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Series:   Elements in the Structure and Dynamics of Complex Networks

Higher-Order Networks

Published online by Cambridge University Press:  23 November 2021

Ginestra Bianconi
Ginestra Bianconi
Affiliation:
Queen Mary University of London

Summary

Higher-order networks describe the many-body interactions of a large variety of complex systems, ranging from the the brain to collaboration networks. Simplicial complexes are generalized network structures which allow us to capture the combinatorial properties, the topology and the geometry of higher-order networks. Having been used extensively in quantum gravity to describe discrete or discretized space-time, simplicial complexes have only recently started becoming the representation of choice for capturing the underlying network topology and geometry of complex systems. This Element provides an in-depth introduction to the very hot topic of network theory, covering a wide range of subjects ranging from emergent hyperbolic geometry and topological data analysis to higher-order dynamics. This Elements aims to demonstrate that simplicial complexes provide a very general mathematical framework to reveal how higher-order dynamics depends on simplicial network topology and geometry.
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Keywords

Higher-order networksSimplicial complexesHigher-order dynamicsSimplicial network topology and geometryTopological data analysis
Type
Element
Information
Series: Elements in the Structure and Dynamics of Complex Networks
Online ISBN: 9781108770996
Publisher: Cambridge University Press
Print publication: 23 December 2021

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References

Barabási, A.-L., Network Science. Cambridge University Press, 2016.Google Scholar
Dorogovtsev, S. N. and Mendes, J. F., Evolution of Networks: From Biological Nets to the Internet and WWW. Oxford University Press, 2003.Google Scholar
Barrat, A., Barthelemy, M. and Vespignani, A., Dynamical Processes on Complex Networks. Cambridge University Press, 2008.CrossRefGoogle Scholar
Menczer, F., Fortunato, S. and Davis, C. A., A First Course in Network Science. Cambridge University Press, 2020.Google Scholar
Dorogovtsev, S. N., Goltsev, A. V. and Mendes, J. F., “Critical phenomena in complex networks,” Reviews of Modern Physics, vol. 80, no. 4, p. 1275, 2008.Google Scholar
Bianconi, G., Multilayer Networks: Structure and Function. Oxford University Press, 2018.Google Scholar
Battiston, F., Cencetti, G., Iacopini, I., et al., “Networks beyond pairwise interactions: structure and dynamics,” Physics Reports, vol. 874, pp. 1–92, 2020.CrossRefGoogle Scholar
Giusti, C., Ghrist, R. and Bassett, D. S., “Two’s company, three (or more) is a simplex,” Journal of Computational Neuroscience, vol. 41, no. 1, pp. 1–14, 2016.CrossRefGoogle ScholarPubMed
Bianconi, G., “Interdisciplinary and physics challenges of network theory,” EPL (Europhysics Letters), vol. 111, no. 5, p. 56001, 2015.Google Scholar
Torres, L., Blevins, A. S., Bassett, D. S. and Eliassi-Rad, T., “The why, how, and when of representations for complex systems,” arXiv preprint arXiv:2006.02870, 2020.Google Scholar
Salnikov, V., Cassese, D. and Lambiotte, R., “Simplicial complexes and complex systems,” European Journal of Physics, vol. 40, no. 1, p. 014001, 2018.Google Scholar
Courtney, O. T. and Bianconi, G., “Generalized network structures: the configuration model and the canonical ensemble of simplicial complexes,” Physical Review E, vol. 93, no. 6, p. 062311, 2016.CrossRefGoogle ScholarPubMed
Skardal, P. and Arenas, A., “Abrupt desynchronization and extensive multistability in globally coupled oscillator simplexes,” Physical Review Letters, vol. 122, no. 24, p. 248301, 2019.Google Scholar
Iacopini, I., Petri, G., Barrat, A. and Latora, V., “Simplicial models of social contagion,” Nature Communications, vol. 10, no. 1, pp. 1–9, 2019.CrossRefGoogle ScholarPubMed
Ghrist, R. W., Elementary Applied Topology. Createspace Seattle, 2014, vol. 1.Google Scholar
Ghrist, R., “Barcodes: the persistent topology of data,” Bulletin of the American Mathematical Society, vol. 45, no. 1, pp. 61–75, 2008.Google Scholar
Kahle, M., “Topology of random simplicial complexes: a survey,” AMS Contemporary Mathematics, vol. 620, pp. 201–222, 2014.CrossRefGoogle Scholar
Otter, N., Porter, M. A., Tillmann, U., Grindrod, P. and Harrington, H. A., “A roadmap for the computation of persistent homology,” EPJ Data Science, vol. 6, no. 1, p. 17, 2017.Google Scholar
Edelsbrunner, H., A Short Course in Computational Geometry and Topology. Springer, 2014.Google Scholar
Petri, G., Expert, P., Turkheimer, F., et al., “Homological scaffolds of brain functional networks,” Journal of The Royal Society Interface, vol. 11, no. 101, p. 20140873, 2014.CrossRefGoogle ScholarPubMed
Petri, G., Scolamiero, M., Donato, I. and Vaccarino, F., “Topological strata of weighted complex networks,” PloS One, vol. 8, no. 6, p. e66506, 2013.CrossRefGoogle ScholarPubMed
Saggar, M., Sporns, O., Gonzalez-Castillo, J., et al. “Towards a new approach to reveal dynamical organization of the brain using topological data analysis,” Nature Communications, vol. 9, no. 1, pp. 1–14, 2018.Google Scholar
Reimann, M. W., Nolte, M., Scolamiero, M., et al., “Cliques of neurons bound into cavities provide a missing link between structure and function,” Frontiers in Computational Neuroscience, vol. 11, p. 48, 2017.Google Scholar
Kartun-Giles, A. P. and Bianconi, G., “Beyond the clustering coefficient: a topological analysis of node neighbourhoods in complex networks,” Chaos, Solitons and Fractals: X, vol. 1, p. 100004, 2019.Google Scholar
Benson, A. R., Gleich, D. F. and J. Leskovec, “Higher-order organization of complex networks,” Science, vol. 353, no. 6295, pp. 163–166, 2016.Google Scholar
Palla, G., Derényi, I., Farkas, I. and Vicsek, T., “Uncovering the overlapping community structure of complex networks in nature and society,” Nature, vol. 435, no. 7043, pp. 814–818, 2005.Google Scholar
Millán, A. P., Torres, J. J. and Bianconi, G., “Explosive higher-order Kuramoto dynamics on simplicial complexes,” Physical Review Letters, vol. 124, no. 21, p. 218301, 2020.CrossRefGoogle ScholarPubMed
Barbarossa, S. and Sardellitti, S., “Topological signal processing over simplicial complexes,” IEEE Transactions on Signal Processing, vol. 68, pp. 2992–3007, 2020.Google Scholar
Bianconi, G. and Rahmede, C., “Network geometry with flavor: from complexity to quantum geometry,” Physical Review E, vol. 93, no. 3, p. 032315, 2016.Google Scholar
Bianconi, G. and Rahmede, C., “Emergent hyperbolic geometry,” Scientific Reports, vol. 7, p. 41974, 2017.Google Scholar
Wu, Z., Menichetti, G., Rahmede, C. and Bianconi, G., “Emergent complex network geometry,” Scientific Reports, vol. 5, p. 10073, 2015.Google Scholar
Mulder, D. and Bianconi, G., “Network geometry and complexity,” Journal of Statistical Physics, vol. 173, no. 3–4, pp. 783–805, 2018.CrossRefGoogle Scholar
Torres, J. J. and Bianconi, G., “Simplicial complexes: higher-order spectral dimension and dynamics,” Journal of Physics: Complexity, vol. 1, no. 1, p. 015002, 2020.Google Scholar
Burioni, R. and Cassi, D., “Random walks on graphs: ideas, techniques and results,” Journal of Physics A: Mathematical and General, vol. 38, no. 8, p. R45, 2005.CrossRefGoogle Scholar
Millán, A. P., Torres, J. J. and Bianconi, G., “Complex network geometry and frustrated synchronization,” Scientific Reports, vol. 8, no. 1, pp. 1–10, 2018.Google Scholar
Millán, A. P., Torres, J. J. and Bianconi, G., “Synchronization in network geometries with finite spectral dimension,” Physical Review E, vol. 99, no. 2, p. 022307, 2019.Google Scholar
Bianconi, G. and Ziff, R. M., “Topological percolation on hyperbolic simplicial complexes,” Physical Review E, vol. 98, no. 5, p. 052308, 2018.CrossRefGoogle Scholar
Cinardi, N., Rapisarda, A. and Bianconi, G., “Quantum statistics in network geometry with fractional flavor,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2019, no. 10, p. 103403, 2019.CrossRefGoogle Scholar
Bianconi, G. and Rahmede, C., “Complex quantum network manifolds in dimension d>2 are scale-free,” Scientific Reports, vol. 5, no. 1, pp. 1–10, 2015.CrossRefGoogle ScholarPubMed
Patania, A., Petri, G. and Vaccarino, F., “The shape of collaborations,” EPJ Data Science, vol. 6, no. 1, p. 18, 2017.Google Scholar
Bollobás, B. and Béla, B., Random Graphs. Cambridge University Press, 2001, no. 73.Google Scholar
Bianconi, G. and Marsili, M., “Emergence of large cliques in random scale-free networks,” EPL (Europhysics Letters), vol. 74, no. 4, p. 740, 2006.Google Scholar
MacKay, D. J. C, Information Theory, Inference and Learning Algorithms. Cambridge University Press, 2003.Google Scholar
Cover, T. M., Elements of Information Theory. John Wiley & Sons, 1999.Google Scholar
Anand, K. and Bianconi, G., “Entropy measures for networks: toward an information theory of complex topologies,” Physical Review E, vol. 80, no. 4, p. 045102, 2009.Google Scholar
Kardar, M., Statistical Physics of Particles. Cambridge University Press, 2007.Google Scholar
Anand, K. and Bianconi, G., “Gibbs entropy of network ensembles by cavity methods,” Physical Review E, vol. 82, no. 1, p. 011116, 2010.Google Scholar
Costa, A. and Farber, M., “Random simplicial complexes,” in Configuration Spaces, Callegaro, F., Cohen, F., De Concini, C., M. Feichtner, E., Gaiffi, G., Salvetti, M. (Eds.). Springer, 2016, pp. 129–153.Google Scholar
Zuev, K., Eisenberg, O. and Krioukov, D., “Exponential random simplicial complexes,” Journal of Physics A: Mathematical and Theoretical, vol. 48, no. 46, p. 465002, 2015.Google Scholar
Repository for higher-order network codes. [Online]. Available: https://github.com/ginestrabGoogle Scholar
Bender, E. A. and Canfield, E. R., “The asymptotic number of labeled graphs with given degree sequences,” Journal of Combinatorial Theory, Series A, vol. 24, no. 3, pp. 296–307, 1978.Google Scholar
Ghoshal, G., Zlatić, V., Caldarelli, G. and Newman, M. E., “Random hypergraphs and their applications,” Physical Review E, vol. 79, no. 6, p. 066118, 2009.CrossRefGoogle ScholarPubMed
Wegner, A. E. and Olhede, S., “Atomic subgraphs and the statistical mechanics of networks,” Physical Review E, vol. 103, no. 04, p. 042311, 2021.Google Scholar
Klimm, F., Deane, C. M. and Reinert, G., “Hypergraphs for predicting essential genes using multiprotein complex data,” bioRxiv, 2020.CrossRefGoogle Scholar
Sun, H. and Bianconi, G., “Higher-order percolation processes on multiplex hypergraphs,” arXiv preprint arXiv:2104.05457, 2021.Google Scholar
Zhao, K., Stehlé, J., Bianconi, G. and Barrat, A., “Social network dynamics of face-to-face interactions,” Physical Review E, vol. 83, no. 5, p. 056109, 2011.Google Scholar
Zhao, K., Karsai, M. and Bianconi, G., “Entropy of dynamical social networks,” PloS One, vol. 6, no. 12, p. e28116, 2011.Google Scholar
Petri, G. and Barrat, A., “Simplicial activity driven model,” Physical Review Letters, vol. 121, no. 22, p. 228301, 2018.CrossRefGoogle ScholarPubMed
Holme, P. and Saramäki, J., “Temporal networks,” Physics Reports, vol. 519, no. 3, pp. 97–125, 2012.Google Scholar
Karsai, M., Kaski, K., Barabási, A.-L. and Kertész, J., “Universal features of correlated bursty behaviour,” Scientific Reports, vol. 2, p. 397, 2012.Google Scholar
Karsai, M., Jo, H.-H., Kaski, K., et al., Bursty Human Dynamics. Springer, 2018.Google Scholar
Cattuto, C., Van den Broeck, W., Barrat, A., et al., “Dynamics of person-to-person interactions from distributed rfid sensor networks,” PloS One, vol. 5, no. 7, p. e11596, 2010.CrossRefGoogle ScholarPubMed
Stehlé, J., Barrat, A. and Bianconi, G., “Dynamical and bursty interactions in social networks,” Physical Review E, vol. 81, no. 3, p. 035101, 2010.Google Scholar
Cencetti, G., Battiston, F., Lepri, B. and Karsai, M., “Temporal properties of higher-order interactions in social networks,” arXiv preprint arXiv:2010.03404, 2020.Google Scholar
Link to SocioPattern project webpage. [Online]. Available: http://www.sociopatterns.org/Google Scholar
Jost, J., Mathematical Concepts. Springer, 2015.Google Scholar
Lee, S. H., Fricker, M. D. and Porter, M. A., “Mesoscale analyses of fungal networks as an approach for quantifying phenotypic traits,” Journal of Complex Networks, Apr 2016. [Online]. Available:Google Scholar
Link to license. [Online]. Available: https://creativecommons.org/licenses/by/4.0/Google Scholar
Watts, D. J. and Strogatz, S. H., “Collective dynamics of ‘small-world’ networks,” Nature, vol. 393, no. 6684, pp. 440–442, 1998.Google Scholar
Patania, A., Vaccarino, F. and Petri, G., “Topological analysis of data,” EPJ Data Science, vol. 6, no. 1, pp. 1–6, 2017.Google Scholar
Bianconi, G., Darst, R. K., Iacovacci, J. and Fortunato, S., “Triadic closure as a basic generating mechanism of communities in complex networks,” Physical Review E, vol. 90, no. 4, p. 042806, 2014.Google Scholar
Ollivier, Y., “Ricci curvature of metric spaces,” Comptes Rendus Mathematique, vol. 345, no. 11, pp. 643–646, 2007.Google Scholar
Bauer, F., Jost, J. and Liu, S., “Ollivier-Ricci curvature and the spectrum of the normalized graph Laplace operator,” arXiv preprint arXiv:1105.3803, 2011.CrossRefGoogle Scholar
Jost, J. and Liu, S., “Ollivier’s Ricci curvature, local clustering and curvature-dimension inequalities on graphs,” Discrete and Computational Geometry, vol. 51, no. 2, pp. 300–322, 2014.CrossRefGoogle Scholar
Sreejith, R., Mohanraj, K., Jost, J., Saucan, E. and Samal, A., “Forman curvature for complex networks,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2016, no. 6, p. 063206, 2016.Google Scholar
Regge, T., “General relativity without coordinates,” Il Nuovo Cimento (1955–1965), vol. 19, no. 3, pp. 558–571, 1961.Google Scholar
Dittrich, B., Freidel, L. and Speziale, S., “Linearized dynamics from the 4-simplex Regge action,” Physical Review D, vol. 76, no. 10, p. 104020, 2007.Google Scholar
Ambjørn, J., Jurkiewicz, J. and Loll, R., “Emergence of a 4d world from causal quantum gravity,” Physical Review Letters, vol. 93, no. 13, p. 131301, 2004.Google Scholar
Gromov, M., “Hyperbolic groups,” in Essays in Group Theory, Gersten, S. M. (Ed.). Springer, 1987, pp. 75–263.Google Scholar
Jonckheere, E., Lohsoonthorn, P. and Bonahon, F., “Scaled Gromov hyperbolic graphs,” Journal of Graph Theory, vol. 57, no. 2, pp. 157–180, 2008.Google Scholar
Albert, R., DasGupta, B. and Mobasheri, N., “Topological implications of negative curvature for biological and social networks,” Physical Review E, vol. 89, no. 3, p. 032811, 2014.CrossRefGoogle ScholarPubMed
Kennedy, W. S., Narayan, O. and Saniee, I., “On the hyperbolicity of large-scale networks,” arXiv preprint arXiv:1307.0031, 2013.Google Scholar
Calcagni, G., Eichhorn, A. and Saueressig, F., “Probing the quantum nature of spacetime by diffusion,” Physical Review D, vol. 87, no. 12, p. 124028, 2013.Google Scholar
Benedetti, D. and Henson, J., “Spectral geometry as a probe of quantum spacetime,” Physical Review D, vol. 80, no. 12, p. 124036, 2009.CrossRefGoogle Scholar
Jonsson, T. and Wheater, J. F., “The spectral dimension of the branched polymer phase of two-dimensional quantum gravity,” Nuclear Physics B, vol. 515, no. 3, pp. 549–574, 1998.Google Scholar
Durhuus, B., Jonsson, T. and Wheater, J. F., “The spectral dimension of generic trees,” Journal of Statistical Physics, vol. 128, no. 5, pp. 1237–1260, 2007.Google Scholar
Burioni, R. and Cassi, D., “Universal properties of spectral dimension,” Physical Review Letters, vol. 76, no. 7, p. 1091–1093, 1996.Google Scholar
Reitz, M. and Bianconi, G., “The higher-order spectrum of simplicial complexes: a renormalization group approach,” Journal of Physics A: Mathematical and Theoretical, vol. 53, p. 295001, 2020.CrossRefGoogle Scholar
Wedeen, V. J., Rosene, D. L., Wang, R., et al., “The geometric structure of the brain fiber pathways,” Science, vol. 335, no. 6076, pp. 1628–1634, 2012.Google Scholar
Katifori, E., Szöllősi, G. J. and Magnasco, M. O., “Damage and fluctuations induce loops in optimal transport networks,” Physical Review Letters, vol. 104, no. 4, p. 048704, 2010.CrossRefGoogle ScholarPubMed
Rocks, J. W., Liu, A. J. and Katifori, E., “Hidden topological structure of flow network functionality,” Physical Review Letters, vol. 126, no. 2, p. 028102, 2021.Google Scholar
Á. Serrano, M., Boguná, M. and Sagués, F., “Uncovering the hidden geometry behind metabolic networks,” Molecular Biosystems, vol. 8, no. 3, pp. 843–850, 2012.Google Scholar
Radicchi, F., Krioukov, D., Hartle, H. and Bianconi, G., “Classical information theory of networks,” Journal of Physics: Complexity, vol. 1, no. 2, p. 025001, 2020.Google Scholar
Penrose, R., “On the nature of quantum geometry,” Magic Without Magic, Klauder, J. R. (Ed.). W. H. Freeman & Co. Ltd. pp. 333–354, 1972.Google Scholar
Smolin, L., The Life of the Cosmos. Oxford University Press, 1999.Google Scholar
Bianconi, G. and Barabási, A.-L., “Bose–Einstein condensation in complex networks,” Physical Review Letters, vol. 86, no. 24, p. 5632, 2001.Google Scholar
Fountoulakis, N., Iyer, T., Mailler, C. and Sulzbach, H., “Dynamical models for random simplicial complexes,” arXiv preprint arXiv:1910.12715, 2019.Google Scholar
Barabási, A.-L. and Albert, R., “Emergence of scaling in random networks,” Science, vol. 286, no. 5439, pp. 509–512, 1999.Google Scholar
Dorogovtsev, S. N., Mendes, J. F. and Samukhin, A. N., “Size-dependent degree distribution of a scale-free growing network,” Physical Review E, vol. 63, no. 6, p. 062101, 2001.CrossRefGoogle ScholarPubMed
Nokkala, J., Piilo, J. and Bianconi, G., “Probing the spectral dimension of quantum network geometries,” Journal of Physics: Complexity, vol. 2, no. 1, p. 015001, 2020.Google Scholar
da Silva, D. C., Bianconi, G., da Costa, R. A., Dorogovtsev, S. N. and Mendes, J. F., “Complex network view of evolving manifolds,” Physical Review E, vol. 97, no. 3, p. 032316, 2018.Google Scholar
Šuvakov, M., Andjelković, M. and Tadić, B., “Hidden geometries in networks arising from cooperative self-assembly,” Scientific Reports, vol. 8, no. 1, pp. 1–10, 2018.Google Scholar
Dankulov, M. M., Tadić, B. and Melnik, R., “Spectral properties of hyperbolic nanonetworks with tunable aggregation of simplexes,” Physical Review E, vol. 100, no. 1, p. 012309, 2019.Google Scholar
Millán, A. P., Ghorbanchian, R., Defenu, N., Battiston, F., and Bianconi, G., “Local topological moves determine global diffusion properties of hyperbolic higher-order networks,” arXiv preprint arXiv:2102.12885, 2021.Google Scholar
Girvan, M. and Newman, M. E., “Community structure in social and biological networks,” Proceedings of the National Academy of Sciences, vol. 99, no. 12, pp. 7821–7826, 2002.Google Scholar
Courtney, O. T. and Bianconi, G., “Weighted growing simplicial complexes,” Physical Review E, vol. 95, no. 6, p. 062301, 2017.Google Scholar
Kovalenko, K., Sendiña-Nadal, I., Khalil, N., et al., “rowing scale-free simplices,” arXiv preprint arXiv:2006.12899, 2020.Google Scholar
Bianconi, G. and Barabási, A.-L., “Competition and multiscaling in evolving networks,” EPL (Europhysics Letters), vol. 54, no. 4, p. 436, 2001.Google Scholar
Bianconi, G., Rahmede, C. and Wu, Z., “Complex quantum network geometries: evolution and phase transitions,” Physical Review E, vol. 92, no. 2, p. 022815, 2015.Google Scholar
Strogatz, S., Sync: The Emerging Science of Spontaneous Order. Penguin UK, 2004.Google Scholar
Kuramoto, Y., “Self-entrainment of a population of coupled non-linear oscillators,” in International Symposium on Mathematical Problems in Theoretical Physics, Araki, H. (Ed.). Springer, 1975, pp. 420–422.Google Scholar
Strogatz, S. H., “From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators,” Physica D: Nonlinear Phenomena, vol. 143, no. 1–4, pp. 1–20, 2000.Google Scholar
Pikovsky, A., Rosenblum, M. and Kurths, J., Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, 2003, no. 12.Google Scholar
Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y. and Zhou, C., “Synchronization in complex networks,” Physics Reports, vol. 469, no. 3, pp. 93–153, 2008.Google Scholar
Boccaletti, S., Pisarchik, A. N., Del Genio, C. I. and Amann, A., Synchronization: from Coupled Systems to Complex Networks. Cambridge University Press, 2018.Google Scholar
Restrepo, J. G., Ott, E. and Hunt, B. R., “Onset of synchronization in large networks of coupled oscillators,” Physical Review E, vol. 71, no. 3, p. 036151, 2005.Google Scholar
Millán, A. P., Restrepo, J. G., Torres, J. J. and Bianconi, G., “Geometry, topology and simplicial synchronization,” arXiv preprint arXiv:2105.00943, 2021.Google Scholar
Kuehn, C. and Bick, C., “A universal route to explosive phenomena,” Science Advances, vol. 7, no. 16, p. eabe3824, 2021.Google Scholar
Severino, F. P. U., Ban, J., Song, Q., et al., “The role of dimensionality in neuronal network dynamics,” Scientific Reports, vol. 6, p. 29640, 2016.CrossRefGoogle Scholar
Gambuzza, L., Di Patti, F., Gallo, L., et al., “The master stability function for synchronization in simplicial complexes,” arXiv preprint arXiv:2004.03913, 2020.Google Scholar
Mulas, R., Kuehn, C. and Jost, J., “Coupled dynamics on hypergraphs: master stability of steady states and synchronization,” Physical Review E, vol. 101, no. 6, p. 062313, 2020.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 preprint arXiv:2010.00613, 2020.Google Scholar
Salova, A. and D’Souza, R. M., “Cluster synchronization on hypergraphs,” arXiv preprint arXiv:2101.05464, 2021.Google Scholar
Ghorbanchian, R., Restrepo, J. G., Torres, J. J. and Bianconi, G., “Higher-order simplicial synchronization of coupled topological signals,” Communications Physics, vol. 4, p. 120, 2021.Google Scholar
Skardal, P. S. and Arenas, A., “Memory selection and information switching in oscillator networks with higher-order interactions,” Journal of Physics: Complexity, vol. 2, no. 1, p. 015003, 2020.Google Scholar
Skardal, P. and Arenas, A., “Higher order interactions in complex networks of phase oscillators promote abrupt synchronization switching,” Communications Physics, vol. 3, no. 1, pp. 1–6, 2020.Google Scholar
Dai, X., Kovalenko, K., Molodyk, M., et al., “D-dimensional oscillators in simplicial structures: odd and even dimensions display different synchronization scenarios,” arXiv preprint arXiv:2010.14976, 2020.Google Scholar
Lucas, M., Cencetti, G. and Battiston, F., “Multiorder Laplacian for synchronization in higher-order networks,” Physical Review Research, vol. 2, no. 3, p. 033410, 2020.Google Scholar
Zhang, X., Boccaletti, S., Guan, S. and Liu, Z., “Explosive synchronization in adaptive and multilayer networks,” Physical Review Letters, vol. 114, no. 3, p. 038701, 2015.CrossRefGoogle ScholarPubMed
DeVille, L., “Consensus on simplicial complexes, or: The nonlinear simplicial Laplacian,” arXiv preprint arXiv:2010.07421, 2020.Google Scholar
Benjamini, I. and Schramm, O., “Percolation in the hyperbolic plane,” Journal of the American Mathematical Society, vol. 14, no. 2, pp. 487–507, 2001.Google Scholar
Boettcher, S., Singh, V. and Ziff, R. M., “Ordinary percolation with discontinuous transitions,” Nature Communications, vol. 3, no. 1, pp. 1–5, 2012.Google Scholar
Kryven, I., Ziff, R. M. and Bianconi, G., “Renormalization group for link percolation on planar hyperbolic manifolds,” Physical Review E, vol. 100, no. 2, p. 022306, 2019.CrossRefGoogle Scholar
Bianconi, G., Kryven, I. and Ziff, R. M., “Percolation on branching simplicial and cell complexes and its relation to interdependent percolation,” Physical Review E, vol. 100, no. 6, p. 062311, 2019.Google Scholar
Sun, H., Ziff, R. M. and Bianconi, G., “Renormalization group theory of percolation on pseudo-fractal simplicial and cell complexes,” Physical Review E, vol. 102, p. 012308, 2020.Google Scholar
Bobrowski, O. and Skraba, P., “Homological percolation and the Euler characteristic,” Physical Review E, vol. 101, no. 3, p. 032304, 2020.Google Scholar
Lee, Y., Lee, J., Oh, S. M., Lee, D. and Kahng, B., “Homological percolation transitions in evolving coauthorship complexes,” arXiv preprint arXiv:2010.12224, 2020.Google Scholar
Coutinho, B. C., Wu, A.-K., Zhou, H.-J. and Liu, Y.-Y., “Covering problems and core percolations on hypergraphs,” Physical Review Letters, vol. 124, no. 24, p. 248301, 2020.Google Scholar
Amburg, I., Kleinberg, J. and Benson, A., “Planted hitting set recovery in hypergraphs,” Journal of Physics: Complexity, vol. 2, p. 035004, 2021.Google Scholar
Auto, D. M., Moreira, A. A., Herrmann, H. J. and Andrade Jr, J. S., “Finite-size effects for percolation on Apollonian networks,” Physical Review E, vol. 78, no. 6, p. 066112, 2008.CrossRefGoogle ScholarPubMed
Watts, D. J., “A simple model of global cascades on random networks,” Proceedings of the National Academy of Science, vol. 99, no. 9, p. 5766, 2002.CrossRefGoogle ScholarPubMed
Granovetter, M., “Threshold models of collective behavior,” American Journal of Sociology, vol. 83, no. 6, p. 1420, 1978.Google Scholar
Landry, N. W. and Restrepo, J. G., “The effect of heterogeneity on hypergraph contagion models,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 30, no. 10, p. 103117, 2020.Google Scholar
St-Onge, G., Sun, H., Allard, A., Hébert-Dufresne, L. and Bianconi, G., “Universal nonlinear infection kernel from heterogeneous exposure on higher-order networks.” arXiv preprint arXiv:2101.07229, 2021.Google Scholar
St-Onge, G., Allard, A. and Hébert-Dufresne, L., “Localization, bistability and optimal seeding of contagions on higher-order networks,” in Artificial Life Conference Proceedings. MIT Press, 2020, pp. 567–569.Google Scholar
St-Onge, G., Thibeault, V., Allard, A., Dubé, L. J. and Hébert-Dufresne, L., “Master equation analysis of mesoscopic localization in contagion dynamics on higher-order networks,” arXiv preprint arXiv:2004.10203, 2020.Google Scholar
de Arruda, G. F., Petri, G. and Moreno, Y., “Social contagion models on hypergraphs,” Physical Review Research, vol. 2, no. 2, p. 023032, 2020.Google Scholar
Jhun, B., Jo, M. and Kahng, B., “Simplicial SIS model in scale-free uniform hypergraph,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2019, no. 12, p. 123207, 2019.Google Scholar
de Arruda, G. F., Tizzani, M. and Moreno, Y., “Phase transitions and stability of dynamical processes on hypergraphs,” arXiv preprint arXiv:2005.10891, 2020.Google Scholar
Taylor, D., Klimm, F., A. Harrington, H., et al., “Topological data analysis of contagion maps for examining spreading processes on networks,” Nature Communications, vol. 6, p. 7723, 2015.Google Scholar
Massara, G. P., Di Matteo, T. and Aste, T., “Network filtering for big data: Triangulated maximally filtered graph,” Journal of Complex Networks, vol. 5, no. 2, pp. 161–178, 2016.Google Scholar
Andrade, J. S. Jr, Herrmann, H. J., Andrade, R. F. and Da Silva, L. R., “Apollonian networks: simultaneously scale-free, small world, Euclidean, space filling, and with matching graphs,” Physical Review Letters, vol. 94, no. 1, p. 018702, 2005.Google Scholar
Dorogovtsev, S. N., Goltsev, A. V. and Mendes, J. F. F., “Pseudofractal scale-free web,” Physical Review E, vol. 65, no. 6, p. 066122, 2002.Google Scholar
Zhang, Z., Comellas, F., G. Fertin and Rong, L., “High-dimensional Apollonian networks,” Journal of Physics A: Mathematical and General, vol. 39, no. 8, pp. 1811–1818, 2006.Google Scholar
Bianconi, G. and Dorogovstev, S. N., “The spectral dimension of simplicial complexes: a renormalization group theory,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2020, no. 1, p. 014005, 2020.Google Scholar
Söderberg, B., “Apollonian tiling, the Lorentz group, and regular trees,” Physical Review A, vol. 46, no. 4, pp. 1859–1866, 1992.Google Scholar
Graham, R. L., Lagarias, J. C., Mallows, C. L., Wilks, A. R. and Yan, C. H., “Apollonian circle packings: geometry and group theory. I. The Apollonian group,” Discrete and Computational Geometry, vol. 34, no. 4, pp. 547–585, 2005.Google Scholar

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