62:2222–2231PubMedPubMedCentralCrossRefGoogle Scholar
  4. 4.
    Bohland JW et al (2009) A proposal for a coordinated effort for the determination of brainwide neuroanatomical connectivity in model organisms at a mesoscopic scale. PLoS Comput Biol 5:e1000334PubMedPubMedCentralCrossRefGoogle Scholar
  5. 5.
    Kandel ER, Markram H, Matthews PM, Yuste R, Koch C (2013) Neuroscience thinks big (and collaboratively). Nat Rev Neurosci 14:659–664PubMedCrossRefGoogle Scholar
  6. 6.
    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 314:1–340PubMedCrossRefGoogle Scholar
  7. 7.
    Lichtman JW, Pfister H, Shavit N (2014) The big data challenges of connectomics. Nat Neurosci 17:1448–1454PubMedPubMedCentralCrossRefGoogle Scholar
  8. 8.
    Chiang A-S et al (2011) Three-dimensional reconstructionof brain-wide wiring networks in Drosophila at single-cell resolution. Curr Biol 21:1–11PubMedCrossRefGoogle Scholar
  9. 9.
    Scannell JW, Young MP (1993) The connectional organization of neural systems in the cat cerebral cortex. Curr Biol 3:191–200PubMedCrossRefGoogle Scholar
  10. 10.
    Shanahan M, Bingman VP, Shimizu T, Gunturkun O (2013) Large-scale network organization in the avian forebrain: a connectivity matrix and theoretical analysis. Front Comput Neurosci 7:1–17CrossRefGoogle Scholar
  11. 11.
    Stephan KE (2013) The history of CoCoMac. NeuroImage 80:46–52Google Scholar
  12. 12.
    Hagmann P et al (2007) Mapping human whole-brain structural networks with diffusion MRI. PLoS One 2:e597PubMedPubMedCentralCrossRefGoogle Scholar
  13. 13.
    Newman MJE (2003) The structure and function of complex networks. SIAM Rev 45:167–256CrossRefGoogle Scholar
  14. 14.
    Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang DU (2006) Complex networks: structure and dynamics. Phys Rep 424:175–308CrossRefGoogle Scholar
  15. 15.
    Axer M et al (2011) A novel approach to the human connectome: ultra-high resolution mapping of fiber tracts in the brain. Neuroimage 54:1091–1101PubMedCrossRefGoogle Scholar
  16. 16.
    Chung K, Deisseroth K (2013) CLARITY for mapping the nervous system. Nat Meth 10:508–513CrossRefGoogle Scholar
  17. 17.
    Bullmore E, Sporns O (2009) Complex brain networks: graph theoretical analysis of structural and functional systems. Nat Rev Neurosci 10:186–198PubMedCrossRefGoogle Scholar
  18. 18.
    Fornito A, Zalesky A, Breakspear M (2013) Graph analysis of the human connectome: promise, progress, and pitfalls. Neuroimage 80:426–444PubMedCrossRefGoogle Scholar
  19. 19.
    Euler L (1736) Solutio problematis ad geometriam situs pertinentis. Commentarii Academiae Scientiarum Imperialis Petropolitanae 8:128–140Google Scholar
  20. 20.
    Luce RD, Perry AD (1949) A method of matrix analysis of group structure. Psychometrika 14:95–116PubMedCrossRefGoogle Scholar
  21. 21.
    Katz L (1947) On the matric analysis of sociometric data. Sociometry 10:233–241Google Scholar
  22. 22.
    Forsyth E, Katz L (1946) A matrix approach to the analysis of sociometric data: preliminary report. Sociometry 9:340–347CrossRefGoogle Scholar
  23. 23.
    Harary F, Norman RZ (1953) Graph theory as a mathematical model in social science. University of Michigan PressGoogle Scholar
  24. 24.
    Erdos P, Renyi A (1959) On random graphs. Publ Math Debrecen 6:290–297Google Scholar
  25. 25.
    Barabasi A, Albert R (1999) Emergence of scaling in random networks. Science 286:509–512PubMedCrossRefGoogle Scholar
  26. 26.
    Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world’ networks. Nature 393:440–442PubMedCrossRefGoogle Scholar
  27. 27.
    Mesulam MM (1998) From sensation to cognition. Brain 121:1013–1052Google Scholar
  28. 28.
    Tononi G, Sporns O, Edelman GM (1994) A measure for brain complexity: relating functional segregation and integration in the nervous system. Proc Natl Acad Sci U S A 91:5033–5037PubMedPubMedCentralCrossRefGoogle Scholar
  29. 29.
    Friston KJ (2011) Functional and effective connectivity: a review. Brain Connect 1:13–36PubMedCrossRefGoogle Scholar
  30. 30.
    Felleman DJ, Van Essen DC (1991) Distributed hierarchical processing in the primate cerebral cortex. Cereb Cortex 1:1–47PubMedCrossRefGoogle Scholar
  31. 31.
    Scannell JW, Blakemore C, Young MP (1995) Analysis of connectivity in the cat cerebral cortex. J Neurosci 15:1463–1483PubMedGoogle Scholar
  32. 32.
    Hilgetag CC, Burns GA, O'Neill MA, Scannell JW, Young MP (2000) Anatomical connectivity defines the organization of clusters of cortical areas in the macaque monkey and the cat. Philos Trans R Soc Lond B Biol Sci 355:91–110PubMedPubMedCentralCrossRefGoogle Scholar
  33. 33.
    Sporns O, Tononi G, Edelman GM (2000) Theoretical neuroanatomy: relating anatomical and functional connectivity in graphs and cortical connection matrices. Cereb Cortex 10:127–141PubMedCrossRefGoogle Scholar
  34. 34.
    Scannell JW, Burns GA, Hilgetag CC, O'Neil MA, Young MP (1999) The connectional organization of the cortico-thalamic system of the cat. Cereb Cortex 9:277–299PubMedCrossRefGoogle Scholar
  35. 35.
    Sporns O, Chialvo DR, Kaiser M, Hilgetag CC (2004) Organization, development and function of complex brain networks. Trends Cogn Sci 8:418–425PubMedCrossRefGoogle Scholar
  36. 36.
    Stam CJ (2004) Functional connectivity patterns of human magnetoencephalographic recordings: a ‘small-world’ network? Neurosci Lett 355:25–28PubMedCrossRefGoogle Scholar
  37. 37.
    Eguiluz VM, Chialvo DR, Cecchi GA, Baliki M, Apkarian AV (2005) Scale-free brain functional networks. Phys Rev Lett 94:018102PubMedCrossRefGoogle Scholar
  38. 38.
    Salvador R, Suckling J, Schwarzbauer C, Bullmore E (2005) Undirected graphs of frequency-dependent functional connectivity in whole brain networks. Philos Trans R Soc Lond B Biol Sci 360:937–946PubMedPubMedCentralCrossRefGoogle Scholar
  39. 39.
    Salvador R et al (2005) Neurophysiological architecture of functional magnetic resonance images of human brain. Cereb Cortex 15:1332–1342PubMedCrossRefGoogle Scholar
  40. 40.
    Achard S, Salvador R, Whitcher B, Suckling J, Bullmore E (2006) A resilient, low-frequency, small-world human brain functional network with highly connected association cortical hubs. J Neurosci 26:63–72PubMedCrossRefGoogle Scholar
  41. 41.
    Iturria-Medina Y et al (2007) Characterizing brain anatomical connections using diffusion weighted MRI and graph theory. NeuroImage 36:645–660PubMedCrossRefGoogle Scholar
  42. 42.
    Zalesky A, Fornito A (2009) A DTI-derived measure of cortico-cortical connectivity. IEEE Trans Med Imaging 28:1023–1036PubMedCrossRefGoogle Scholar
  43. 43.
    Skudlarski P et al (2008) Measuring brain connectivity: diffusion tensor imaging validates resting state temporal correlations. Neuroimage 43:554–561PubMedPubMedCentralCrossRefGoogle Scholar
  44. 44.
    Albert R, Barabasi AL (2002) Statistical mechanics of complex networks. Rev Mod Phys 74:47–97CrossRefGoogle Scholar
  45. 45.
    Logothetis NK (2008) What we can do and what we cannot do with fMRI. Nature 453:869–878PubMedCrossRefGoogle Scholar
  46. 46.
    Butts CT (2009) Revisiting the foundations of network analysis. Science 325:414–416PubMedCrossRefGoogle Scholar
  47. 47.
    Smith SM et al (2011) Network modelling methods for FMRI. Neuroimage 54:875–891PubMedCrossRefGoogle Scholar
  48. 48.
    Fornito A, Zalesky A, Bullmore ET (2010) Network scaling effects in graph analytic studies of human resting-state FMRI data. Front Syst Neurosci 4:22PubMedPubMedCentralGoogle Scholar
  49. 49.
    Hayasaka S, Laurienti PJ (2010) Comparison of characteristics between region-and voxel-based network analyses in resting-state fMRI data. Neuroimage 50:499–508PubMedCrossRefGoogle Scholar
  50. 50.
    Zalesky A et al (2010) Whole-brain anatomical networks: does the choice of nodes matter? Neuroimage 50:970–983PubMedCrossRefGoogle Scholar
  51. 51.
    Mountcastle VB (1997) The columnar organization of the neocortex. Brain 120(Pt 4):701–722PubMedCrossRefGoogle Scholar
  52. 52.
    Rademacher J, Caviness VS Jr, Steinmetz H, Galaburda AM (1993) Topographical variation of the human primary cortices: implications for neuroimaging, brain mapping, and neurobiology. Cereb Cortex 3:313–329PubMedCrossRefGoogle Scholar
  53. 53.
    Welker W (1990) 8b: Comparative structure and evolution of cerebral cortex. In: Jones EG, Peters A (eds) Cerebral cortex. Plenum, New York, pp 3–136CrossRefGoogle Scholar
  54. 54.
    Bassett DS et al (2010) Efficient physical embedding of topologically complex information processing networks in brains and computer circuits. PLoS Comput Biol 6:e1000748PubMedPubMedCentralCrossRefGoogle Scholar
  55. 55.
    Bullmore E, Sporns O (2012) The economy of brain network organization. Nat Rev Neurosci 13:336–349PubMedGoogle Scholar
  56. 56.
    Passingham RE, Stephan KE, Kotter R (2002) The anatomical basis of functional localization in the cortex. Nat Rev Neurosci 3:606–616PubMedCrossRefGoogle Scholar
  57. 57.
    Eliasmith C et al (2012) A large-scale model of the functioning brain. Science 338:1202–1205PubMedCrossRefGoogle Scholar
  58. 58.
    van den Heuvel MP, Stam CJ, Boersma M, Pol HEH (2008) Small-world and scale-free organization of voxel-based resting-state functional connectivity in the human brain. Neuroimage 43:528–539PubMedCrossRefGoogle Scholar
  59. 59.
    Power JD et al (2011) Functional network organization of the human brain. Neuron 72:665–678PubMedPubMedCentralCrossRefGoogle Scholar
  60. 60.
    Wig GS, Schlaggar BL, Petersen SE (2011) Concepts and principles in the analysis of brain networks. Ann N Y Acad Sci 1224:126–146PubMedCrossRefGoogle Scholar
  61. 61.
    Tzourio-Mazoyer N et al (2002) Automated anatomical labeling of activations in SPM using a macroscopic anatomical parcellation of the MNI MRI single-subject brain. NeuroImage 15:273–289PubMedCrossRefGoogle Scholar
  62. 62.
    Desikan RS et al (2006) An automated labeling system for subdividing the human cerebral cortex on MRI scans into gyral based regions of interest. NeuroImage 31:968–980PubMedCrossRefGoogle Scholar
  63. 63.
    Salvador R et al (2008) A simple view of the brain through a frequency-specific functional connectivity measure. NeuroImage 39:279–289PubMedCrossRefGoogle Scholar
  64. 64.
    Dosenbach NU et al (2010) Prediction of individual brain maturity using fMRI. Science 329:1358–1361PubMedPubMedCentralCrossRefGoogle Scholar
  65. 65.
    Fair DA et al (2007) Development of distinct control networks through segregation and integration. Proc Natl Acad Sci U S A 104:13507–13512PubMedPubMedCentralCrossRefGoogle Scholar
  66. 66.
    Andrews-Hanna JR, Reidler JS, Sepulcre J, Poulin R, Buckner RL (2010) Functional-anatomic fractionation of the brain’s default network. Neuron 65:550–562PubMedPubMedCentralCrossRefGoogle Scholar
  67. 67.
    Dwyer DB et al (2014) Large-scale brain network dynamics supporting adolescent cognitive control. J Neurosci 34:14096–14107PubMedCrossRefGoogle Scholar
  68. 68.
    Cocchi L et al (2014) Complexity in relational processing predicts changes in functional brain network dynamics. Cereb Cortex 24:2283–2296Google Scholar
  69. 69.
    Fornito A, Harrison BJ, Zalesky A, Simons JS (2012) Competitive and cooperative dynamics of large-scale brain functional networks supporting recollection. Proc Natl Acad Sci U S A 109:12788–12793PubMedPubMedCentralCrossRefGoogle Scholar
  70. 70.
    Beckmann CF, DeLuca M, Devlin JT, Smith SM (2005) Investigations into resting-state connectivity using independent component analysis. Philos Trans R Soc Lond B Biol Sci 360:1001–1013PubMedPubMedCentralCrossRefGoogle Scholar
  71. 71.
    Calhoun VD, Adali T, Pekar JJ (2004) A method for comparing group fMRI data using independent component analysis: application to visual, motor and visuomotor tasks. Magn Reson Imaging 22:1181–1191PubMedCrossRefGoogle Scholar
  72. 72.
    Smith SM et al (2009) Correspondence of the brain's functional architecture during activation and rest. Proc Natl Acad Sci U S A 106:13040–13045PubMedPubMedCentralCrossRefGoogle Scholar
  73. 73.
    Yu Q et al (2011) Altered topological properties of functional network connectivity in schizophrenia during resting state: a small-world brain network study. PLoS One 6:e25423PubMedPubMedCentralCrossRefGoogle Scholar
  74. 74.
    Kiviniemi V et al (2009) Functional segmentation of the brain cortex using high model order group PICA. Hum Brain Mapp 30:3865–3886PubMedCrossRefGoogle Scholar
  75. 75.
    Nelson SM et al (2010) A parcellation scheme for human left lateral parietal cortex. Neuron 67:156–170PubMedPubMedCentralCrossRefGoogle Scholar
  76. 76.
    Cohen AL et al (2008) Defining functional areas in individual human brains using resting functional connectivity MRI. NeuroImage 41:45–57PubMedPubMedCentralCrossRefGoogle Scholar
  77. 77.
    Yeo BT et al (2011) The organization of the human cerebral cortex estimated by intrinsic functional connectivity. J Neurophysiol 106:1125–1165PubMedCrossRefGoogle Scholar
  78. 78.
    Craddock RC, James GA, Holtzheimer PE, Hu XP, Mayberg HS (2012) A whole brain fMRI atlas generated via spatially constrained spectral clustering. Hum Brain Mapp 33:1914–1928PubMedCrossRefGoogle Scholar
  79. 79.
    Johansen-Berg H et al (2004) Changes in connectivity profiles define functionally distinct regions in human medial frontal cortex. Proc Natl Acad Sci U S A 101:13335–13340PubMedPubMedCentralCrossRefGoogle Scholar
  80. 80.
    Behrens TE et al (2003) Non-invasive mapping of connections between human thalamus and cortex using diffusion imaging. Nat Neurosci 6:750–757PubMedCrossRefGoogle Scholar
  81. 81.
    Anwander A, Tittgemeyer M, von Cramon DY, Friederici AD, Knosche TR (2007) Connectivity-based parcellation of Broca's area. Cereb Cortex 17:816–825PubMedCrossRefGoogle Scholar
  82. 82.
    Glasser MF, Van Essen DC (2011) Mapping human cortical areas in vivo based on myelin content as revealed by T1- and T2-weighted MRI. J Neurosci 31:11597–11616PubMedPubMedCentralCrossRefGoogle Scholar
  83. 83.
    Eickhoff SB et al (2005) A new SPM toolbox for combining probabilistic cytoarchitectonic maps and functional imaging data. NeuroImage 25:1325–1335PubMedCrossRefGoogle Scholar
  84. 84.
    Zilles K et al (2002) Architectonics of the human cerebral cortex and transmitter receptor fingerprints: reconciling functional neuroanatomy and neurochemistry. Eur Neuropsychopharmacol 12:587–599PubMedCrossRefGoogle Scholar
  85. 85.
    Alexander-Bloch A, Giedd JN, Bullmore E (2013) Imaging structural co-variance between human brain regions. Nat Rev Neurosci 14:322–336PubMedPubMedCentralCrossRefGoogle Scholar
  86. 86.
    Lerch JP et al (2006) Mapping anatomical correlations across cerebral cortex (MACACC) using cortical thickness from MRI. NeuroImage 31:993–1003PubMedCrossRefGoogle Scholar
  87. 87.
    Bastiani M, Shah NJ, Goebel R, Roebroeck A (2012) Human cortical connectome reconstruction from diffusion weighted MRI: the effect of tractography algorithm. Neuroimage 62:1732–1749PubMedCrossRefGoogle Scholar
  88. 88.
    Jones DK, Knösche TR, Turner R (2013) White matter integrity, fiber count, and other fallacies: the do‘s and dont’s of diffusion MRI. Neuroimage 73:239–254PubMedCrossRefGoogle Scholar
  89. 89.
    van den Heuvel MP, Mandl RCW, Stam CJ, Kahn RS, Hulshoff Pol HE (2010) Aberrant frontal and temporal complex network structure in schizophrenia: a graph theoretical analysis. J Neurosci 30:15915–15926PubMedCrossRefGoogle Scholar
  90. 90.
    Alexander DC et al (2010) Orientationally invariant indices of axon diameter and density from diffusion MRI. Neuroimage 52:1374–1389PubMedCrossRefGoogle Scholar
  91. 91.
    Friston KJ (1994) Functional and effective connectivity in neuroimaging: a synthesis. Hum Brain Mapping 2:56–78CrossRefGoogle Scholar
  92. 92.
    Vincent JL et al (2007) Intrinsic functional architecture in the anaesthetized monkey brain. Nature 447:83–86PubMedCrossRefGoogle Scholar
  93. 93.
    Honey CJ et al (2009) Predicting human resting-state functional connectivity from structural connectivity. Proc Natl Acad Sci U S A 106:2035–2040PubMedPubMedCentralCrossRefGoogle Scholar
  94. 94.
    Zalesky A, Fornito A, Bullmore E (2012) On the use of correlation as a measure of network connectivity. Neuroimage 60:2096–2106PubMedCrossRefGoogle Scholar
  95. 95.
    Fox MD, Raichle ME (2007) Spontaneous fluctuations in brain activity observed with functional magnetic resonance imaging. Nat Rev Neurosci 8:700–711PubMedCrossRefGoogle Scholar
  96. 96.
    Fornito A, Bullmore ET (2010) What can spontaneous fluctuations of the blood oxygenation-level-dependent signal tell us about psychiatric disorders? Curr Opin Psychiatry 23:239–249PubMedCrossRefGoogle Scholar
  97. 97.
    Zalesky A, Fornito A, Cocchi L, Gollo LL, Breakspear M (2014) Time-resolved resting-state brain networks. Proc Natl Acad Sci U S A 111:10341–10346PubMedPubMedCentralCrossRefGoogle Scholar
  98. 98.
    Smith SM et al (2012) Temporally-independent functional modes of spontaneous brain activity. Proc Natl Acad Sci U S A 109:3131–3136PubMedPubMedCentralCrossRefGoogle Scholar
  99. 99.
    Hutchison RM et al (2013) Dynamic functional connectivity: promise, issues, and interpretations. NeuroImage 80:360–378PubMedCrossRefGoogle Scholar
  100. 100.
    Feinberg DA et al (2010) Multiplexed echo planar imaging for sub-second whole brain FMRI and fast diffusion imaging. PLoS One 5:e15710PubMedPubMedCentralCrossRefGoogle Scholar
  101. 101.
    Rissman J, Gazzaley A, D'Esposito M (2004) Measuring functional connectivity during distinct stages of a cognitive task. NeuroImage 23:752–763PubMedCrossRefGoogle Scholar
  102. 102.
    Fornito A, Yoon J, Zalesky A, Bullmore ET, Carter CS (2011) General and specific functional connectivity disturbances in first-episode schizophrenia during cognitive control performance. Biol Psychiatry 70:64–72PubMedPubMedCentralCrossRefGoogle Scholar
  103. 103.
    Friston KJ et al (1997) Psychophysiological and modulatory interactions in neuroimaging. NeuroImage 6:218–229PubMedCrossRefGoogle Scholar
  104. 104.
    Cole MW et al (2013) Multi-task connectivity reveals flexible hubs for adaptive task control. Nat Neurosci 16:1348–1355PubMedPubMedCentralCrossRefGoogle Scholar
  105. 105.
    Friston K, Moran R, Seth AK (2013) Analysing connectivity with Granger causality and dynamic causal modelling. Curr Opin Neurobiol 23:172–178PubMedPubMedCentralCrossRefGoogle Scholar
  106. 106.
    Friston KJ, Harrison L, Penny W (2003) Dynamic causal modelling. NeuroImage 19:1273–1302PubMedCrossRefGoogle Scholar
  107. 107.
    Seghier ML, Friston KJ (2013) Network discovery with large DCMs. Neuroimage 68:181–191PubMedPubMedCentralCrossRefGoogle Scholar
  108. 108.
    Friston KJ, Kahan J, Biswal B, Razi A (2014) A DCM for resting state fMRI. Neuroimage 94:396–407PubMedPubMedCentralCrossRefGoogle Scholar
  109. 109.
    Fornito A, Zalesky A, Pantelis C, Bullmore ET (2012) Schizophrenia, neuroimaging and connectomics. Neuroimage 62:2296–2314PubMedCrossRefGoogle Scholar
  110. 110.
    van Wijk BCM, Stam CJ, Daffertshofer A (2010) Comparing brain networks of different size and connectivity density using graph theory. Plos One 5:e13701PubMedPubMedCentralCrossRefGoogle Scholar
  111. 111.
    Irimia A, Chambers MC, Torgerson CM, Van Horn JD (2012) Circular representation of human cortical networks for subject and population-level connectomic visualization. Neuroimage 60:1340–1351PubMedPubMedCentralCrossRefGoogle Scholar
  112. 112.
    Fornito A, Bullmore ET (2015) Connectomics: a new paradigm for understanding brain disease. Eur Neuropsychopharmacol. 25: 733–748Google Scholar
  113. 113.
    Fornito A et al (2013) Functional dysconnectivity of corticostriatal circuitry as a risk phenotype for psychosis. JAMA Psychiatry 70:1143–1151PubMedCrossRefGoogle Scholar
  114. 114.
    Meskaldji DE et al (2011) Adaptive strategy for the statistical analysis of connectomes. Plos One 6:e23009PubMedPubMedCentralCrossRefGoogle Scholar
  115. 115.
    Ginestet CE, Simmons A (2011) Statistical parametric network analysis of functional connectivity dynamics during a working memory task. Neuroimage 55:688–704PubMedCrossRefGoogle Scholar
  116. 116.
    Zalesky A, Fornito A, Bullmore ET (2010) Network-based statistic: Identifying differences in brain networks. Neuroimage 53:1197–1207PubMedCrossRefGoogle Scholar
  117. 117.
    Nichols TE, Holmes AP (2002) Nonparametric permutation tests for functional neuroimaging: a primer with examples. Hum Brain Mapp 15:1–25PubMedCrossRefGoogle Scholar
  118. 118.
    Benjamini Y, Hochberg Y (1995) Controlling the false discovery rate: a practical and powerful approach to multiple testing. J R Stat Soc Ser B 57:289–300Google Scholar
  119. 119.
    Zalesky A, Cocchi L, Fornito A, Murray MM, Bullmore E (2012) Connectivity differences in brain networks. Neuroimage 60:1055–1062PubMedCrossRefGoogle Scholar
  120. 120.
    Tomasi D, Volkow VD (2010) Functional connectivity density mapping. Proc Natl Acad Sci U S A, 107:9885–9890Google Scholar
  121. 121.
    Cole MW, Anticevic A, Repovs G, Barch D (2011) Variable global dysconnectivity and individual differences in schizophrenia. Biol Psychiatry 70:43–50PubMedPubMedCentralCrossRefGoogle Scholar
  122. 122.
    Rubinov M, Sporns O (2011) Weight-conserving characterization of complex functional brain networks. Neuroimage 56:2068–2079PubMedCrossRefGoogle Scholar
  123. 123.
    Rubinov M, Sporns O (2010) Complex network measures of brain connectivity: uses and interpretations. Neuroimage 52:1059–1069PubMedCrossRefGoogle Scholar
  124. 124.
    Amaral LA, Scala A, Barthelemy M, Stanley HE (2000) Classes of small-world networks. Proc Natl Acad Sci U S A 97:11149–11152PubMedPubMedCentralCrossRefGoogle Scholar
  125. 125.
    van den Heuvel MP, Sporns O (2011) Rich-club organization of the human connectome. J Neurosci 31:15775–15786PubMedCrossRefGoogle Scholar
  126. 126.
    Buckner RL et al (2009) Cortical hubs revealed by intrinsic functional connectivity: mapping, assessment of stability, and relation to Alzheimer’s Disease. J Neurosci 29:1860–1873PubMedPubMedCentralCrossRefGoogle Scholar
  127. 127.
    Power JD, Schlaggar BL, Lessov-Schlaggar CN, Petersen SE (2013) Evidence for hubs in human functional brain networks. Neuron 79:798–813PubMedCrossRefGoogle Scholar
  128. 128.
    van den Heuvel MP, Kahn RS, Goni J, Sporns O (2012) High-cost, high-capacity backbone for global brain communication. Proc Natl Acad Sci U S A 109:11372–11377PubMedPubMedCentralCrossRefGoogle Scholar
  129. 129.
    Mišić B, Sporns O, McIntosh AR (2014) Communication efficiency and congestion of signal traffic in large-scale brain networks. PLoS Comput Biol 10:e1003427PubMedPubMedCentralCrossRefGoogle Scholar
  130. 130.
    van den Heuvel MP, Sporns O (2013) An anatomical substrate for integration among functional networks in human cortex. J Neurosci 33:14489–14500PubMedCrossRefGoogle Scholar
  131. 131.
    Albert R, Jeong H, Barabasi AL (2000) Error and attack tolerance of complex networks. Nature 406:378–382PubMedCrossRefGoogle Scholar
  132. 132.
    Fornito A, Breakspear M, Zalesky A (2015) The connectomics of brain disorders. Nat Rev Neurosci 16:159–172Google Scholar
  133. 133.
    Crossley NA et al (2014) The hubs of the human connectome are generally implicated in the anatomy of brain disorders. Brain 137:2382–2395PubMedPubMedCentralCrossRefGoogle Scholar
  134. 134.
    Maslov S, Sneppen K (2002) Specificity and stability in topology of protein networks. Science 296:910–913PubMedCrossRefGoogle Scholar
  135. 135.
    Humphries MD, Gurney K, Prescott TJ (2006) The brainstem reticular formation is a small-world, not scale-free, network. Proc Biol Sci 273:503–511PubMedCrossRefGoogle Scholar
  136. 136.
    Latora V, Marchiori M (2001) Efficient behavior of small-world networks. Phys Rev Lett 87:198701PubMedCrossRefGoogle Scholar
  137. 137.
    Latora V, Marchiori M (2003) Economic small-world behavior in weighted networks. Eur Phys J B 32:249–263CrossRefGoogle Scholar
  138. 138.
    Kaiser M, Hilgetag CC (2006) Nonoptimal component placement, but short processing paths, due to long-distance projections in neural systems. PLoS Comput Biol 2:e95PubMedPubMedCentralCrossRefGoogle Scholar
  139. 139.
    Chen Y, Wang S, Hilgetag CC, Zhou C (2013) Trade-off between multiple constraints enables simultaneous formation of modules and hubs in neural systems. PLoS Comput Biol 9:e1002937PubMedPubMedCentralCrossRefGoogle Scholar
  140. 140.
    Fornito A et al (2011) Genetic influences on cost-efficient organization of human cortical functional networks. J Neurosci 31:3261–3270PubMedCrossRefGoogle Scholar
  141. 141.
    Fortunato S (2010) Community detection in graphs. Phys Rep 486:75–174CrossRefGoogle Scholar
  142. 142.
    Newman M, Girvan M (2004) Finding and evaluating community structure in networks. Phys Rev 69:026113Google Scholar
  143. 143.
    Lancichinetti A, Fortunato S (2009) Community detection algorithms: a comparative analysis. Phys Rev E 80:056117CrossRefGoogle Scholar
  144. 144.
    Good BH, de Montjoye YA, Clauset A (2010) Performance of modularity maximization in practical contexts. Phys Rev E 81:046106CrossRefGoogle Scholar
  145. 145.
    Lancichinetti A, Fortunato S (2012) Consensus clustering in complex networks. Sci Rep 2:336Google Scholar
  146. 146.
    Guimerà R, Sales-Pardo M, Amaral L (2004) Modularity from fluctuations in random graphs and complex networks. Phys Rev E 70:025101CrossRefGoogle Scholar
  147. 147.
    Hagmann P et al (2008) Mapping the structural core of human cerebral cortex. PLoS Biol 6:e159PubMedPubMedCentralCrossRefGoogle Scholar
  148. 148.
    Meunier D, Achard S, Morcom A, Bullmore E (2009) Age-related changes in modular organization of human brain functional networks. NeuroImage 44:715–723PubMedCrossRefGoogle Scholar
  149. 149.
    Buzsaki G, Geisler C, Henze DA, Wang XJ (2004) Interneuron diversity series: circuit complexity and axon wiring economy of cortical interneurons. Trends Neurosci 27:186–193PubMedCrossRefGoogle Scholar
  150. 150.
    Ercsey-Ravasz M et al (2013) A predictive network model of cerebral cortical connectivity based on a distance rule. Neuron 80:184–197PubMedPubMedCentralCrossRefGoogle Scholar
  151. 151.
    Crossley NA, Mechelli A, Vertes PE (2013) Cognitive relevance of the community structure of the human brain functional coactivation network. Proc Natl Acad Sci U S A 110:11583–11588PubMedPubMedCentralCrossRefGoogle Scholar
  152. 152.
    Fodor JA (1983) Modularity of mind: an essay on faculty psychology. MIT PressGoogle Scholar
  153. 153.
    Simon HA (1962) The architecture of complexity. Proc Am Philos Soc 106:467–482Google Scholar
  154. 154.
    Kitano H (2004) Biological robustness. Nat Rev Genet 5:826–837PubMedCrossRefGoogle Scholar
  155. 155.
    Guimera R, Nunes Amaral LA (2005) Functional cartography of complex metabolic networks. Nature 433:895–900PubMedPubMedCentralCrossRefGoogle Scholar
  156. 156.
    Xie J, Kelley S, Szymanski BK (2013) Overlapping community detection in networks. ACM Comput Surv 45:1–35CrossRefGoogle Scholar
  157. 157.
    Meunier D, Lambiotte R, Bullmore ET (2011) Modular and hierarchically modular organization of brain networks. Front Neurosci 4:200Google Scholar
  158. 158.
    Meunier D, Lambiotte R, Fornito A, Ersche KD, Bullmore ET (2009) Hierarchical modularity in human brain functional networks. Front Neuroinform 3:37PubMedPubMedCentralCrossRefGoogle Scholar
  159. 159.
    Newman MEJ (2010) Networks. a Introduction. Oxford University PressGoogle Scholar
  160. 160.
    Goñi J et al (2014) Resting-brain functional connectivity predicted by analytic measures of network communication. Proc Natl Acad Sci U S A 111:833–838PubMedCrossRefGoogle Scholar
  161. 161.
    Betzel RF et al (2014) Multi-scale community organization of the human structural connectome and its relationship with resting-state functional connectivity. Net Sci 1:353–373CrossRefGoogle Scholar
  162. 162.
    Bassett DS et al (2011) Dynamic reconfiguration of human brain networks during learning. Proc Natl Acad Sci U S A 108:7641–7646PubMedPubMedCentralCrossRefGoogle Scholar
  163. 163.
    Deco G, Jirsa VK, McIntosh AR (2011) Emerging concepts for the dynamical organization of resting-state activity in the brain. Nat Publ Group 12:43–56Google Scholar
  164. 164.
    Alstott J, Breakspear M, Hagmann P, Cammoun L, Sporns O (2009) Modeling the impact of lesions in the human brain. PLoS Comput Biol 5:e1000408PubMedPubMedCentralCrossRefGoogle Scholar
  165. 165.
    Honey CJ, Sporns O (2008) Dynamical consequences of lesions in cortical networks. Hum Brain Mapp 29:802–809PubMedCrossRefGoogle Scholar
  166. 166.
    Raj A, Kuceyeski A, Weiner M (2012) A network diffusion model of disease progression in dementia. Neuron 73:1204–1215PubMedPubMedCentralCrossRefGoogle Scholar
  167. 167.
    de Haan W, Mott K, van Straaten ECW, Scheltens P, Stam CJ (2012) Activity dependent degeneration explains hub vulnerability in Alzheimer's disease. PLoS Comput Biol 8:e1002582PubMedPubMedCentralCrossRefGoogle Scholar
  168. 168.
    Vertes PE et al (2012) Simple models of human brain functional networks. Proc Natl Acad Sci U S A 109:5868–5873PubMedPubMedCentralCrossRefGoogle Scholar
  169. 169.
    Song HF, Kennedy H, Wang X-J (2014) Spatial embedding of structural similarity in the cerebral cortex. Proc Natl Acad Sci U S A 111:16580–16585PubMedPubMedCentralCrossRefGoogle Scholar
  170. 170.
    Goni J et al (2013) Exploring the morphospace of communication efficiency in complex networks. PLoS One 8, e58070PubMedPubMedCentralCrossRefGoogle Scholar
  171. 171.
    Avena-Koenigsberger A et al (2014) Using Pareto optimality to explore the topology and dynamics of the human connectome. Philos Trans R Soc Lond B Biol Sci 369:20130530PubMedPubMedCentralCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Brain and Mental Health Laboratory, Monash Institute of Cognitive and Clinical Neuroscience, School of Psychological Sciences and Monash Biomedical ImagingMonash UniversityMelbourneAustralia

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