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  • Cited by 135
Publisher:
Cambridge University Press
Online publication date:
November 2021
Print publication year:
2021
Online ISBN:
9781108770996
Series:
Elements in the Structure and Dynamics of Complex Networks

Book description

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.

References

[1]Barabási, A.-L., Network Science. Cambridge University Press, 2016.
[2]Dorogovtsev, S. N. and Mendes, J. F., Evolution of Networks: From Biological Nets to the Internet and WWW. Oxford University Press, 2003.
[3]Barrat, A., Barthelemy, M. and Vespignani, A., Dynamical Processes on Complex Networks. Cambridge University Press, 2008.
[4]Menczer, F., Fortunato, S. and Davis, C. A., A First Course in Network Science. Cambridge University Press, 2020.
[5]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.
[6]Bianconi, G., Multilayer Networks: Structure and Function. Oxford University Press, 2018.
[7]Battiston, F., Cencetti, G., Iacopini, I., et al., “Networks beyond pairwise interactions: structure and dynamics,” Physics Reports, vol. 874, pp. 192, 2020.
[8]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. 114, 2016.
[9]Bianconi, G., “Interdisciplinary and physics challenges of network theory,” EPL (Europhysics Letters), vol. 111, no. 5, p. 56001, 2015.
[10]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.
[11]Salnikov, V., Cassese, D. and Lambiotte, R., “Simplicial complexes and complex systems,” European Journal of Physics, vol. 40, no. 1, p. 014001, 2018.
[12]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.
[13]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.
[14]Iacopini, I., Petri, G., Barrat, A. and Latora, V., “Simplicial models of social contagion,” Nature Communications, vol. 10, no. 1, pp. 19, 2019.
[15]Ghrist, R. W., Elementary Applied Topology. Createspace Seattle, 2014, vol. 1.
[16]Ghrist, R., “Barcodes: the persistent topology of data,” Bulletin of the American Mathematical Society, vol. 45, no. 1, pp. 6175, 2008.
[17]Kahle, M., “Topology of random simplicial complexes: a survey,” AMS Contemporary Mathematics, vol. 620, pp. 201222, 2014.
[18]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.
[19]Edelsbrunner, H., A Short Course in Computational Geometry and Topology. Springer, 2014.
[20]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.
[21]Petri, G., Scolamiero, M., Donato, I. and Vaccarino, F., “Topological strata of weighted complex networks,” PloS One, vol. 8, no. 6, p. e66506, 2013.
[22]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. 114, 2018.
[23]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.
[24]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.
[25]Benson, A. R., Gleich, D. F. and J. Leskovec, “Higher-order organization of complex networks,” Science, vol. 353, no. 6295, pp. 163166, 2016.
[26]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. 814818, 2005.
[27]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.
[28]Barbarossa, S. and Sardellitti, S., “Topological signal processing over simplicial complexes,” IEEE Transactions on Signal Processing, vol. 68, pp. 29923007, 2020.
[29]Bianconi, G. and Rahmede, C., “Network geometry with flavor: from complexity to quantum geometry,” Physical Review E, vol. 93, no. 3, p. 032315, 2016.
[30]Bianconi, G. and Rahmede, C., “Emergent hyperbolic geometry,” Scientific Reports, vol. 7, p. 41974, 2017.
[31]Wu, Z., Menichetti, G., Rahmede, C. and Bianconi, G., “Emergent complex network geometry,” Scientific Reports, vol. 5, p. 10073, 2015.
[32]Mulder, D. and Bianconi, G., “Network geometry and complexity,” Journal of Statistical Physics, vol. 173, no. 3–4, pp. 783805, 2018.
[33]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.
[34]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.
[35]Millán, A. P., Torres, J. J. and Bianconi, G., “Complex network geometry and frustrated synchronization,” Scientific Reports, vol. 8, no. 1, pp. 110, 2018.
[36]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.
[37]Bianconi, G. and Ziff, R. M., “Topological percolation on hyperbolic simplicial complexes,” Physical Review E, vol. 98, no. 5, p. 052308, 2018.
[38]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.
[39]Bianconi, G. and Rahmede, C., “Complex quantum network manifolds in dimension d>2 are scale-free,” Scientific Reports, vol. 5, no. 1, pp. 110, 2015.
[40]Patania, A., Petri, G. and Vaccarino, F., “The shape of collaborations,” EPJ Data Science, vol. 6, no. 1, p. 18, 2017.
[41]Bollobás, B. and Béla, B., Random Graphs. Cambridge University Press, 2001, no. 73.
[42]Bianconi, G. and Marsili, M., “Emergence of large cliques in random scale-free networks,” EPL (Europhysics Letters), vol. 74, no. 4, p. 740, 2006.
[43]MacKay, D. J. C, Information Theory, Inference and Learning Algorithms. Cambridge University Press, 2003.
[44]Cover, T. M., Elements of Information Theory. John Wiley & Sons, 1999.
[45]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.
[46]Kardar, M., Statistical Physics of Particles. Cambridge University Press, 2007.
[47]Anand, K. and Bianconi, G., “Gibbs entropy of network ensembles by cavity methods,” Physical Review E, vol. 82, no. 1, p. 011116, 2010.
[48]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. 129153.
[49]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.
[50]Repository for higher-order network codes. [Online]. Available: https://github.com/ginestrab
[51]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. 296307, 1978.
[52]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.
[53]Wegner, A. E. and Olhede, S., “Atomic subgraphs and the statistical mechanics of networks,” Physical Review E, vol. 103, no. 04, p. 042311, 2021.
[54]Klimm, F., Deane, C. M. and Reinert, G., “Hypergraphs for predicting essential genes using multiprotein complex data,” bioRxiv, 2020.
[55]Sun, H. and Bianconi, G., “Higher-order percolation processes on multiplex hypergraphs,” arXiv preprint arXiv:2104.05457, 2021.
[56]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.
[57]Zhao, K., Karsai, M. and Bianconi, G., “Entropy of dynamical social networks,” PloS One, vol. 6, no. 12, p. e28116, 2011.
[58]Petri, G. and Barrat, A., “Simplicial activity driven model,” Physical Review Letters, vol. 121, no. 22, p. 228301, 2018.
[59]Holme, P. and Saramäki, J., “Temporal networks,” Physics Reports, vol. 519, no. 3, pp. 97125, 2012.
[60]Karsai, M., Kaski, K., Barabási, A.-L. and Kertész, J., “Universal features of correlated bursty behaviour,” Scientific Reports, vol. 2, p. 397, 2012.
[61]Karsai, M., Jo, H.-H., Kaski, K., et al., Bursty Human Dynamics. Springer, 2018.
[62]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.
[63]Stehlé, J., Barrat, A. and Bianconi, G., “Dynamical and bursty interactions in social networks,” Physical Review E, vol. 81, no. 3, p. 035101, 2010.
[64]Cencetti, G., Battiston, F., Lepri, B. and Karsai, M., “Temporal properties of higher-order interactions in social networks,” arXiv preprint arXiv:2010.03404, 2020.
[65]Link to SocioPattern project webpage. [Online]. Available: http://www.sociopatterns.org/
[66]Jost, J., Mathematical Concepts. Springer, 2015.
[67]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:
[68]Link to license. [Online]. Available: https://creativecommons.org/licenses/by/4.0/
[69]Watts, D. J. and Strogatz, S. H., “Collective dynamics of ‘small-world’ networks,” Nature, vol. 393, no. 6684, pp. 440442, 1998.
[70]Patania, A., Vaccarino, F. and Petri, G., “Topological analysis of data,” EPJ Data Science, vol. 6, no. 1, pp. 16, 2017.
[71]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.
[72]Ollivier, Y., “Ricci curvature of metric spaces,” Comptes Rendus Mathematique, vol. 345, no. 11, pp. 643646, 2007.
[73]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.
[74]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. 300322, 2014.
[75]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.
[76]Regge, T., “General relativity without coordinates,” Il Nuovo Cimento (1955–1965), vol. 19, no. 3, pp. 558571, 1961.
[77]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.
[78]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.
[79]Gromov, M., “Hyperbolic groups,” in Essays in Group Theory, Gersten, S. M. (Ed.). Springer, 1987, pp. 75263.
[80]Jonckheere, E., Lohsoonthorn, P. and Bonahon, F., “Scaled Gromov hyperbolic graphs,” Journal of Graph Theory, vol. 57, no. 2, pp. 157180, 2008.
[81]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.
[82]Kennedy, W. S., Narayan, O. and Saniee, I., “On the hyperbolicity of large-scale networks,” arXiv preprint arXiv:1307.0031, 2013.
[83]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.
[84]Benedetti, D. and Henson, J., “Spectral geometry as a probe of quantum spacetime,” Physical Review D, vol. 80, no. 12, p. 124036, 2009.
[85]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. 549574, 1998.
[86]Durhuus, B., Jonsson, T. and Wheater, J. F., “The spectral dimension of generic trees,” Journal of Statistical Physics, vol. 128, no. 5, pp. 12371260, 2007.
[87]Burioni, R. and Cassi, D., “Universal properties of spectral dimension,” Physical Review Letters, vol. 76, no. 7, p. 10911093, 1996.
[88]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.
[89]Wedeen, V. J., Rosene, D. L., Wang, R., et al., “The geometric structure of the brain fiber pathways,” Science, vol. 335, no. 6076, pp. 16281634, 2012.
[90]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.
[91]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.
[92]Á. Serrano, M., Boguná, M. and Sagués, F., “Uncovering the hidden geometry behind metabolic networks,” Molecular Biosystems, vol. 8, no. 3, pp. 843850, 2012.
[93]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.
[94]Penrose, R., “On the nature of quantum geometry,” Magic Without Magic, Klauder, J. R. (Ed.). W. H. Freeman & Co. Ltd. pp. 333354, 1972.
[95]Smolin, L., The Life of the Cosmos. Oxford University Press, 1999.
[96]Bianconi, G. and Barabási, A.-L., “Bose–Einstein condensation in complex networks,” Physical Review Letters, vol. 86, no. 24, p. 5632, 2001.
[97]Fountoulakis, N., Iyer, T., Mailler, C. and Sulzbach, H., “Dynamical models for random simplicial complexes,” arXiv preprint arXiv:1910.12715, 2019.
[98]Barabási, A.-L. and Albert, R., “Emergence of scaling in random networks,” Science, vol. 286, no. 5439, pp. 509512, 1999.
[99]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.
[100]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.
[101]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.
[102]Šuvakov, M., Andjelković, M. and Tadić, B., “Hidden geometries in networks arising from cooperative self-assembly,” Scientific Reports, vol. 8, no. 1, pp. 110, 2018.
[103]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.
[104]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.
[105]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. 78217826, 2002.
[106]Courtney, O. T. and Bianconi, G., “Weighted growing simplicial complexes,” Physical Review E, vol. 95, no. 6, p. 062301, 2017.
[107]Kovalenko, K., Sendiña-Nadal, I., Khalil, N., et al., “rowing scale-free simplices,” arXiv preprint arXiv:2006.12899, 2020.
[108]Bianconi, G. and Barabási, A.-L., “Competition and multiscaling in evolving networks,” EPL (Europhysics Letters), vol. 54, no. 4, p. 436, 2001.
[109]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.
[110]Strogatz, S., Sync: The Emerging Science of Spontaneous Order. Penguin UK, 2004.
[111]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. 420422.
[112]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. 120, 2000.
[113]Pikovsky, A., Rosenblum, M. and Kurths, J., Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, 2003, no. 12.
[114]Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y. and Zhou, C., “Synchronization in complex networks,” Physics Reports, vol. 469, no. 3, pp. 93153, 2008.
[115]Boccaletti, S., Pisarchik, A. N., Del Genio, C. I. and Amann, A., Synchronization: from Coupled Systems to Complex Networks. Cambridge University Press, 2018.
[116]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.
[117]Millán, A. P., Restrepo, J. G., Torres, J. J. and Bianconi, G., “Geometry, topology and simplicial synchronization,” arXiv preprint arXiv:2105.00943, 2021.
[118]Kuehn, C. and Bick, C., “A universal route to explosive phenomena,” Science Advances, vol. 7, no. 16, p. eabe3824, 2021.
[119]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.
[120]Gambuzza, L., Di Patti, F., Gallo, L., et al., “The master stability function for synchronization in simplicial complexes,” arXiv preprint arXiv:2004.03913, 2020.
[121]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.
[122]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.
[123]Salova, A. and D’Souza, R. M., “Cluster synchronization on hypergraphs,” arXiv preprint arXiv:2101.05464, 2021.
[124]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.
[125]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.
[126]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. 16, 2020.
[127]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.
[128]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.
[129]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.
[130]DeVille, L., “Consensus on simplicial complexes, or: The nonlinear simplicial Laplacian,” arXiv preprint arXiv:2010.07421, 2020.
[131]Benjamini, I. and Schramm, O., “Percolation in the hyperbolic plane,” Journal of the American Mathematical Society, vol. 14, no. 2, pp. 487507, 2001.
[132]Boettcher, S., Singh, V. and Ziff, R. M., “Ordinary percolation with discontinuous transitions,” Nature Communications, vol. 3, no. 1, pp. 15, 2012.
[133]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.
[134]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.
[135]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.
[136]Bobrowski, O. and Skraba, P., “Homological percolation and the Euler characteristic,” Physical Review E, vol. 101, no. 3, p. 032304, 2020.
[137]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.
[138]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.
[139]Amburg, I., Kleinberg, J. and Benson, A., “Planted hitting set recovery in hypergraphs,” Journal of Physics: Complexity, vol. 2, p. 035004, 2021.
[140]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.
[141]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.
[142]Granovetter, M., “Threshold models of collective behavior,” American Journal of Sociology, vol. 83, no. 6, p. 1420, 1978.
[143]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.
[144]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.
[145]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. 567569.
[146]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.
[147]de Arruda, G. F., Petri, G. and Moreno, Y., “Social contagion models on hypergraphs,” Physical Review Research, vol. 2, no. 2, p. 023032, 2020.
[148]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.
[149]de Arruda, G. F., Tizzani, M. and Moreno, Y., “Phase transitions and stability of dynamical processes on hypergraphs,” arXiv preprint arXiv:2005.10891, 2020.
[150]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.
[151]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. 161178, 2016.
[152]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.
[153]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.
[154]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. 18111818, 2006.
[155]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.
[156]Söderberg, B., “Apollonian tiling, the Lorentz group, and regular trees,” Physical Review A, vol. 46, no. 4, pp. 18591866, 1992.
[157]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. 547585, 2005.

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