zbMATH — the first resource for mathematics

Correlation between graphs with an application to brain network analysis. (English) Zbl 06917821
Summary: The global functional brain network (graph) is more suitable for characterizing brain states than local analysis of the connectivity of brain regions. Therefore, graph-theoretic approaches are natural methods to use for studying the brain. However, conventional graph theoretical analyses are limited due to the lack of formal statistical methods of estimation and inference. For example, the concept of correlation between two vectors of graphs has not yet been defined. Thus, the introduction of a notion of correlation between graphs becomes necessary to better understand how brain sub-networks interact. To develop a framework to infer correlation between graphs, one may assume that they are generated by models and that the parameters of the models are the random variables. Then, it is possible to define that two graphs are independent when the random variables representing their parameters are independent. In the real world, however, the model is rarely known, and consequently, the parameters cannot be estimated. By analyzing the graph spectrum, it is shown that the spectral radius is highly associated with the parameters of the graph model. Based on this, a framework for correlation inference between graphs is constructed and the approach illustrated on functional magnetic resonance imaging data on 814 subjects comprising 529 controls and 285 individuals diagnosed with autism spectrum disorder (ASD). Results show that correlations between the default-mode and control, default-mode and somatomotor, and default-mode and visual sub-networks are higher in individuals with ASD than in the controls.
62 Statistics
Full Text: DOI
[1] Abrahams, B. S.; Geschwind, D. H., Connecting genes to brain in the autism spectrum disorders, Arch. Neurol., 67, 4, 395-399, (2010)
[2] Alon, N., Eigenvalues and expanders, Combinatorica, 6, 2, 83-96, (1986) · Zbl 0661.05053
[3] Assaf, M.; Jagannathan, K.; Calhoun, V. D.; Miller, L.; Stevens, M. C.; Sahl, R.; O’Boyle, J. G.; Schultz, R. T.; Pearlson, G. D., Abnormal functional connectivity of default mode sub-networks in autism spectrum disorder patients, Neuroimage, 53, 1, 247-256, (2010)
[4] Barabási, A.-L.; Albert, R., Emergence of scaling in random networks, Science, 286, 5439, 509-512, (1999) · Zbl 1226.05223
[5] Bassett, D. S.; Wymbs, N. F.; Porter, M. A.; Mucha, P. J.; Carlson, J. M.; Grafton, S. T., Dynamic reconfiguration of human brain networks during learning, Proc. Natl. Acad. Sci., 108, 18, 7641-7646, (2011)
[6] Bassett, D. S.; Yang, M.; Wymbs, N. F.; Grafton, S. T., Learning-induced autonomy of sensorimotor systems, Nature Neurosci., 18, 5, 744-751, (2015)
[7] Benjamini, Y.; Hochberg, Y., Controlling the false discovery rate: a practical and powerful approach to multiple testing, J. R. Stat. Soc. Ser. B Stat. Methodol., 57, 1, 289-300, (1995) · Zbl 0809.62014
[8] Betancur, C., Etiological heterogeneity in autism spectrum disorders: more than 100 genetic and genomic disorders and still counting, Brain Res., 1380, 42-77, (2011)
[9] Bordenave, C., Eigenvalues of Euclidean random matrices, Random Struct. Algorithms, 33, 4, 515-532, (2008) · Zbl 1158.15020
[10] Borkowf, C. B., Computing the nonnull asymptotic variance and the asymptotic relative efficiency of searman’s rank correlation, Comput. Statist. Data Anal., 39, 3, 271-286, (2002) · Zbl 0990.62042
[11] Buckner, R. L.; Andrews-Hanna, J. R.; Schacter, D. L., The brain’s default network, Ann. New York Acad. Sci., 1124, 1, 1-38, (2008)
[12] Cassidy, C. M.; Van Snellenberg, J. X.; Benavides, C.; Slifstein, M.; Wang, Z.; Moore, H.; Abi-Dargham, A.; Horga, G., Dynamic connectivity between brain networks supports working memory: relationships to dopamine release and schizophrenia, J. Neurosci., 36, 15, 4377-4388, (2016)
[13] Craddock, R. C.; James, G. A.; Holtzheimer, P. E.; Hu, X. P.; Mayberg, H. S., A whole brain fMRI atlas generated via spatially constrained spectral clustering, Hum. Brain Mapp., 33, 8, 1914-1928, (2012)
[14] Di Martino, A.; Ross, K.; Uddin, L. Q.; Sklar, A. B.; Castellanos, F. X.; Milham, M. P., Functional brain correlates of social and nonsocial processes in autism spectrum disorders: an activation likelihood estimation meta-analysis, Biol. Psychiatry, 65, 1, 63-74, (2009)
[15] Ding, X.; Jiang, T., Spectral distributions of adjacency and Laplacian matrices of random graphs, Ann. Appl. Probab., 20, 6, 2086-2117, (2010) · Zbl 1231.05236
[16] Dorogovtsev, S. N.; Goltsev, A. V.; Mendes, J. F.F.; Samukhin, A. N., Spectra of complex networks, Phys. Rev. E, 68, (2003), URL http://link.aps.org/doi/10.1103/PhysRevE.68.046109 · Zbl 1010.05073
[17] Ecker, C.; Bookheimer, S. Y.; Murphy, D. G., Neuroimaging in autism spectrum disorder: brain structure and function across the lifespan, Lancet Neurol, (2015)
[18] Ecker, C.; Marquand, A.; Mourão-Miranda, J.; Johnston, P.; Daly, E. M.; Brammer, M. J.; Maltezos, S.; Murphy, C. M.; Robertson, D.; Williams, S. C., Describing the brain in autism in five dimensions—magnetic resonance imaging-assisted diagnosis of autism spectrum disorder using a multiparameter classification approach, J. Neurosci., 30, 32, 10612-10623, (2010)
[19] Ecker, C.; Spooren, W.; Murphy, D., Translational approaches to the biology of autism: false dawn or a new era?, Mol. Psychiatry, 18, 4, 435-442, (2013)
[20] Erdös, P.; Rényi, A., On random graphs i, Publ. Math. Debrecen, 6, 290-297, (1959) · Zbl 0092.15705
[21] Fisher, R., On the “probable error” of a coefficient of correlation deduced from a small sample, Metron, 1, Pt 4, 1-32, (1921)
[22] Füredi, Z.; Komlós, J., The eigenvalues of random symmetric matrices, Combinatorica, 1, 3, 233-241, (1981) · Zbl 0494.15010
[23] Frith, C., What do imaging studies tell us about the neural basis of autism, Autism: Neural Basis Treat. Possibilities, 149-176, (2003)
[24] Fujita, A.; Takahashi, D. Y.; Patriota, A. G., A non-parametric method to estimate the number of clusters, Comput. Statist. Data Anal., 73, 27-39, (2014)
[25] Garcés, P.; Pereda, E.; Hernández-Tamames, J. A.; Del-Pozo, F.; Maestú, F.; Ángel Pineda-Pardo, J., Multimodal description of whole brain connectivity: A comparison of resting state meg, fMRI, and dwi, Hum. Brain Mapp., 37, 1, 20-34, (2016)
[26] Hallmayer, J.; Cleveland, S.; Torres, A.; Phillips, J.; Cohen, B.; Torigoe, T.; Miller, J.; Fedele, A.; Collins, J.; Smith, K., Genetic heritability and shared environmental factors among twin pairs with autism, Arch. Gen. Psychiatry, 68, 11, 1095-1102, (2011)
[27] Heller, R.; Heller, Y.; Gorfine, M., A consistent multivariate test of association based on ranks of distances, Biometrika, (2012), arXiv:http://biomet.oxfordjournals.org/content/early/2012/12/04/biomet.ass070.full.pdf+html, http://dx.doi.org/10.1093/biomet/ass070, URL http://biomet.oxfordjournals.org/content/early/2012/12/04/biomet.ass070.abstract · Zbl 1348.62162
[28] Hoeffding, W., A non-parametric test of independence, Ann. Math. Statist., 546-557, (1948) · Zbl 0032.42001
[29] Just, M. A.; Keller, T. A.; Malave, V. L.; Kana, R. K.; Varma, S., Autism as a neural systems disorder: a theory of frontal-posterior underconnectivity, Neurosci. Biobehav. Rev., 36, 4, 1292-1313, (2012)
[30] Kana, R. K.; Keller, T. A.; Minshew, N. J.; Just, M. A., Inhibitory control in high-functioning autism: decreased activation and underconnectivity in inhibition networks, Biol. Psychiatry, 62, 3, 198-206, (2007)
[31] Kana, R. K.; Uddin, L. Q.; Kenet, T.; Chugani, D.; Müller, R.-A., Brain connectivity in autism, (2014), Frontiers E-books
[32] Kennedy, D. P.; Courchesne, E., Functional abnormalities of the default network during self-and other-reflection in autism, Soc. Cogn. Affect. Neurosci., 3, 2, 177-190, (2008)
[33] Kolev, N.; Anjos, U.d.; Mendes, B. V.d. M., Copulas: a review and recent developments, Stoch. Models, 22, 4, 617-660, (2006) · Zbl 1120.60006
[34] McGrath, J.; Johnson, K.; Ecker, C.; O’Hanlon, E.; Gill, M.; Gallagher, L.; Garavan, H., Atypical visuospatial processing in autism: insights from functional connectivity analysis, Autism Res., 5, 5, 314-330, (2012)
[35] Meringer, M., Fast generation of regular graphs and construction of cages, J. Graph Theory, 30, 2, 137-146, (1999) · Zbl 0918.05062
[36] Nebel, M. B.; Eloyan, A.; Barber, A. D.; Mostofsky, S. H., Precentral gyrus functional connectivity signatures of autism, Front. Syst. Neurosci., 8, (2014)
[37] Ng, A. Y.; Jordan, M. I.; Weiss, Y., On spectral clustering: analysis and an algorithm, Adv. Neural Inf. Process. Syst., 2, 849-856, (2002)
[38] Nomi, J. S.; Uddin, L. Q., Developmental changes in large-scale network connectivity in autism, NeuroImage: Clin., 7, 732-741, (2015)
[39] Penrose, M., Random geometric graphs, vol. 5, (2003), Oxford University Press Oxford
[40] Power, J. D.; Barnes, K. A.; Snyder, A. Z.; Schlaggar, B. L.; Petersen, S. E., Spurious but systematic correlations in functional connectivity mri networks arise from subject motion, Neuroimage, 59, 3, 2142-2154, (2012)
[41] Reynolds, A. P.; Richards, G.; de la Iglesia, B.; Rayward-Smith, V. J., Clustering rules: a comparison of partitioning and hierarchical clustering algorithms, J. Math. Model. Algorithms, 5, 4, 475-504, (2006) · Zbl 1104.62073
[42] Richiardi, J.; Eryilmaz, H.; Schwartz, S.; Vuilleumier, P.; Van De Ville, D., Decoding brain states from fMRI connectivity graphs, Neuroimage, 56, 2, 616-626, (2011)
[43] Shannon, C. E.; Weaver, W., The mathematical theory of communication, (2015), University of Illinois Press
[44] Spearman, C., “general intelligence,” objectively determined and measured, Am. J. Psychol., 15, 2, 201-292, (1904)
[45] Sporns, O., Network attributes for segregation and integration in the human brain, Curr. Opin. Neurobiol., 23, 2, 162-171, (2013)
[46] Stam, C. J., Modern network science of neurological disorders, Nat. Rev. Neurosci., 15, 10, 683-695, (2014)
[47] Stevenson, R. A., Using functional connectivity analyses to investigate the bases of autism spectrum disorders and other clinical populations, J. Neurosci., 32, 50, 17933-17934, (2012)
[48] Supekar, K.; Uddin, L. Q.; Khouzam, A.; Phillips, J.; Gaillard, W. D.; Kenworthy, L. E.; Yerys, B. E.; Vaidya, C. J.; Menon, V., Brain hyperconnectivity in children with autism and its links to social deficits, Cell Rep., 5, 3, 738-747, (2013)
[49] Székely, G. J.; Rizzo, M. L.; Bakirov, N. K., Measuring and testing dependence by correlation of distances, Ann. Statist., 35, 6, 2769-2794, (2007) · Zbl 1129.62059
[50] Takahashi, D. Y.; Sato, J. R.; Ferreira, C. E.; Fujita, A., Discriminating different classes of biological networks by analyzing the graphs spectra distribution, PLoS One, 7, 12, e49949, (2012)
[51] Uddin, L. Q.; Supekar, K.; Menon, V., Reconceptualizing functional brain connectivity in autism from a developmental perspective, Front. Hum. Neurosci., 7, 458, (2013)
[52] Van Mieghem, P., Graph spectra for complex networks, (2010), Cambridge University Press
[53] Wass, S., Distortions and disconnections: disrupted brain connectivity in autism, Brain Cogn., 75, 1, 18-28, (2011)
[54] Watts, D.; Strogatz, S., Collective dynamics of ‘small-world’ networks, Nature, 393, 440-442, (1998) · Zbl 1368.05139
[55] Weng, S.-J.; Wiggins, J. L.; Peltier, S. J.; Carrasco, M.; Risi, S.; Lord, C.; Monk, C. S., Alterations of resting state functional connectivity in the default network in adolescents with autism spectrum disorders, Brain Res., 1313, 202-214, (2010)
[56] Wing, L., The autistic spectrum, Lancet, 350, 9093, 1761-1766, (1997)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.