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Multilayer Brain Networks

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Abstract

The field of neuroscience is facing an unprecedented expanse in the volume and diversity of available data. Traditionally, network models have provided key insights into the structure and function of the brain. With the advent of big data in neuroscience, both more sophisticated models capable of characterizing the increasing complexity of the data and novel methods of quantitative analysis are needed. Recently, multilayer networks, a mathematical extension of traditional networks, have gained increasing popularity in neuroscience due to their ability to capture the full information of multi-model, multi-scale, spatiotemporal data sets. Here, we review multilayer networks and their applications in neuroscience, showing how incorporating the multilayer framework into network neuroscience analysis has uncovered previously hidden features of brain networks. We specifically highlight the use of multilayer networks to model disease, structure–function relationships, network evolution, and link multi-scale data. Finally, we close with a discussion of promising new directions of multilayer network neuroscience research and propose a modified definition of multilayer networks designed to unite and clarify the use of the multilayer formalism in describing real-world systems.

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Fig. 1

(Reprinted with permission from Battiston et al. 2017)

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(Reprinted with permission from Yu et al. 2017)

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(Reprinted with permission from Bentley and Branicky 2016)

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(Reprinted with permission from Bassett et al. 2015)

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(Reprinted with permission from Virkar et al. 2016)

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References

  • Allen, E.A., Damaraju, E., Plis, S.M., Erhardt, E.B., Eichele, T., Calhoun, V.D.: Tracking whole-brain connectivity dynamics in the resting state. Cereb. Cortex 24(3), 663–676 (2014)

    Google Scholar 

  • Arenas, A., De Domenico, M.: Nonlinear dynamics on interconnected networks. Phys. D Nonlinear Phenom. 323, 1–4 (2016)

    MathSciNet  MATH  Google Scholar 

  • Bassett, D.S., Bullmore, E.T.: Human brain networks in health and disease. Curr. Opin. Neurol. 22(4), 340–347 (2009)

    Google Scholar 

  • Bassett, D.S., Sporns, O.: Network neuroscience. Nat. Neurosci. 20(3), 353–364 (2017)

    Google Scholar 

  • Bassett, D.S., Meyer-Lindenberg, A., Achard, S., Duke, T., Bullmore, E.: Adaptive reconfiguration of fractal small-world human brain functional networks. Proc. Natl. Acad. Sci. 103(51), 19518–19523 (2006)

    Google Scholar 

  • 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)

    Google Scholar 

  • Bassett, D.S., Wymbs, N.F., Rombach, M.P., Porter, M.A., Mucha, P.J., Grafton, S.T.: Task-based core-periphery organization of human brain dynamics. PLoS Comput. Biol. 9(9), e1003171 (2013a)

    Google Scholar 

  • Bassett, D.S., Porter, M.A., Wymbs, N.F., Grafton, S.T., Carlson, J.M., Peter J, M.: Robust detection of dynamic community structure in networks. Chaos Interdiscip. J. Nonlinear Sci. 23(1), 013142 (2013b)

    MathSciNet  Google Scholar 

  • Bassett, D.S., Yang, M., Wymbs, N.F., Grafton, S.T.: Learning-induced autonomy of sensorimotor systems. Nat. Neurosci. 18(5), 744–751 (2015)

    Google Scholar 

  • Battiston, F., Nicosia, V., Latora, V.: Structural measures for multiplex networks. Phys. Rev. E 89(3), 032804 (2014)

    Google Scholar 

  • Battiston, F., Nicosia, V., Chavez, M., Latora, V.: Multilayer motif analysis of brain networks. Chaos Interdiscip. J. Nonlinear Sci. 27(4), 047404 (2017)

    MathSciNet  Google Scholar 

  • Bazzi, M., Jeub, L.G, Arenas, A., Howison, S.D., Porter, M.A.: Generative benchmark models for mesoscale structure in multilayer networks. arXiv preprint arXiv:1608.06196 (2016a)

  • Bazzi, M., Porter, M.A., Williams, S., McDonald, M., Fenn, D.J., Howison, S.D.: Community detection in temporal multilayer networks, with an application to correlation networks. Multiscale Model. Simul. 14, 1–41 (2016b)

    MathSciNet  MATH  Google Scholar 

  • Bentley, B., Branicky, R., Barnes, C.L., Chew, Y.L., Yemini, E., Bullmore, E.T., Vértes, P.E., Schafer, W.R.: The multilayer connectome of caenorhabditis elegans. PLoS Comput. Biol. 12(12), e1005283 (2016)

    Google Scholar 

  • Blondel, V.D., Guillaume, J.-L., Lambiotte, R., Lefebvre, E.: Fast unfolding of communities in large networks. J. Stat. Mech. Theory Exp. 2008(10), P10008 (2008)

    MATH  Google Scholar 

  • Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.-U.: Complex networks: structure and dynamics. Phys. Rep. 424(4), 175–308 (2006)

    MathSciNet  MATH  Google Scholar 

  • Boccaletti, S., Bianconi, G., Criado, R., del Genio, C.I., Gómez-Gardeñes, J., Romance, M., Sendiña-Nadal, I., Wang, Z., Zanin, M.: The structure and dynamics of multilayer networks. Phys. Rep. 544(1), 1–122 (2014)

    MathSciNet  Google Scholar 

  • Boutemine, O., Bouguessa, M.: Mining community structures in multidimensional networks. ACM Trans. Knowl. Discov. Data (TKDD) 11(4), 51 (2017)

    Google Scholar 

  • Braun, U., Muldoon, S.F., Bassett, D.S.: On human brain networks in health and disease. eLS. 1–9 (2015a)

  • Braun, U., Schäfer, A., Walter, H., Erk, S., Romanczuk-Seiferth, N., Haddad, L., Schweiger, J.I., Grimm, O., Heinz, A., Tost, H., Meyer-Lindenberg, A., Bassett, D.S.: Dynamic reconfiguration of frontal brain networks during executive cognition in humans. Proc. Natl. Acad. Sci. 112(37), 11678–11683 (2015b)

  • Brookes, M.J., Tewarie, P.K., Hunt, B.A.E., Robson, S.E., Gascoyne, L.E., Liddle, E.B., Liddle, P.F., Morris, P.G.: A multi-layer network approach to meg connectivity analysis. Neuroimage 132, 425–438 (2016)

    Google Scholar 

  • Brovelli, A., Badier, J.-M., Bonini, F., Bartolomei, F., Coulon, O., Auzias, G.: Dynamic reconfiguration of visuomotor-related functional connectivity networks. J. Neurosci. 37(4), 839–853 (2017)

    Google Scholar 

  • Buldú, J.M., Porter, M. A.: Frequency-based brain networks: From a multiplex framework to a full multilayer description. arXiv.org (2017)

  • Bullmore, E., Sporns, O.: Complex brain networks: graph theoretical analysis of structural and functional systems. Nat. Rev. Neurosci. 10(3), 186–198 (2009)

    Google Scholar 

  • Calhoun, V.D., Eichele, T., Pearlson, G.: Functional brain networks in schizophrenia: a review. Front. Hum. Neurosci. 3, 17 (2009)

  • Cellai, D., López, E., Zhou, J., Gleeson, J.P., Bianconi, G.: Percolation in multiplex networks with overlap. Phys. Rev. E 88(5), 052811 (2013)

    Google Scholar 

  • Chen, J.E., Glover, G.H.: Bold fractional contribution to resting-state functional connectivity above 0.1 Hz. NeuroImage 107, 207–218 (2015)

    Google Scholar 

  • Chen, T., Cai, W., Ryali, S., Supekar, K., Menon, V.: Distinct global brain dynamics and spatiotemporal organization of the salience network. PLoS Biol. 14(6), e1002469 (2016)

    Google Scholar 

  • Crofts, J.J., Forrester, M., O’Dea, R.D.: Structure-function clustering in multiplex brain networks. EPL (Europhys. Lett.) 116(1), 18003–8 (2016)

    Google Scholar 

  • Danziger, M.M., Shekhtman, L.M., Berezin, Y., Havlin, S.: The effect of spatiality on multiplex networks. EPL (Europhys. Lett.) 115(3), 36002 (2016)

    Google Scholar 

  • de Arruda, G.F., Cozzo, E., Moreno, Y.: On degree–degree correlations in multilayer networks. Phys. D Nonlinear Phenom. 323, 5–11 (2016)

    MathSciNet  MATH  Google Scholar 

  • De Domenico, M.: Manlio: multilayer modeling and analysis of human brain networks. Giga Sci. 6(5), 1–8 (2017)

    Google Scholar 

  • De Domenico, M., Solé-Ribalta, A., Cozzo, E., Kivelä, M., Moreno, Y., Porter, M.A., Gómez, S., Arenas, A.: Mathematical formulation of multilayer networks. Phys. Rev. X 3(4), 041022 (2013)

    Google Scholar 

  • De Domenico, M., Solé-Ribalta, A., Omodei, E., Gómez, S., Arenas, A.: Ranking in interconnected multilayer networks reveals versatile nodes. Nat. Commun. 6, 6868 (2015a)

    Google Scholar 

  • De Domenico, M., Porter, M.A., Arenas, A.: Muxviz: a tool for multilayer analysis and visualization of networks. J. Complex Netw. 3(2), 159–176 (2015b)

    Google Scholar 

  • De Domenico, M., Nicosia, V., Arenas, A., Latora, V.: Structural reducibility of multilayer networks. Nat. Commun. 6, 6864 (2015c)

    Google Scholar 

  • De Domenico, M., Lancichinetti, A., Arenas, A., Rosvall, M.: Identifying modular flows on multilayer networks reveals highly overlapping organization in interconnected systems. Phys. Rev. X 5(1), 011027 (2015d)

    Google Scholar 

  • De Domenico, M., Nicosia, V., Arenas, A., Latora, V.: Structural reducibility of multilayer networks. Nat. Commun. 6, 6864 (2015e)

    Google Scholar 

  • De Domenico, M., Granell, C., Porter, M.A., Arenas, A.: The physics of spreading processes in multilayer networks. Nat. Phys. 12(10), 901–906 (2016a)

    Google Scholar 

  • De Domenico, M., Sasai, S., Arenas, A.: Mapping multiplex hubs in human functional brain networks. Front. Neurosci. 10, 326 (2016b)

  • Decelle, A., Krzakala, F., Moore, C., Zdeborová, L.: Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications. Phys. Rev. E 84(6), 066106 (2011)

    Google Scholar 

  • Delamillieure, P., Doucet, G., Mazoyer, B., Turbelin, M.-R., Delcroix, N., Mellet, E., Zago, L., Crivello, F., Petit, L., Tzourio-Mazoyer, N., et al.: The resting state questionnaire: an introspective questionnaire for evaluation of inner experience during the conscious resting state. Brain Res. Bull. 81(6), 565–573 (2010)

    Google Scholar 

  • Erdos, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5(1), 17–60 (1960)

    MathSciNet  MATH  Google Scholar 

  • Feldt, S., Bonifazi, P., Cossart, R.: Dissecting functional connectivity of neuronal microcircuits: experimental and theoretical insights. Trends Neurosci. 34(5), 225–236 (2011)

    Google Scholar 

  • Fornito, A., Zalesky, A., Breakspear, M.: The connectomics of brain disorders. Nat. Rev. Neurosci. 16(3), 159–172 (2015)

    Google Scholar 

  • Gallotti, R., Barthelemy, M.: The multilayer temporal network of public transport in great Britain. Sci. Data 2, 140056 (2015)

  • Gao, C., Sun, M., Shen, B.: Features and evolution of international fossil energy trade relationships: a weighted multilayer network analysis. Appl. Energy 156, 542–554 (2015)

    Google Scholar 

  • Gauvin, L., Panisson, A., Cattuto, C.: Detecting the community structure and activity patterns of temporal networks: a non-negative tensor factorization approach. PLoS ONE 9(1), e86028 (2014)

    Google Scholar 

  • Good, B.H., de Montjoye, Y.A., Clauset, A.: Performance of modularity maximization in practical contexts. Phys. Rev. E 81(4), 046106 (2010)

    MathSciNet  Google Scholar 

  • Gosak, M., Dolenšek, J., Markovič, R., Rupnik, M.S., Marhl, M., Stožer, A.: Multilayer network representation of membrane potential and cytosolic calcium concentration dynamics in beta cells. Chaos Solitons Fractals 80, 76–82 (2015)

    MATH  Google Scholar 

  • Guillon, J., Attal, Y., Colliot, O., La Corte, V., Dubois, B., Schwartz, D., Chavez, M., De Vico Fallani, F:. Loss of brain inter-frequency hubs in Alzheimer’s disease. Sci. Rep. 7, 1–13 (2017)

  • Halu, A., De Domenico, M., Arenas, A., Sharma, A.: The multiplex network of human diseases. bioRxiv 100370 (2017)

  • Iacovacci, J., Bianconi, G.: Extracting information from multiplex networks. Chaos: an interdisciplinary. J. Nonlinear Sci. 26(6), 065306 (2016)

    Google Scholar 

  • Iacovacci, J., Zhihao, W., Bianconi, G.: Mesoscopic structures reveal the network between the layers of multiplex data sets. Phys. Rev. E 92(4), 042806 (2015)

    Google Scholar 

  • Jeub, L.G.S., Bazzi, M., Jutla, I.S., Mucha, P.J.: A generalized Louvain method for community detection implemented in MATLAB. http://netwiki.amath.unc.edu/GenLouvain (2011–2017)

  • Khambhati, A.N., Sizemore, A.E., Betzel, R.F., Bassett, D.S.: Modeling and interpreting mesoscale network dynamics. NeuroImage (2017)

  • Kivelä, M., Porter, M.A.: Isomorphisms in multilayer networks. arXiv preprint arXiv:1506.00508 (2015)

  • Kivela, M., Arenas, A., Barthelemy, M., Gleeson, J.P., Moreno, Y., Porter, M.A.: Multilayer networks. J. Complex Netw. 2(3), 203–271 (2014)

    Google Scholar 

  • Lambiotte, R., Delvenne, J.-C., Barahona, M.: Random walks, Markov processes and the multiscale modular organization of complex networks. IEEE Trans. Netw. Sci. Eng. 1(2), 76–90 (2014)

    MathSciNet  Google Scholar 

  • Lee, K.-H., Farrow, T.F.D., Spence, S.A., Woodruff, P.W.R.: Social cognition, brain networks and schizophrenia. Psychol. Med. 34(3), 391–400 (2004)

    Google Scholar 

  • Liao, X.-H., Xia, M.-R., Xu, T., Dai, Z.-J., Cao, X.-Y., Niu, H.-J., Zuo, X.-N., Zang, Y.-F., He, Y.: Functional brain hubs and their test–retest reliability: a multiband resting-state functional mri study. Neuroimage 83, 969–982 (2013)

    Google Scholar 

  • Luczak, T.: Sparse random graphs with a given degree sequence. In: Proceedings of the Symposium on Random Graphs, Poznan, pp. 165–182 (1989)

  • Lynall, M.-E., Bassett, D.S., Kerwin, R., McKenna, P.J., Kitzbichler, M., Muller, U., Bullmore, E.: Functional connectivity and brain networks in schizophrenia. J. Neurosci. 30(28), 9477–9487 (2010)

    Google Scholar 

  • Majhi, S., Perc, M., Ghosh, D.: Chimera states in uncoupled neurons induced by a multilayer structure. Sci. Rep. 6, 39033 (2016)

    Google Scholar 

  • Makarov, V.V., Koronovskii, A.A., Maksimenko, V.A., Hramov, A.E., Moskalenko, O.I., Buldu, J.M., Boccaletti, S.: Emergence of a multilayer structure in adaptive networks of phase oscillators. Chaos Solitons Fractals 84, 23–30 (2016)

    MATH  Google Scholar 

  • Maksimenko, V.A., Makarov, V.V., Bera, B.K., Ghosh, D., Dana, S.K., Goremyko, M.V., Frolov, N.S., Koronovskii, A.A., Hramov, A.E.: Excitation and suppression of chimera states by multiplexing. Phys. Rev. E 94(5–1), 052205 (2016)

    Google Scholar 

  • Martinčić-Ipšić, S., Margan, D., Meštrović, A.: Multilayer network of language: a unified framework for structural analysis of linguistic subsystems. Phys. A Stat. Mech. Appl. 457, 117–128 (2016)

    Google Scholar 

  • Matias, C., Miele, V.: Statistical clustering of temporal networks through a dynamic stochastic block model. J. R Stat. Soc. Seri. B (Stat. Methodol.) 79(4), 1119–1141 (2017)

  • Mattar, M.G., Bassett, D.S.: Brain network architecture: implications for human learning. arXiv preprintarXiv:1609.01790 (2016)

  • Medaglia, J.D., Lynall, M.-E., Bassett, D.S.: Cognitive network neuroscience. J. Cognit. Neurosci. 27(8), 1471–1491 (2015)

    Google Scholar 

  • Mucha, P.J., Richardson, T., Macon, K., Porter, M.A., Onnela, J.-P.: Community structure in time-dependent, multiscale, and multiplex networks. Science 328(5980), 876–878 (2010)

    MathSciNet  MATH  Google Scholar 

  • Muldoon, S.F., Bassett, D.S.: Why network neuroscience? Compelling evidence and current frontiers. Comment on “understanding brain networks and brain organization” by Luiz Pessoa. Phys. Life Rev. 11(3), 455–457 (2014)

    Google Scholar 

  • Muldoon, S.F., Bassett, D.S.: Network and multilayer network approaches to understanding human brain dynamics. Philos. Sci. 83(5), 710–720 (2016)

    MathSciNet  Google Scholar 

  • Muldoon, S.F., Bridgeford, E.W., Bassett, D.S.: Small-world propensity and weighted brain networks. Sci. Rep. 6(1), 22057 (2016)

    Google Scholar 

  • Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45(2), 167–256 (2003)

    MathSciNet  MATH  Google Scholar 

  • Newman, M.E.J.: Modularity and community structure in networks. Proc. Natl. Acad. Sci. 103(23), 8577–8582 (2006)

    Google Scholar 

  • Nicosia, V., Latora, V.: Measuring and modeling correlations in multiplex networks. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 92(3), 032805–20 (2015)

    Google Scholar 

  • Nicosia, V., Skardal, P.S., Arenas, A., Latora, V.: Collective phenomena emerging from the interactions between dynamical processes in multiplex networks. Phys. Rev. Lett. 118(13), 138302 (2017)

    Google Scholar 

  • Ou-Yang, L., Dai, D.-Q., Li, X.-L., Min, W., Zhang, X.-F., Yang, P.: Detecting temporal protein complexes from dynamic protein–protein interaction networks. BMC Bioinform. 15(1), 335 (2014)

    Google Scholar 

  • Page, L., Brin, S., Motwani, R., Winograd, T.: The pagerank citation ranking: bringing order to the web. Technical Report, Stanford InfoLab (1999)

  • Papadopoulos, L., Puckett, J.G., Daniels, K.E., Bassett, D.S.: Evolution of network architecture in a granular material under compression. Phys. Rev. E 94(3), 032908 (2016)

    Google Scholar 

  • Paul, S., Chen, Y.: Null models and modularity based community detection in multi-layer networks. arXiv preprintarXiv:1608.00623 (2016)

  • Pilosof, S., Porter, M.A., Pascual, M., Kéfi, S.: The multilayer nature of ecological networks. Nat. Ecol. Evol. 1, 0101 (2017)

    Google Scholar 

  • Porter, M.A., Onnela, J.-P., Mucha, P.J.: Communities in networks. Not. AMS 56(9), 1082–1097 (2009)

    MathSciNet  MATH  Google Scholar 

  • Rombach, P., Porter, M.A., Fowler, J.H., Mucha, P.J.: Core-periphery structure in networks (revisited). SIAM Rev. 59(3), 619–646 (2017)

    MathSciNet  MATH  Google Scholar 

  • Sarzynska, M., Leicht, E.A., Chowell, G., Porter, M.A.: Null models for community detection in spatially embedded, temporal networks. J. Complex Netw. 4(3), 363–406 (2015)

    MathSciNet  Google Scholar 

  • Sasai, S., Homae, F., Watanabe, H., Sasaki, A.T., Tanabe, H.C., Sadato, N., Taga, G.: Frequency-specific network topologies in the resting human brain. Front. Hum. Neurosci. 8, 1022 (2014)

  • Solá, L., Romance, M., Criado, R., Flores, J., del Amo, A.G., Boccaletti, S.: Eigenvector centrality of nodes in multiplex networks. Chaos Interdiscip. J. Nonlinear Sci. 23(3), 033131 (2013)

    MATH  Google Scholar 

  • Sporns, O., Betzel, R.F.: Modular brain networks. Ann. Rev. Psychol. 67(1), 613–640 (2016)

    Google Scholar 

  • Sporns, O., Chialvo, D., Kaiser, M., Hilgetag, C.: Organization, development and function of complex brain networks. Trends Cognit. Sci. 8(9), 418–425 (2004)

    Google Scholar 

  • Supekar, K., Menon, V., Rubin, D., Musen, M., Greicius, M.D.: Network analysis of intrinsic functional brain connectivity in alzheimer’s disease. PLoS Comput. Biol. 4(6), e1000100 (2008)

    Google Scholar 

  • Taylor, D., Shai, S., Stanley, N., Mucha, P.J.: Enhanced detectability of community structure in multilayer networks through layer aggregation. Phys. Rev. Lett. 116(22), 228301 (2016)

    Google Scholar 

  • Telesford, Q.K., Lynall, M.-E., Vettel, J., Miller, M.B., Grafton, S.T., Bassett, D.S.: Detection of functional brain network reconfiguration during task-driven cognitive states. NeuroImage 142, 198–210 (2016)

    Google Scholar 

  • Telesford, Q.K., Ashourvan, A., Wymbs, N.F., Grafton, S.T., Vettel, J.M., Bassett, D.S.: Cohesive network reconfiguration accompanies extended training. Hum. Brain Map. 38(9), 4744–4759 (2017a)

    Google Scholar 

  • Telesford, Q.K., Ashourvan, A., Wymbs, N.F., Grafton, S.T., Vettel, J.M., Bassett, D.S.: Cohesive network reconfiguration accompanies extended training. Hum. Brain Map. 38(9), 4744–4759 (2017b)

    Google Scholar 

  • Tewarie, P., Hillebrand, A., van Dijk, B.W., Stam, C.J., O’Neill, G.C., Van Mieghem, P., Meier, J.M., Woolrich, M.W., Morris, P.G., Brookes, M.J.: Integrating cross-frequency and within band functional networks in resting-state MEG: a multi-layer network approach. NeuroImage 142(C), 324–336 (2016)

    Google Scholar 

  • Thompson, W.H., Fransson, P.: The frequency dimension of fmri dynamic connectivity: network connectivity, functional hubs and integration in the resting brain. NeuroImage 121, 227–242 (2015)

    Google Scholar 

  • Tijms, B.M., Wink, A.M., de Haan, W., van der Flier, W.M., Stam, C.J., Scheltens, P., Barkhof, F.: Alzheimer’s disease: connecting findings from graph theoretical studies of brain networks. Neurobiol. Aging 34(8), 2023–2036 (2013)

    Google Scholar 

  • van den Martijn, M.P., Fornito, A.: Brain networks in schizophrenia. Neuropsychol. Rev. 24(1), 32–48 (2014)

    Google Scholar 

  • Virkar, Y.S., Shew, W.L., Restrepo, J.G., Ott, E.: Feedback control stabilization of critical dynamics via resource transport on multilayer networks: how glia enable learning dynamics in the brain. Phys. Rev. E 94(4), 042310 (2016)

    Google Scholar 

  • Weir, W.H., Emmons, S., Gibson, R., Taylor, D., Mucha, P.J.: Post-processing partitions to identify domains of modularity optimization. Algorithms 10(3), 93–21 (2017)

    MathSciNet  MATH  Google Scholar 

  • White, J.G., Southgate, E., Thomson, J.N., Brenner, S.: The structure of the nervous system of the nematode caenorhabditis elegans: the mind of a worm. Philos. Trans. R Soc. Lond. 314, 1–340 (1986)

    Google Scholar 

  • Wymbs, N.F., Bassett, D.S., Mucha, P.J., Porter, M.A., Grafton, S.T.: Differential recruitment of the sensorimotor putamen and frontoparietal cortex during motor chunking in humans. Neuron 74(5), 936–946 (2012)

    Google Scholar 

  • Yang, Z., Algesheimer, R., Tessone, C.J.: A comparative analysis of community detection algorithms on artificial networks. Sci. Rep. 6, 30750 (2016)

    Google Scholar 

  • Yu, M., Engels, M., Hillebrand, A., van Straaten, E.C.W., Gouw, A.A., Teunissen, C., van der Flier, W.M., Scheltens, P., Stam, C.J.: Selective impairment of hippocampus and posterior hub areas in alzheimer’s disease: an meg-based multiplex network study. Brain 140(5), 1466–1485 (2017)

    Google Scholar 

  • Zhang, H., Wang, C.-D., Lai, J.-H., Philip, S.Y.: Modularity in complex multilayer networks with multiple aspects: a static perspective. Appl. Inform. 4 (2017). https://doi.org/10.1186/s40535-017-0035-4

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Acknowledgements

SFM would like to acknowledge support from the National Science Foundation (SMA-1734795) and the Army Research Laboratory (Contract Number: W911NF-10-2-0022). The content is solely the responsibility of the authors and does not necessarily represent the official views of any of the funding agencies.

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  1. Department of Mathematics and CDSE Program, University at Buffalo, SUNY, Buffalo, NY, USA

    Michael Vaiana & Sarah Feldt Muldoon

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  1. Michael Vaiana
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Correspondence to Sarah Feldt Muldoon.

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Communicated by Danielle S. Bassett.

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Vaiana, M., Muldoon, S.F. Multilayer Brain Networks. J Nonlinear Sci 30, 2147–2169 (2020). https://doi.org/10.1007/s00332-017-9436-8

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