Element contents
Multilayer Network Science
From Cells to Societies
Published online by Cambridge University Press: 23 August 2022
Summary
Networks are convenient mathematical models to represent the structure of complex systems, from cells to societies. In the last decade, multilayer network science – the branch of the field dealing with units interacting in multiple distinct ways, simultaneously – was demonstrated to be an effective modeling and analytical framework for a wide spectrum of empirical systems, from biopolymers networks (such as interactome and metabolomes) to neuronal networks (such as connectomes), from social networks to urban and transportation networks. In this Element, a decade after one of the most seminal papers on this topic, the authors review the most salient features of multilayer network science, covering both theoretical aspects and direct applications to real-world coupled/interdependent systems, from the point of view of multilayer structure, dynamics and function. The authors discuss potential frontiers for this topic and the corresponding challenges in the field for the next future.
Keywords
Information
- Type
- Element
- Information
- Online ISBN: 9781009085809Publisher: Cambridge University PressPrint publication: 15 September 2022
References
, L. G. Alves, A., Mangioni, G., Rodrigues, F. A., Panzarasa, P., and Moreno., Y. Unfolding the complexity of the global value chain: Strength and entropy in the single-layer, multiplex, and multi-layer international trade networks. Entropy, 20 (12): 909, 2018.Google Scholar
Acebrón, J. A., Bonilla, L. L., Vicente, C. J. P., Ritort, F., and Spigler, R.. The Kuramoto model: A simple paradigm for synchronization phenomena. Reviews of Modern Physics, 77 (1): 137, 2005.CrossRefGoogle Scholar
Achard, S. and Bullmore, E.. Efficiency and cost of economical brain functional networks. PLoS Computational Biology, 3 (2): e17, 2007.CrossRefGoogle ScholarPubMed
Adamic, L. A. and Adar, E.. Friends and neighbors on the Web. Social Networks, 25 (3): 211–230, 2003.CrossRefGoogle Scholar
Airoldi, E. M., Blei, D. M., Fienberg, S. E., and Xing, E. P.. Mixed membership stochastic blockmodels. Journal of Machine Learning Research, 9: 1981–2014, 2008.Google ScholarPubMed
Akbarzadeh, M. and Estrada, E.. Communicability geometry captures traffic flows in cities. Nature Human Behaviour, 2 (9): 645–652, 2018.Google Scholar
Albert, R., Jeong, H., and Barabási, A.-L.. Error and attack tolerance of complex networks. Nature, 406 (6794): 378, 2000.CrossRefGoogle ScholarPubMed
Aleta, A., Tuninetti, M., Paolotti, D., Moreno, Y., and Starnini, M.. Link prediction in multiplex networks via triadic closure. Physical Review Research, 2 (4): 042029, 2020.CrossRefGoogle Scholar
Alves, L. G. A., Mangioni, G., Cingolani, I., Rodrigues, F. A., Panzarasa, P., and Moreno, Y.. The nested structural organization of the worldwide trade multi-layer network. Scientific Reports, 9 (1): 1–14, 2019.Google Scholar
Amato, R., Díaz-Guilera, A., and Kleineberg, K.-K.. Interplay between social influence and competitive strategical games in multiplex networks. Scientific Reports, 7 (1): 1–8, 2017.Google Scholar
Amato, R., Kouvaris, N. E., San Miguel, M., and Díaz-Guilera, A.. Opinion competition dynamics on multiplex networks. New Journal of Physics, 19 (12): 123019, 2017.CrossRefGoogle Scholar
Amelio, A., Mangioni, G., and Tagarelli, A.. Modularity in multilayer networks using redundancy-based resolution and projection-based inter-layer coupling. IEEE Transactions on Network Science and Engineering, 7(3):1198–1214, 1 July–Sept. 2020. https://doi.org/10.1109/TNSE.2019.2913325.Google Scholar
Anandkumar, A., Ge, R., Hsu, D., and Kakade, S. M.. A tensor approach to learning mixed membership community models. Journal of Machine Learning Research, 15 (1): 2239–2312, 2014.Google Scholar
Antonopoulos, C. G. and Shang, Y.. Opinion formation in multiplex networks with general initial distributions. Scientific Reports, 8 (1): 2852, 2018.Google Scholar
Arenas, A., Díaz-Guilera, A., and Pérez-Vicente, C. J.. Synchronization reveals topological scales in complex networks. Physical Review Letters, 96: 114102, 2006.Google Scholar
Arenas, A., Diaz-Guilera, A., Kurths, J., Moreno, Y., and Zhou, C.. Synchronization in complex networks. Physics Reports, 469(3):93–153, 2008a. https://doi.org/10.1016/j.physrep.2008.09.002CrossRefGoogle Scholar
Arenas, A., Fernandez, A., and Gomez, S.. Analysis of the structure of complex networks at different resolution levels. New Journal of Physics, 10 (5): 053039, 2008b.CrossRefGoogle Scholar
Artime, O. and De Domenico, M.. Abrupt transition due to non-local cascade propagation in multiplex systems. New Journal of Physics, 22 (9): 093035, 2020.Google Scholar
Artime, O. and De Domenico., M. Percolation on feature-enriched interconnected systems. Nature Communications, 12 (1): 1–12, 2021.CrossRefGoogle ScholarPubMed
Artime, O., Fernández-Gracia, J., Ramasco, J. J., and San Miguel, M.. Joint effect of ageing and multilayer structure prevents ordering in the voter model. Scientific Reports, 7 (1): 7166, 2017.Google Scholar
Artime, O., d’Andrea, V., Gallotti, R., Sacco, P. L., and De Domenico, M.. Effectiveness of dismantling strategies on moderated vs. unmoderated online social platforms. Scientific Reports, 10 (1): 1–11, 2020.Google Scholar
Aslak, U., Rosvall, M., and Lehmann, S.. Constrained information flows in temporal networks reveal intermittent communities. Physical Review E, 97 (6): 062312, 2018.CrossRefGoogle ScholarPubMed
Asllani, M., Busiello, D. M., Carletti, T., Fanelli, D., and Planchon, G.. Turing patterns in multiplex networks. Physical Review E, 90 (4): 042814, 2014.Google Scholar
Asllani, M., Busiello, D. M., Carletti, T., Fanelli, D., and Planchon, G.. Turing instabilities on Cartesian product networks. Scientific Reports, 5 (1): 1–10, 2015.Google Scholar
Azimi-Tafreshi, N.. Cooperative epidemics on multiplex networks. Physical Review E, 93 (4): 042303, 2016.Google Scholar
Azimi-Tafreshi, N., Gómez-Gardenes, J., and Dorogovtsev, S. N.. k—Core percolation on multiplex networks. Physical Review E, 90 (3): 032816, 2014.CrossRefGoogle ScholarPubMed
Baggio, J. A., BurnSilver, S. B., Arenas, A., Magdanz, J. S., Kofinas, G. P., and De Domenico, M.. Multiplex social ecological network analysis reveals how social changes affect community robustness more than resource depletion. Proceedings of the National Academy of Sciences, 113 (48): 13708–13713, 2016.Google Scholar
Bak, P., Tang, C., and Wiesenfeld., K. Self-organized criticality. Physical Review A, 38 (1): 364, 1988.Google Scholar
Barrat, A., Barthelemy, M., and Vespignani, A.. Dynamical processes on complex networks. Cambridge University Press, 2008.Google Scholar
Bashan, A., Berezin, Y., Buldyrev, S. V., and Havlin, S.. The extreme vulnerability of interdependent spatially embedded networks. Nature Physics, 9 (10): 667–672, 2013.Google Scholar
Bassett, D. S. and Sporns, O.. Network neuroscience. Nature Neuroscience, 20 (3): 353, 2017.CrossRefGoogle ScholarPubMed
Battiston, F., Nicosia, V., and Latora, V.. Structural measures for multiplex networks. Physical Review E, 89 (3): 032804, 2014.Google Scholar
Battiston, F., Nicosia, V., and Latora, V.. Efficient exploration of multiplex networks. New Journal of Physics, 18 (4): 043035, 2016.Google Scholar
Battiston, F., Nicosia, V., and Latora, V.. The new challenges of multiplex networks: measures and models. European Physical Journal Special Topics, 226 (3): 401–416, 2017a.Google Scholar
Battiston, F., Perc, M., and Latora, V.. Determinants of public cooperation in multiplex networks. New Journal of Physics, 19 (7): 073017, 2017b.Google Scholar
Battiston, F., Cencetti, G., Iacopini, I., et al. Networks beyond pairwise interactions: Structure and dynamics. Physics Reports, 874:1–92, 2020.Google Scholar
Baxter, G. J., Dorogovtsev, S. N., Goltsev, A. V., and Mendes, J. F. F.. Avalanche collapse of interdependent networks. Physical Review Letters, 109 (24): 248701, 2012.CrossRefGoogle ScholarPubMed
Baxter, G. J., Bianconi, G., da Costa, R. A., Dorogovtsev, S. N., and Mendes, J. F. F.. Correlated edge overlaps in multiplex networks. Physical Review E, 94 (1): 012303, 2016.Google Scholar
Bazzi, M., Porter, M. A., Williams, S., McDonald, M., Fenn, D. J., and Howison, S. D.. Community detection in temporal multilayer networks, with an application to correlation networks. Multiscale Modeling & Simulation, 14 (1): 1–41, 2016.CrossRefGoogle Scholar
Bazzi, M., Jeub, L. G. S., Arenas, A., Howison, S. D., and Porter, M. A.. A framework for the construction of generative models for mesoscale structure in multilayer networks. Physical Review Research, 2 (2): 023100, 2020.Google Scholar
Beisner, B., Braun, N., Pósfai, M., Vandeleest, J., D’Souza, R., and McCowan, B.. A multiplex centrality metric for complex social networks: Sex, social status, and family structure predict multiplex centrality in rhesus macaques. PeerJ, 8: e8712, 2020.Google Scholar
Bentley, B., Branicky, R., Barnes, C. L., et al. The multilayer connectome of Caenorhabditis elegans. PLoS Computational Biology, 12 (12): 1005283, 2016.Google Scholar
Berezin, Y., Bashan, A., and Havlin, S.. Comment on “Percolation transitions are not always sharpened by making networks interdependent.” Physical Review Letters, 111 (18): 189601, 2013.Google Scholar
Bertagnolli, G. and De Domenico, M.. Diffusion geometry of multiplex and interdependent systems. Physical Review E, 103: 042301, 2021.Google Scholar
Bertagnolli, G., Gallotti, R., and De Domenico, M.. Quantifying efficient information exchange in real network flows. Communications Physics, 4 (1): 1–10, 2021.Google Scholar
Biamonte, J., Faccin, M., and De Domenico, M.. Complex networks from classical to quantum. Communications Physics, 2 (1): 1–10, 2019.Google Scholar
Bianconi, G.. Statistical mechanics of multiplex networks: Entropy and overlap. Physical Review E, 87: 062806, 2013.Google Scholar
Bianconi, G.. Epidemic spreading and bond percolation on multilayer networks. Journal of Statistical Mechanics, 2017 (3): 034001, 2017.Google Scholar
Bianconi, G.. Multilayer networks: Structure and function. Oxford University Press, 2018.Google Scholar
Bianconi, G. and Radicchi, F.. Percolation in real multiplex networks. Physical Review E, 94 (6): 060301, 2016.Google Scholar
Boccaletti, S., Bianconi, G., Criado, R., et al. The structure and dynamics of multilayer networks. Physics Reports, 544 (1): 1–122, 2014.Google Scholar
Boccaletti, S., Pisarchik, A. N., del Genio, C. I., and Amann, A.. Synchronization. Cambridge University Press, 2018.CrossRefGoogle Scholar
Boguñá, M., Krioukov, D., and Claffy, K. C.. Navigability of complex networks. Nature Physics, 5 (1): 74, 2009.Google Scholar
Boguñá, M., Bonamassa, I., De Domenico, M., Havlin, S., Krioukov, D., and M. Á. Serrano. Network geometry. Nature Reviews Physics, 3:114–135, 2021.CrossRefGoogle Scholar
Bonacich, P.. Power and centrality: A family of measures. American Journal of Sociology, 92 (5): 1170–1182, 1987.CrossRefGoogle Scholar
Borgatti, S. P. and Everett, M. G.. A graph-theoretic perspective on centrality. Social Networks, 28 (4): 466–484, 2006.Google Scholar
Bosetti, P., Poletti, P., Stella, M., Lepri, B., Merler, S., and De Domenico, M.. Heterogeneity in social and epidemiological factors determines the risk of measles outbreaks. PNAS, 117:30118, 2020.Google Scholar
Bottcher, L. and Porter, M. A.. Classical and quantum random-walk centrality measures in multilayer networks. arxiv preprint arXiv:2012. 07157, 2020.Google Scholar
Brechtel, A., Gramlich, P., Ritterskamp, D., Drossel, B., and Gross., T. Master stability functions reveal diffusion-driven pattern formation in networks. Physical Review E, 97 (3), 2018.Google Scholar
Brin, S. and Page, L.. The anatomy of a large-scale hypertextual web search engine. Computer Networks and ISDN Systems, 30 (1–7): 107–117, 1998.Google Scholar
Bródka, P., Chmiel, A., Magnani, M., and Ragozini, G.. Quantifying layer similarity in multiplex networks: A systematic study. Royal Society Open Science, 5 (8): 171747, 2018.Google Scholar
Brummitt, C. D., D’Souza, R. M., and Leicht, E. A.. Suppressing cascades of load in interdependent networks. PNAS, 109 (12): E680–E689, 2012 a.Google Scholar
Brummitt, C. D., Lee, K.-M., and Goh, K.-I.. Multiplexity-facilitated cascades in networks. Physical Review E, 85 (4): 045102, 2012 b.Google Scholar
Buccafurri, F., Lax, G., Nicolazzo, S., Nocera, A., and Ursino, D.. Measuring betweenness centrality in social internetworking scenarios. In Demey, Y. T. and Panetto, H. (eds.), On the move to meaningful internet systems: OTM 2013 Workshops. OTM 2013. Lecture Notes in Computer Science, vol. 8186. Springer, 2013. https://doi.org/10.1007/978-3-642-41033-8_84Google Scholar
Buendía, V., Villegas, P., Burioni, R., and Muñoz, M. A.. The broad edge of synchronisation: Griffiths effects and collective phenomena in brain networks. arXiv preprint arXiv:2109.11783, 2021.Google Scholar
Buldyrev, S. V., Parshani, R., Paul, G., Stanley, H. E., and Havlin, S.. Catastrophic cascade of failures in interdependent networks. Nature, 464 (7291): 1025–1028, 2010.Google Scholar
Buono, C., Alvarez-Zuzek, L. G., Macri, P. A., and Braunstein, L. A.. Epidemics in partially overlapped multiplex networks. PloS One, 9 (3): e92200, 2014.Google Scholar
Burda, Z., Duda, J., Luck, J.-M., and Waclaw, B.. Localization of the maximal entropy random walk. Physical Review Letters, 102 (16): 160602, 2009.Google Scholar
Busiello, D. M., Carletti, T., and Fanelli, D.. Homogeneous-per-layer patterns in multiplex networks. Europhysics Letters, 121 (4): 48006, 2018.Google Scholar
Carchiolo, V., Longheu, A., Malgeri, M., and Mangioni, G.. Communities unfolding in multislice networks. In Complex Networks, pages 187–195. Springer, 2011.Google Scholar
Cardillo, A., Gómez-Gardeñes, J., Zanin, M., et al. Emergence of network features from multiplexity. Scientific Reports, 3 (1), 2013.CrossRefGoogle ScholarPubMed
Cellai, D., López, E., Zhou, J., Gleeson, J. P., and Bianconi, G.. Percolation in multiplex networks with overlap. Physical Review E, 88 (5): 052811, 2013.Google Scholar
Cellai, D., Dorogovtsev, S. N., and Bianconi, G.. Message passing theory for percolation models on multiplex networks with link overlap. Physical Review E, 94 (3): 032301, 2016.Google Scholar
Centola, D.. The social origins of networks and diffusion. American Journal of Sociology, 120 (5): 1295–1338, 2015.Google Scholar
Chodrow, P. S., Al-Awwad, Z., Jiang, S., and González, M. C.. Demand and congestion in multiplex transportation networks. PloS One, 11 (9): e0161738, 2016.Google Scholar
Chung, F. R. K.. Spectral graph theory. 2nd edition. American Mathematical Society, 1997.Google Scholar
Cimini, G., Squartini, T., Saracco, F., et al. The statistical physics of real-world networks. Nature Reviews Physics, 1 (1): 58–71, 2019.Google Scholar
Cohen, R., Erez, K., Ben-Avraham, D., and Havlin, S.. Breakdown of the Internet under intentional attack. Physical Review Letters, 86 (16): 3682, 2001.Google Scholar
Cozzo, E., Baños, R. A., Meloni, S., and Moreno, Y.. Contact-based social contagion in multiplex networks. Physical Review E, 88 (5): 050801, 2013.Google Scholar
Cozzo, E., Kivelä, M., De Domenico, M., et al. Structure of triadic relations in multiplex networks. New Journal of Physics, 17 (7): 073029, 2015.Google Scholar
Cozzo, E., De Arruda, G. F., Rodrigues, F. A., and Moreno, Y.. Multiplex networks: Basic formalism and structural properties. Springer, 2018.Google Scholar
Criado, R., Flores, J., García del Amo, A., Gómez-Gardeñes, J., and Romance, M.. A mathematical model for networks with structures in the mesoscale. International Journal of Computer Mathematics, 89 (3): 291–309, 2012.CrossRefGoogle Scholar
Czaplicka, A., Toral, R., and San Miguel, M.. Competition of simple and complex adoption on interdependent networks. Physical Review E, 94 (6): 062301, 2016.Google Scholar
O’Brien, J. D., Dassios, I. K., and Gleeson, J. P.. Spreading of memes on multiplex networks. New Journal of Physics, 21 (2): 025001, 2019.Google Scholar
Danziger, M. M., Shekhtman, L. M., Bashan, A., Berezin, Y., and Havlin, S.. Vulnerability of interdependent networks and networks of networks. In Interconnected Networks, pages 79–99. Springer, 2016.CrossRefGoogle Scholar
Danziger, M. M., Bonamassa, I., Boccaletti, S., and Havlin, S.. Dynamic interdependence and competition in multilayer networks. Nature Physics, 15 (2): 178–185, 2019.Google Scholar
de Arruda, G. F., Cozzo, E., Peixoto, T. P., Rodrigues, F. A., and Moreno, Y.. Disease localization in multilayer networks. Physical Review X, 7 (1): 011014, 2017.Google Scholar
De Domenico, M.. Diffusion geometry unravels the emergence of functional clusters in collective phenomena. Physical Review Letters, 118 (16): 168301, 2017.Google Scholar
De Domenico, M.. Multilayer modeling and analysis of human brain networks. GigaScience, 6 (5): 1–8, 2017.Google Scholar
De Domenico, M.. Multilayer network modeling of integrated biological systems. Comment on “Network science of biological systems at different scales: A review” by Gosak et al. Physics of Life Reviews, 2018.CrossRefGoogle Scholar
De Domenico, M.. Multilayer Networks Illustrated, 2020. http://doi.org/10.17605/OSF.IO/GY53K. Accessed November 25, 2020.Google Scholar
De Domenico, M.. Multilayer networks: Analysis and visualization. Introduction to muxViz with R. Springer-Verlag, 2021.Google Scholar
De Domenico, M. and Biamonte, J.. Spectral entropies as information-theoretic tools for complex network comparison. Physical Review X, 6 (4): 041062, 2016.Google Scholar
De Domenico, M. et al. Complexity explained. OSF, 2019. osf.io/tqgnw. Accessed November 25, 2020.Google Scholar
De Domenico, M., Solé-Ribalta, A., Cozzo, E., et al. Mathematical formulation of multilayer networks. Physical Review X, 3 (4): 041022, 2013.CrossRefGoogle Scholar
De Domenico, M., Solé-Ribalta, A., Gómez, S., and Arenas, A.. Navigability of interconnected networks under random failures. PNAS, 111 (23): 8351–8356, 2014.Google Scholar
De Domenico, M., Lancichinetti, A., Arenas, A., and Rosvall, M.. Identifying modular flows on multilayer networks reveals highly overlapping organization in interconnected systems. Physical Review X, 5 (1): 011027, 2015.Google Scholar
De Domenico, M., Nicosia, V., Arenas, A., and Latora, V.. Structural reducibility of multilayer networks. Nature Communications, 6: 6864, 2015.Google Scholar
De Domenico, M., Porter, M. A., and Arenas, A.. MuxViz: A tool for multilayer analysis and visualization of networks. Journal of Complex Networks, 3 (2): 159–176, 2015.Google Scholar
De Domenico, M., Solé-Ribalta, A., Omodei, E., Gómez, S., and Arenas, A.. Ranking in interconnected multilayer networks reveals versatile nodes. Nature Communications, 6: 6868, 2015.Google Scholar
De Domenico, M., Granell, C., Porter, M. A., and Arenas, A.. The physics of spreading processes in multilayer networks. Nature Physics, 12 (10): 901, 2016.Google Scholar
De Domenico, M., Sasai, S., and Arenas, A.. Mapping multiplex hubs in human functional brain networks. Frontiers in Neuroscience, 10: 326, 2016.Google Scholar
Del Genio, C. I., J. Gómez-Gardeñes, Bonamassa, I., and Boccaletti, S.. Synchronization in networks with multiple interaction layers. Science Advances, 2 (11): 1–10, 2016.Google Scholar
del Rio-Chanona, R. M., Korniyenko, Y., Patnam, M., and Porter, M. A.. The multiplex nature of global financial contagions. Applied Network Science, 5 (1): 1–23, 2020.Google Scholar
Della Rossa, F., Pecora, L., Blaha, K., et al. Symmetries and cluster synchronization in multilayer networks. Nature Communications, 11 (1): 1–17, 2020.CrossRefGoogle ScholarPubMed
Diakonova, M., Nicosia, V., Latora, V., and San Miguel, M.. Irreducibility of multilayer network dynamics: The case of the voter model. New Journal of Physics, 18 (2): 023010, 2016.Google Scholar
Dickison, M., Havlin, S., and Stanley, H. E.. Epidemics on interconnected networks. Physical Review E, 85 (6): 066109, 2012.Google Scholar
Dorogovtsev, S. N., Goltsev, A. V., and Mendes, J. F. F.. Critical phenomena in complex networks. Reviews of Modern Physics, 80 (4): 1275, 2008.Google Scholar
Duh, M., Gosak, M., Slavinec, M., and Perc, M.. Assortativity provides a narrow margin for enhanced cooperation on multilayer networks. New Journal of Physics, 21: 123016, 2019.Google Scholar
Edler, D., Bohlin, L., and Rosvall, M.. Mapping higher-order network flows in memory and multilayer networks with infomap. Algorithms, 10 (4): 112, 2017.CrossRefGoogle Scholar
Esquivel, A. V. and Rosvall, M.. Compression of flow can reveal overlapping-module organization in networks. Physical Review X, 1 (2): 021025, 2011.Google Scholar
Estrada, E.. The structure of complex networks: Theory and applications. Oxford University Press, 2012.Google Scholar
Estrada, E.. Communicability geometry of multiplexes. New Journal of Physics, 21 (1): 015004, 2019.CrossRefGoogle Scholar
Estrada, E. and Gómez-Gardeñes, J.. Communicability reveals a transition to coordinated behavior in multiplex networks. Physical Review E, 89 (4): 042819, 2014.Google Scholar
EUROCONTROL. Ash-cloud of April and May 2010: Impact on air traffic, 2010. https://www.eurocontrol.int/publication/ash-cloud-april-and-may-2010-impact-air-traffic. Accessed March 17, 2020.Google Scholar
Fortunato, S.. Community detection in graphs. Physics Reports, 486 (3–5): 75–174, 2010.Google Scholar
Fortunato, S. and Barthelemy, M.. Resolution limit in community detection. PNAS, 104 (1): 36–41, 2007.Google Scholar
Fortunato, S. and Hric, D.. Community detection in networks: A user guide. Physics Reports, 659: 1–44, 2016.Google Scholar
Freeman, L. C., Borgatti, S. P., and White, D. R.. Centrality in valued graphs: A measure of betweenness based on network flow. Social Networks, 13 (2): 141–154, 1991.Google Scholar
Funk, S., Gilad, E., Watkins, C., and Jansen, V. A. A.. The spread of awareness and its impact on epidemic outbreaks. PNAS, 106 (16): 6872–6877, 2009.Google Scholar
Funk, S., Bansal, S., Bauch, C. T., et al. Nine challenges in incorporating the dynamics of behaviour in infectious diseases models. Epidemics, 10: 21–25, 2015.Google Scholar
Galimberti, E., Bonchi, F., Gullo, F., and Lanciano, T.. Core decomposition in multilayer networks: Theory, algorithms, and applications. ACM Transactions on Knowledge Discovery from Data (TKDD), 14 (1): 1–40, 2020.Google Scholar
Gallotti, R. and Barthelemy, M.. Anatomy and efficiency of urban multimodal mobility. Scientific Reports, 4 (1): 1–9, 2014.Google Scholar
Gallotti, R. and Barthelemy, M.. The multilayer temporal network of public transport in Great Britain. Scientific Data, 2 (1): 1–8, 2015.Google Scholar
Gallotti, R., Bertagnolli, G., and De Domenico, M.. Unraveling the hidden organisation of urban systems and their mobility flows. EPJ Data Science, 10 (1): 1–17, 2021.Google Scholar
Gambuzza, L. V., Frasca, M., and Gómez-Gardeñes, J.. Intra-layer synchronization in multiplex networks. Europhysics Letters, 110 (2): 20010, 2015.Google Scholar
Gao, J., Buldyrev, S. V., Stanley, H. E., and Havlin, S.. Networks formed from interdependent networks. Nature Physics, 8 (1): 40, 2012.Google Scholar
Gauvin, L., Panisson, A., and 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
Ghavasieh, A. and De Domenico, M.. Enhancing transport properties in interconnected systems without altering their structure. Physical Review Research, 2: 013155, 2020.Google Scholar
Ghavasieh, A., Nicolini, C., and De Domenico, M.. Statistical physics of complex information dynamics. Physical Review E, 102: 052304, 2020.Google Scholar
Girvan, M. and Newman, M. E. J.. Community structure in social and biological networks. PNAS, 99 (12): 7821–7826, 2002.CrossRefGoogle ScholarPubMed
Goldenberg, A., Zheng, A. X., Fienberg, S. E., et al. A survey of statistical network models. Foundations and Trends® in Machine Learning, 2 (2): 129–233, 2010.Google Scholar
Goldenfeld, N.. Lectures on phase transitions and the renormalization group. CRC Press, 2018.Google Scholar
Gomez, S., Diaz-Guilera, A., Gomez-Gardenes, J., et al. Diffusion dynamics on multiplex networks. Physical Review Letters, 110 (2): 028701, 2013.Google Scholar
Gómez-Gardeñes, J., Gómez, S., Arenas, A., and Y. Moreno. Explosive synchronization transitions in scale-free networks. Physical Review Letters, 106 (12): 1–6, 2011a.Google Scholar
Gómez-Gardeñes, J., Romance, M., Criado, R., Vilone, D., and Sánchez, A.. Evolutionary games defined at the network mesoscale: The public goods game. Chaos, 21 (1): 1–10, 2011b.Google Scholar
Gómez-Gardenes, J., Reinares, I., Arenas, A., and Floría, L. M.. Evolution of cooperation in multiplex networks. Scientific Reports, 2: 620, 2012.Google Scholar
Gomez-Gardenes, J., de Domenico, M., Gutierrez, G., Arenas, A., and Gomez, S.. Layer-layer competition in multiplex complex networks. Philosophical Transactions of the Royal Society A, 373 (2056): 20150117, 2015.Google Scholar
Granell, C., Gómez, S., and Arenas, A.. Dynamical interplay between awareness and epidemic spreading in multiplex networks. Physical Review Letters, 111 (12), 2013.Google Scholar
Granell, C., Gómez, S., and Arenas, A.. Competing spreading processes on multiplex networks: Awareness and epidemics. Physical Review E, 90 (1): 012808, 2014.Google Scholar
Grassberger, P.. Percolation transitions in the survival of interdependent agents on multiplex networks, catastrophic cascades, and solid-on-solid surface growth. Physical Review E, 91 (6): 062806, 2015.Google Scholar
Guimera, R. and Amaral, L. A. N.. Functional cartography of complex metabolic networks. Nature, 433 (7028): 895, 2005.Google Scholar
Hackett, A., Cellai, D., Gómez, S., Arenas, A., and Gleeson, J. P.. Bond percolation on multiplex networks. Physical Review X, 6 (2): 021002, 2016.Google Scholar
Halu, A., Mondragón, R. J., P. Panzarasa, and G. Bianconi. Multiplex PageRank. PloS One, 8 (10): e78293, 2013.Google Scholar
Holland, P. W., Laskey, K. B., and Leinhardt, S.. Stochastic blockmodels: First steps. Social Networks, 5 (2): 109–137, 1983.Google Scholar
Holme, P.. Modern temporal network theory: A colloquium. European Physical Journal B, 88 (9): 234, 2015.Google Scholar
Holme, P. and Saramäki, J.. Temporal networks. Physics Reports, 519 (3): 97–125, 2012.Google Scholar
Hu, Y., Zhou, D., Zhang, R., et al. Percolation of interdependent networks with intersimilarity. Physical Review E, 88 (5): 052805, 2013.Google Scholar
Hu, Y., Havlin, S., and Makse, H. A.. Conditions for viral influence spreading through multiplex correlated social networks. Physical Review X, 4 (2): 021031, 2014.Google Scholar
Huang, X., Shao, S., Wang, H., et al. The robustness of interdependent clustered networks. Europhysics Letters, 101 (1): 18002, 2013a.Google Scholar
Huang, X., Vodenska, I., Havlin, S., and Stanley, H. E.. Cascading failures in bi-partite graphs: Model for systemic risk propagation. Scientific Reports, 3: 1219, 2013b.Google Scholar
Iacovacci, J., Rahmede, C., Arenas, A., and Bianconi, G.. Functional multiplex PageRank. Europhysics Letters, 116 (2): 28004, 2016.Google Scholar
Jalan, S. and Singh, A.. Cluster synchronization in multiplex networks. Europhysics Letters, 113 (3): 2–7, 2016.Google Scholar
Jang, S., Lee, J. S., Hwang, S., and Kahng, B.. Ashkin-Teller model and diverse opinion phase transitions on multiplex networks. Physical Review E, 92 (2): 022110, 2015.Google Scholar
Jensen, H. J.. Self-organized criticality: Emergent complex behavior in physical and biological systems, volume 10. Cambridge University Press, 1998.Google Scholar
Katz, L.. A new status index derived from sociometric analysis. Psychometrika, 18 (1): 39–43, 1953.Google Scholar
Kempe, D., Kleinberg, J., and Tardos, É. Maximizing the spread of influence through a social network. In Proceedings of the 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 137–146, 2003.Google Scholar
Kenett, D. Y., Gao, J., Huang, X., et al. Network of interdependent networks: Overview of theory and applications. In Networks of Networks: The Last Frontier of Complexity, pages 3–36. Springer, 2014.Google Scholar
Kivelä, M., Arenas, A., Barthelemy, M., et al. Multilayer networks. Journal of Complex Networks, 2 (3): 203–271, 2014.Google Scholar
Kleinberg, J. M.. Authoritative sources in a hyperlinked environment. Journal of the ACM, 46 (5): 604–632, 1999.Google Scholar
Kleineberg, K. K. and Helbing, D.. Topological enslavement in evolutionary games on correlated multiplex networks. New Journal of Physics, 20 (5), 2018.Google Scholar
Kolda, T. G. and Bader, B. W.. Tensor decompositions and applications. SIAM Review, 51 (3): 455–500, 2009.Google Scholar
Kouvaris, N. E., Hata, S., and Díaz-Guilera, A.. Pattern formation in multiplex networks. Scientific Reports, 5 (1): 1–9, 2015.Google Scholar
Kryven, I.. Bond percolation in coloured and multiplex networks. Nature Communications, 10 (1): 1–16, 2019.Google Scholar
Lacasa, L., Mariño, I. P., Miguez, J., et al. Multiplex decomposition of non-Markovian dynamics and the hidden layer reconstruction problem. Physical Review X, 8 (3): 031038, 2018.Google Scholar
Lacasa, L., Stramaglia, S., and Marinazzo, D.. Beyond pairwise network similarity: Exploring mediation and suppression between networks. Communications Physics, 4 (1): 1–8, 2021.Google Scholar
Lambiotte, R., Delvenne, J.-C., and Barahona, M.. Random walks, Markov processes and the multiscale modular organization of complex networks. IEEE Transactions on Network Science and Engineering, 1 (2): 76–90, 2014.Google Scholar
Lambiotte, R., Rosvall, M., and Scholtes, I.. From networks to optimal higher-order models of complex systems. Nature Physics, 15 (4): 313–320, 2019.Google Scholar
Latora, V. and Marchiori, M.. Efficient behavior of small-world networks. Physical Review Letters, 87 (19): 198701, 2001.Google Scholar
Latora, V. and Marchiori, M.. Economic small-world behavior in weighted networks. European Physical Journal B, 32 (2): 249–263, 2003.Google Scholar
Latora, V., Nicosia, V., and Russo, G.. Complex networks: Principles, methods and applications. Cambridge University Press, 2017.Google Scholar
Lee, K.-M., Goh, K.-I., and Kim, I.-M.. Sandpiles on multiplex networks. Journal of the Korean Physical Society, 60 (4): 641–647, 2012 a.Google Scholar
Lee, K.-M., Kim, J. Y., Cho, W.-k., Goh, K.-I., and Kim., I. M. Correlated multiplexity and connectivity of multiplex random networks. New Journal of Physics, 14 (3): 033027, 2012 b.Google Scholar
Lee, K.-M., Min, B., and Goh, K.-I.. Towards real-world complexity: An introduction to multiplex networks. European Physical Journal B, 88 (2): 48, 2015.Google Scholar
Leicht, E. A. and D’Souza, R. M.. Percolation on interacting networks. arXiv:0907.0894, 2009.Google Scholar
Leyva, I., Navas, A., Sendiña-Nadal, I., et al. Explosive transitions to synchronization in networks of phase oscillators. Scientific Reports, 3: 1–5, 2013.Google Scholar
Leyva, I., Sendiña-Nadal, I., Sevilla-Escoboza, R., et al. Relay synchronization in multiplex networks. Scientific Reports, 8, 2018.CrossRefGoogle ScholarPubMed
Li, W., Bashan, A., Buldyrev, S. V., Stanley, H. E., and Havlin, S.. Cascading failures in interdependent lattice networks: The critical role of the length of dependency links. Physical Review Letters, 108 (22): 228702, 2012.CrossRefGoogle ScholarPubMed
Lima, A., De Domenico, M., Pejovic, V., and Musolesi, M.. Exploiting cellular data for disease containment and information campaigns strategies in country-wide epidemics. In Proc. of 3rd Intern. Conf. on the Analysis of Mobile Phone Datasets, Boston, USA, page 1. NETMOB, 2013.Google Scholar
Lima, A., De Domenico, M., Pejovic, V., and Musolesi, M.. Disease containment strategies based on mobility and information dissemination. Scientific Reports, 5: 10650, 2015.Google Scholar
Louf, R. and Barthelemy, M.. Patterns of residential segregation. PloS One, 11 (6): e0157476, 2016.Google Scholar
Magnani, M., Micenkova, B., and Rossi, L.. Combinatorial analysis of multiple networks. arXiv:1303.4986, 2013.Google Scholar
Mantel, N.. The detection of disease clustering and a generalized regression approach. Cancer Research, 27 (2 Part 1): 209–220, 1967.Google Scholar
Martens, E. A., Barreto, E., Strogatz, S. H., et al. Exact results for the Kuramoto model with a bimodal frequency distribution. Physical Review E, 79 (2): 026204, 2009.Google Scholar
Massaro, E. and Bagnoli, F.. Epidemic spreading and risk perception in multiplex networks: A self-organized percolation method. Physical Review E, 90 (5): 052817, 2014.Google Scholar
Masuda, N., Porter, M. A., and Lambiotte., R. Random walks and diffusion on networks. Physics Reports, 2017.Google Scholar
Matamalas, J. T., Poncela-Casasnovas, J., Gómez, S., and Arenas, A.. Strategical incoherence regulates cooperation in social dilemmas on multiplex networks. Scientific Reports, 5: 9519, 2015.Google Scholar
Menichetti, G., Remondini, D., Panzarasa, P., Mondragón, R. J., and Bianconi, G.. Weighted multiplex networks. PloS One, 9 (6): e97857, 2014.Google Scholar
Migliano, A. B., Page, A. E., Gómez-Gardeñes, J., et al. Characterization of hunter-gatherer networks and implications for cumulative culture. Nature Human Behaviour, 1 (2): 1–6, 2017.Google Scholar
Min, B. and Goh, K.-I.. Multiple resource demands and viability in multiplex networks. Physical Review E, 89 (4): 040802, 2014.Google Scholar
Min, B., Do Yi, S., Lee, K.-M., and Goh, K.-I.. Network robustness of multiplex networks with interlayer degree correlations. Physical Review E, 89 (4): 042811, 2014.Google Scholar
Min, B., Lee, S., Lee, K.-M., and Goh, K.-I.. Link overlap, viability, and mutual percolation in multiplex networks. Chaos, Solitons & Fractals, 72: 49–58, 2015.Google Scholar
Molloy, M. and Reed, B.. A critical point for random graphs with a given degree sequence. Random Structures & Algorithms, 6 (2–3): 161–180, 1995.Google Scholar
Morris, R. G. and Barthelemy, M.. Transport on coupled spatial networks. Physical Review Letters, 109 (12): 128703, 2012.Google Scholar
Motter, A. E. and Lai, Y.-C.. Cascade-based attacks on complex networks. Physical Review E, 66 (6): 065102, 2002.Google Scholar
Mucha, P. J., Richardson, T., Macon, K., Porter, M. A., and Onnela, J.-P.. Community structure in time-dependent, multiscale, and multiplex networks. Science, 328 (5980): 876–878, 2010.Google Scholar
Nelson, D. R.. Recent developments in phase transitions and critical phenomena. Nature, 269 (5627): 379–383, 1977.Google Scholar
North American Electric Reliability Council Steering Group. Technical a Analysis of the August 14, 2003, blackout: What happened, why, and what did we learn? Technical report, NERC, 2004. Report to the North American Electric Reliability Council Board of Trustees.Google Scholar
Newman, M. E. J.. Modularity and community structure in networks. PNAS, 103 (23): 8577–8582, 2006.Google Scholar
Newman, M. E. J.. Communities, modules and large-scale structure in networks. Nature Physics, 8 (1): 25, 2012.Google Scholar
Newman, M. E. J., Strogatz, S. H., and Watts, D. J.. Random graphs with arbitrary degree distributions and their applications. Physical Review E, 64 (2): 026118, 2001.Google Scholar
Nicosia, V. and Latora, V.. Measuring and modeling correlations in multiplex networks. Physical Review E, 92 (3): 032805, 2015.Google Scholar
Nicosia, V., Bianconi, G., Latora, V., and Barthelemy, M.. Growing multiplex networks. Physical Review Letters, 111: 058701, 2013a.Google Scholar
Nicosia, V., Valencia, M., Chavez, M., Díaz-Guilera, A., and Latora, V.. Remote synchronization reveals network symmetries and functional modules. Physical Review Letters, 110 (17): 1–5, 2013b.Google Scholar
Nicosia, V., Skardal, P. S., Arenas, A., and Latora, V.. Collective phenomena emerging from the interactions between dynamical processes in multiplex networks. Physical Review Letters, 118 (13): 138302, 2017.Google Scholar
Noh, J. D. and Rieger, H.. Random walks on complex networks. Physical Review Letters, 92 (11): 118701, 2004.Google Scholar
North American Electric Reliability Council. 1996 system disturbances. Review of selected 1996 electric system disturbances in North America. Technical report, North American Electric Reliability Council, 2002.Google Scholar
Nowak, M. A. and May, R. M.. Evolutionary games and spatial chaos. Nature, 359: 826–829, 1992.Google Scholar
Nowak, M. A., Tarnita, C. E., and Antal, T.. Evolutionary dynamics in structured populations. Philosophical Transactions of the Royal Society B, 365 (1537): 19–30, 2010.Google Scholar
Nowicki, K. and Snijders, T. A. B.. Estimation and prediction for stochastic blockstructures. Journal of the American Statistical Association, 96 (455): 1077–1087, 2001.Google Scholar
Osat, S., Faqeeh, A., and Radicchi, F.. Optimal percolation on multiplex networks. Nature Communications, 8 (1): 1540, 2017.Google Scholar
Page, L., Brin, S., Motwani, R., and Winograd, T.. The PageRank citation ranking: Bringing order to the Web. Technical report, Stanford InfoLab, 1999.Google Scholar
Pamfil, A. R., Howison, S. D., Lambiotte, R., and Porter, M. A.. Relating modularity maximization and stochastic block models in multilayer networks. SIAM Journal on Mathematics of Data Science, 1 (4): 667–698, 2019.Google Scholar
Pamfil, A. R., Howison, S. D., and Porter, M. A.. Inference of edge correlations in multilayer networks. Physical Review E, 102 (6): 062307, 2020.Google Scholar
Parshani, R., Buldyrev, S. V., and Havlin, S.. Interdependent networks: Reducing the coupling strength leads to a change from a first to second order percolation transition. Physical Review Letters, 105 (4): 048701, 2010.Google Scholar
Pecora, L. M. and Carroll, T. L.. Master stability functions for synchronized coupled systems. Physical Review Letters, 80 (10): 2109–2112, 1998.Google Scholar
Pecora, L. M., Sorrentino, F., Hagerstrom, A. M., Murphy, T. E., and Roy, R.. Cluster synchronization and isolated desynchronization in complex networks with symmetries. Nature Communications, 5 (May), 2014.Google Scholar
Peixoto, T. P.. Inferring the mesoscale structure of layered, edge-valued, and time-varying networks. Physical Review E, 92 (4): 042807, 2015.Google Scholar
Peixoto, T. P.. Nonparametric Bayesian inference of the microcanonical stochastic block model. Physical Review E, 95 (1): 012317, 2017.Google Scholar
Peixoto, T. P.. Bayesian stochastic blockmodeling. In Advances in network clustering and blockmodeling, Wiley, pages 289–332. 2019.Google Scholar
Perc, M., Jordan, J. J., Rand, D. G., et al. Statistical physics of human cooperation. Physics Reports, 687: 1–51, 2017.Google Scholar
Pilosof, S., Porter, M. A., Pascual, M., and Kéfi, S.. The multilayer nature of ecological networks. Nature Ecology & Evolution, 1 (4): 0101, 2017.Google Scholar
Pons, P. and Latapy, M.. Computing communities in large networks using random walks. Journal of Graph Algorithms and Applications, 10 (2): 191–218, 2006.Google Scholar
Pósfai, M., Gao, J., Cornelius, S. P., Barabási, A.-L., and D’Souza, R. M.. Controllability of multiplex, multi-time-scale networks. Physical Review E, 94 (3): 032316, 2016.Google Scholar
Pósfai, M., Braun, N., Beisner, B. A., McCowan, B., and D’Souza, R. M.. Consensus ranking for multi-objective interventions in multiplex networks. New Journal of Physics, 21 (5): 055001, 2019.Google Scholar
Qin, T. and Rohe, K.. Regularized spectral clustering under the degree-corrected stochastic blockmodel. In Advances in neural information processing systems, vol. 2, pages 3120–3128, Curran Associates, 2013.Google Scholar
Radicchi, F.. Percolation in real interdependent networks. Nature Physics, 11 (7): 597–602, 2015.Google Scholar
Radicchi, F. and Arenas, A.. Abrupt transition in the structural formation of interconnected networks. Nature Physics, 9 (11): 717, 2013.Google Scholar
Radicchi, F. and Bianconi, G.. Redundant interdependencies boost the robustness of multiplex networks. Physical Review X, 7: 011013, 2017.Google Scholar
Ramezanian, R., Magnani, M., Salehi, M., and Montesi, D.. Diffusion of innovations over multiplex social networks. In International Symposium on Artificial Intelligence and Signal Processing (AISP), pages 300–304. Institute of Electrical and Electronics Engineers, 2015.Google Scholar
Reis, S. D. S., Hu, Y., Babino, A., et al. Avoiding catastrophic failure in correlated networks of networks. Nature Physics, 10 (10): 762, 2014.Google Scholar
Requejo, R. J. and Díaz-Guilera, A.. Replicator dynamics with diffusion on multiplex networks. Physical Review E, 94 (2): 022301, 2016.Google Scholar
Rosato, V., Issacharoff, L., Tiriticco, F., et al. Modelling interdependent infrastructures using interacting dynamical models. International Journal of Critical Infrastructures, 4 (1–2): 63–79, 2008.Google Scholar
Rosvall, M. and Bergstrom, C. T.. An information-theoretic framework for resolving community structure in complex networks. PNAS, 104 (18): 7327–7331, 2007.Google Scholar
Rosvall, M. and Bergstrom, C. T.. Maps of random walks on complex networks reveal community structure. PNAS, 105 (4): 1118–1123, 2008.Google Scholar
Rosvall, M., Esquivel, A. V., Lancichinetti, A., West, J. D., and Lambiotte, R.. Memory in network flows and its effects on spreading dynamics and community detection. Nature Communications, 5: 4630, 2014.Google Scholar
Rubinov, M. and Sporns, O.. Complex network measures of brain connectivity: Uses and interpretations. Neuroimage, 52 (3): 1059–1069, 2010.Google Scholar
Saa, A.. Symmetries and synchronization in multilayer random networks. Physical Review E, 97 (4): 042304, 2018.Google Scholar
Sahneh, F. D., Scoglio, C., and Van Mieghem, P.. Generalized epidemic mean-field model for spreading processes over multilayer complex networks. IEEE/ACM Transactions on Networking (TON), 21 (5): 1609–1620, 2013.Google Scholar
Salehi, M., Sharma, R., Marzolla, M., et al. Spreading processes in multilayer networks. IEEE Transactions on Network Science and Engineering, 2 (2): 65–83, 2015.Google Scholar
Salnikov, V., Schaub, M. T., and Lambiotte, R.. Using higher-order Markov models to reveal flow-based communities in networks. Scientific Reports, 6: 23194, 2016.Google Scholar
Santoro, A. and Nicosia, V.. Optimal percolation in correlated multilayer networks with overlap. Physical Review Research, 2 (3): 033122, 2020.Google Scholar
Santoro, A., Latora, V., Nicosia, G., and Nicosia, V.. Pareto optimality in multilayer network growth. Physical Review Letters, 121 (12): 128302, 2018.Google Scholar
Santos, F. C., Pacheco, J. M., and Lenaerts, T.. Evolutionary dynamics of social dilemmas in structured heterogeneous populations. PNAS, 103 (9): 3490–3494, 2006.Google Scholar
Santos, F. C., Santos, M. D., and Pacheco, J. M.. Social diversity promotes the emergence of cooperation in public goods games. Nature, 454 (7201): 213–216, 2008.Google Scholar
Sanz, J., Xia, C.-Y., Meloni, S., and Moreno, Y.. Dynamics of interacting diseases. Physical Review X, 4 (4): 041005, 2014.Google Scholar
Schneider, C. M., Yazdani, N., Araújo, N. A. M., Havlin, S., and Herrmann, H. J.. Towards designing robust coupled networks. Scientific Reports, 3: 1969, 2013.Google Scholar
Scott, J.. Popularity, mediation and exclusion. In Social network analysis, pages 95–112, Sage, 2017.Google Scholar
Seidman, S. B.. Network structure and minimum degree. Social Networks, 5 (3): 269–287, 1983.Google Scholar
Shekhtman, L. M., Danziger, M. M., and Havlin, S.. Recent advances on failure and recovery in networks of networks. Chaos, Solitons & Fractals, 90: 28–36, 2016.Google Scholar
Singh, A., Ghosh, S., Jalan, S., and Kurths, J.. Synchronization in delayed multiplex networks. Europhysics Letters, 111 (3): 30010, 2015.Google Scholar
Skardal, P. S. and Arenas, A.. Control of coupled oscillator networks with application to microgrid technologies. Science Advances, 1 (7): e1500339, 2015.Google Scholar
Snijders, T. A. B. and Nowicki, K.. Estimation and prediction for stochastic blockmodels for graphs with latent block structure. Journal of Classification, 14 (1): 75–100, 1997.Google Scholar
Solá, L., Romance, M., Criado, R., et al. Eigenvector centrality of nodes in multiplex networks. Chaos, 23 (3): 033131, 2013.Google Scholar
Sole-Ribalta, A., De Domenico, M., Kouvaris, N. E., et al. Spectral properties of the Laplacian of multiplex networks. Physical Review E, 88 (3): 032807, 2013.Google Scholar
Solé-Ribalta, A., De Domenico, M., Gómez, S., and Arenas, A.. Centrality rankings in multiplex networks. In Proceedings of the 2014 ACM Conference on Web Science, pages 149–155. Association for Computing Machinery, 2014.Google Scholar
Solé-Ribalta, A., De Domenico, M., Gómez, S., and Arenas, A.. Random walk centrality in interconnected multilayer networks. Physica D, 323: 73–79, 2016.Google Scholar
Solé-Ribalta, A., Gómez, S., and Arenas, A.. Congestion induced by the structure of multiplex networks. Physical Review Letters, 116 (10): 108701, 2016.Google Scholar
Son, S.-W., Grassberger, P., and Paczuski, M.. Percolation transitions are not always sharpened by making networks interdependent. Physical Review Letters, 107 (19): 195702, 2011.Google Scholar
Son, S.-W., Bizhani, G., Christensen, C., Grassberger, P., and Paczuski, M.. Percolation theory on interdependent networks based on epidemic spreading. Europhysics Letters, 97 (1): 16006, 2012.Google Scholar
Soriano-Paños, D., Lotero, L., Arenas, A., and Gómez-Gardeñes, J.. Spreading processes in multiplex metapopulations containing different Mobility networks. Physical Review X, 8 (3): 031039, 2018.Google Scholar
Sorrentino, F., Pecora, L. M., Hagerstrom, A. M., Murphy, T. E., and R. Roy. Complete characterization of the stability of cluster synchronization in complex dynamical networks. Science Advances, 2 (4): 1–9, 2016.Google Scholar
Sporns, O.. Network attributes for segregation and integration in the human brain. Current Opinion in Neurobiology, 23 (2): 162–171, 2013.Google Scholar
Stanley, H. E.. Scaling, universality, and renormalization: Three pillars of modern critical phenomena. Reviews of Modern Physics, 71 (2): S358, 1999.Google Scholar
Stella, M., Beckage, N. M., Brede, M., and De Domenico, M.. Multiplex model of mental lexicon reveals explosive learning in humans. Scientific Reports, 8 (1): 2259, 2018.Google Scholar
Strogatz, S. H.. From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators. Physica D, 143 (1–4): 1–20, 2000.Google Scholar
Tan, F., Wu, J., Xia, Y., and Chi, K. T.. Traffic congestion in interconnected complex networks. Physical Review E, 89 (6): 062813, 2014.Google Scholar
Taylor, D., Shai, S., Stanley, N., and Mucha, P. J.. Enhanced detectability of community structure in multilayer networks through layer aggregation. Physical Review Letters, 116 (22): 228301, 2016.Google Scholar
Taylor, D., Caceres, R. S., and Mucha, P. J.. Super-resolution community detection for layer-aggregated multilayer networks. Physical Review X, 7 (3): 031056, 2017.Google Scholar
Taylor, D., Porter, M. A., and Mucha, P. J.. Tunable eigenvector-based centralities for multiplex and temporal networks. Multiscale Modeling & Simulation, 19 (1): 113–147, 2021.Google Scholar
Tejedor, A., Longjas, A., Foufoula-Georgiou, E., Georgiou, T. T., and Moreno., Y. Diffusion dynamics and optimal coupling in multiplex networks with directed layers. Physical Review X, 8 (3): 031071, 2018.Google Scholar
Tewarie, P., Hillebrand, A., van Dijk, B. W., et al. Integrating cross-frequency and within band functional networks in resting-state meg: A multi-layer network approach. Neuroimage, 142: 324–336, 2016.Google Scholar
Timóteo, S., Correia, M., Rodríguez-Echeverría, S., Freitas, H., and Heleno, R.. Multilayer networks reveal the spatial structure of seed-dispersal interactions across the great rift landscapes. Nature Communications, 9 (1): 140, 2018.Google Scholar
Torreggiani, S., Mangioni, G., Puma, M. J., and G. Fagiolo. Identifying the community structure of the food-trade international multi-network. Environmental Research Letters, 13 (5): 054026, 2018.Google Scholar
Traag, V. A.. Complex contagion of campaign donations. PloS One, 11 (4): e0153539, 2016.Google Scholar
Trewavas, A.. A brief history of systems biology: “Every object that biology studies is a system of systems.” Francois Jacob (1974). The Plant Cell, 18 (10): 2420–2430, 2006.Google Scholar
Tucker, L. R.. Some mathematical notes on three-mode factor analysis. Psychometrika, 31 (3): 279–311, 1966.Google Scholar
Valdano, E., Ferreri, L., Poletto, C., and Colizza, V.. Analytical computation of the epidemic threshold on temporal networks. Physical Review X, 5 (2): 021005, 2015.Google Scholar
Valdeolivas, A., Tichit, L., Navarro, C., et al. Random walk with restart on multiplex and heterogeneous biological networks. Bioinformatics, 2018.Google Scholar
Valdez, L. D., Shekhtman, L., La Rocca, C. E., et al. Cascading failures in complex networks. Journal of Complex Networks, 8 (2): cnaa013, 2020.Google Scholar
Valles-Catala, T., Massucci, F. A., Guimera, R., and Sales-Pardo, M.. Multilayer stochastic block models reveal the multilayer structure of complex networks. Physical Review X, 6 (1): 011036, 2016.Google Scholar
Velásquez-Rojas, F.. Interacting opinion and disease dynamics in multiplex networks: Discontinuous phase transition and nonmonotonic consensus times. Physical Review E, 95 (5): 052315, 2017.Google Scholar
Vermeulen, R., Schymanski, E. L., Barabási, A.-L., and Miller, G. W.. The exposome and health: Where chemistry meets biology. Science, 367 (6476): 392–396, 2020.Google Scholar
Verstraete, N., Jurman, G., Bertagnolli, G., et al. CovMulNet19, integrating proteins, diseases, drugs, and symptoms: A network medicine approach to COVID-19. Network and Systems Medicine, 3 (1): 130–141, 2020.Google Scholar
Vespignani, A.. Complex networks: The fragility of interdependency. Nature, 464 (7291): 984, 2010.Google Scholar
Voitalov, I., van der Hoorn, P., Kitsak, M., Papadopoulos, F., and Krioukov, D.. Weighted hypersoft configuration model. Physical Review Research, 2: 043157, 2020.Google Scholar
Wang, H., Li, Q., D’Agostino, G., et al. Effect of the interconnected network structure on the epidemic threshold. Physical Review E, 88 (2): 022801, 2013.Google Scholar
Wang, X., Li, W., Liu, L., et al. Promoting information diffusion through interlayer recovery processes in multiplex networks. Physical Review E, 96 (3): 032304, 2017.Google Scholar
Wang, Z., Wang, L., and Perc, M.. Degree mixing in multilayer networks impedes the evolution of cooperation. Physical Review E, 89: 052813, 2014.Google Scholar
Wang, Z., Andrews, M. A., Wu, Z.-X., Wang, L., and Bauch, C. T.. Coupled disease–behavior dynamics on complex networks: A review. Physics of Life Reviews, 15: 1–29, 2015a.Google Scholar
Wang, Z., Wang, L., Szolnoki, A., and Perc, M.. Evolutionary games on multilayer networks: A colloquium. European Physical Journal B, 88 (5): 1–15, 2015b.Google Scholar
Watts, D. J.. A simple model of global cascades on random networks. PNAS, 99 (9): 5766–5771, 2002.Google Scholar
Watts, D. J. and Strogatz, S. H.. Collective dynamics of small-world networks. Nature, 393 (6684): 440, 1998.Google Scholar
Williamson, B. J., De Domenico, M., and Kadis, D. S.. Multilayer connector hub mapping reveals key brain regions supporting expressive language. Brain Connectivity, 11 (1): 45–55, 2021.Google Scholar
Wu, H., James, R. G., and D’Souza, R. M.. Correlated structural evolution within multiplex networks. Journal of Complex Networks, 8 (2): cnaa014, 2020.Google Scholar
Wu, Q., Fu, X., Small, M., and Xu, X.-J.. The impact of awareness on epidemic spreading in networks. Chaos, 22 (1): 013101, 2012.Google Scholar
Yagan, O. and Gligor, V.. Analysis of complex contagions in random multiplex networks. Physical Review E, 86 (3): 036103, 2012.Google Scholar
Yamamoto, H., Moriya, S., Ide, K., et al. Impact of modular organization on dynamical richness in cortical networks. Science Advances, 4 (11): eaau4914, 2018.Google Scholar
Yuan, Z., Zhao, C., Wang, W.-X., Di, Z., and Lai, Y.-C.. Exact controllability of multiplex networks. New Journal of Physics, 16 (10): 103036, 2014.Google Scholar
Zachary, W. W.. An information flow model for conflict and fission in small groups. Journal of Anthropological Research, 33 (4): 452–473, 1977.Google Scholar
Zhang, X., Boccaletti, S., Guan, S., and Liu, Z.. Explosive synchronization in adaptive and multilayer networks. Physical Review Letters, 114 (3): 1–5, 2015.Google Scholar
Zhang, Y., Latora, V., and Motter, A. E.. Unified treatment of dynamical processes on generalized networks: Higher-order, multilayer, and temporal interactions. arXiv:2010.00613, 2020.Google Scholar
Zhao, D.-W., Wang, L.-H., Zhi, Y.-F., Zhang, J., and Wang, Z.. The robustness of multiplex networks under layer node-based attack. Scientific Reports, 6: 24304, 2016.Google Scholar
Zhao, K. and Bianconi, G.. Percolation on interacting, antagonistic networks. Journal of Statistical Mechanics, 2013 (05): P05005, 2013.Google Scholar
Artime, O. and De Domenico, M.. From the origin of life to pandemics: Emergent phenomena in complex systems. Philosophical Transactions of the Royal Society A, 380 (2227): 20200410, 2022.Google Scholar
Accessibility standard: Unknown
Why this information is here
This section outlines the accessibility features of this content - including support for screen readers, full keyboard navigation and high-contrast display options. This may not be relevant for you.Accessibility Information
Accessibility compliance for the HTML of this element is currently unknown
and may be updated in the future.
- 29
- Cited by