Characterization and optimization of satellite complex networks based on hyperbolic space ^☆

Abstract

In recent years, the rapid advancement of mega-constellations in Low Earth Orbit (LEO) has led to the emergence of satellite communication networks characterized by a complex interplay between high- and low-altitude orbits and by unprecedented scale. Traditional network-representation methodologies in Euclidean space are insufficient to capture the dynamics and evolution of high-dimensional complex networks. By contrast, hyperbolic space offers greater scalability and stronger representational capacity than Euclidean-space methods, thereby providing a more suitable framework for representing large-scale satellite communication networks. This paper aims to address the burgeoning demands of large-scale space-air-ground integrated satellite communication networks by providing a comprehensive review of representation-learning methods for large-scale complex networks and their application within hyperbolic space. First, we briefly introduce several equivalent models of hyperbolic space. Then, we summarize existing representation methods and applications for large-scale complex networks. Building on these advances, we propose representation methods for complex satellite communication networks in hyperbolic space and discuss potential application prospects. Finally, we highlight several pressing directions for future research.

Keywords

Hyperbolic space / Complex network / Network representation / Satellite communication network

References

[1] Y. He, Z. Li, Z. Dou, Chinese satellite frequency and orbit entity relation extraction method based on dynamic integrated learning, Digit. Commun. Netw. 11 (3) (2025) 787-794.

[2] C.J.P. Johansson, R. Tione, T 5 configurations and hyperbolic systems, Commun. Contemp. Math. 26 (3) (2024).

[3] C. Tu, C. Yang, Z. Liu, M. Sun, A survey of network representation learning, Sci. China Inf. Sci. 47 (8) (2017) 17.

[4] Y. Ding, H. Wei, Z. Pan, X. Liu, A survey of network representation learning algo-rithms, Comput. Sci. 47 (9) (2020) 8.

[5] D. Pavlov, S. Kokarev, Hyperbolic statics in space-time, Gravit. Cosmol. 21 (2) (2015) 152-156.

[6] S. Lv, et al., Routing strategy of integrated satellite-terrestrial network based on hyperbolic geometry, IEEE Access 8 (2020) 113003-113010.

[7] S. Spremo, P. Klupar, J. Bregman, et al., Low cost rapid response spacecraft (lcrrs): a case study in small satellite cost optimization for government and commercial use,in:Proceedings of the AIAA SPACE 2008 Conference and Exposition, AIAA, 2008.

[8] M. Boguñá, F. Papadopoulos, D. Krioukov, Sustaining the Internet with hyperbolic mapping, Nat. Commun. 1 (1) (2010) 62.

[9] G. Alanis-Lobato, P. Mier, M.A. Andrade-Navarro, Manifold learning and maximum likelihood estimation for hyperbolic network embedding, Appl. Netw. Sci. 1 (1)(2016) 1-14.

[10] Z. Wang, Y. Wu, Q. Li, F. Jin, W. Xiong, Link prediction based on hyperbolic mapping with community structure for complex networks, Phys. A 450 (2016) 609-623.

[11] Z. Zhu, W. Zhang, X. Guo, et al., Lorentzian graph convolution networks for collabo-rative filtering,in:Proceedings of the 2023 International Joint Conference on Neural Networks, IEEE, 2023, pp. 1-8.

[12] M. Moshiri, F. Safaei, Z. Samei, A novel recovery strategy based on link predic-tion and hyperbolic geometry of complex networks, J. Complex Netw. 9 (4) (2021) cnab007.

[13] S. Yi, et al. A hyperbolic embedding method for weighted networks, IEEE Trans. Netw. Sci. Eng. 8 (1) (2021) 599-612.

[14] Q. Wang, H. Jiang, S. Yi, et al., Learning methods for hyperbolic space representation of complex networks, J. Softw. 32 (1) (2021) 93-117.

[15] W. Peng, T. Varanka, A. Mostafa, et al., Hyperbolic deep neural networks: a survey, IEEE Trans. Pattern Anal. Mach. Intell. 44 (12) (2022) 10023-10044.

[16] M. Yang, M. Zhou, Z. Li, et al., Hyperbolic graph neural networks: a review of meth-ods and applications, 2022.

[17] M. Yang, M. Zhou, H. Xiong, I. King, Hyperbolic temporal network embedding, IEEE Trans. Knowl. Data Eng. 35 (11) (2023) 11489-11502.

[18] M. Yang, M. Zhou, R. Ying, I. King, Hyperbolic representation learning: revisiting and advancing,in:Proceedings of the 40th International Conference on Machine Learning, PMLR, 2023, pp. 39639-39659.

[19] W. Cui, X. Xu, J. Chen, et al., Representation of multirelations of geographic scenes based on hyperbolic space, IEEE Trans. Geosci. Remote Sens. 62 (2024) 1-13.

[20] P. Mettes, M.G. Atigh, M. Keller-Ressel, J. Gu, S. Yeung, Hyperbolic deep learning in computer vision: a survey, Int. J. Comput. Vis. 132 (9) (2024) 3484-3508.

[21] S. Tan, H. Zhao, S. Zhao, et al., Why are hyperbolic neural networks effective? A study on hierarchical representation capability, arXiv:2402.02478.

[22] T. Mikolov, I. Sutskever, K. Chen, G.S. Corrado, J. Dean, Distributed Representations of Words and Phrases and Their Compositionality, Advances in Neural Information Processing Systems, vol. 26, 2013.

[23] G.E. Hinton, J.L. McClelland, D.E. Rumelhart, Distributed representations, Tech. Rep., Carnegie-Mellon University, Pittsburgh, 1984.

[24] Y. Bengio, A. Courville, P. Vincent, Representation learning: a review and new per-spectives, IEEE Trans. Pattern Anal. Mach. Intell. 35 (8) (2013) 1798-1828.

[25] A. Cvetkovski, M. Crovella, Hyperbolic embedding and routing for dynamic graphs, in: Proceedings of the IEEE INFOCOM 2009 Conference on Computer Communica-tions, IEEE, 2009, pp. 1647-1655.

[26] L.S. Goddard, F. Klein, H. Weyi, Vorlesungen über höhere geometrie, Math. Gaz. 36 (315) (1952) 66.

[27] N.I. Lobachevsky, Lobachevskii Geometry and Geometric Foundations Synopsis, Harbin Institute of Technology Press, 2012.

[28] J.W. Cannon, W.J. Floyd, R. Kenyon, W.R. Parry, et al., Hyperbolic geometry, Flavors Geom. 31 (59-115) (1997) 2.

[29] B. Perozzi, R. Al-Rfou, S. Skiena, Deepwalk: online learning of social representations,in:Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, ACM, 2014, pp. 701-710.

[30] A. Ng, M. Jordan, Y. Weiss, On spectral clustering: analysis and an algorithm, in: Advances in Neural Information Processing Systems, vol. 14, 2001.

[31] S.T. Roweis, L.K. Saul, Nonlinear dimensionality reduction by locally linear embed-ding, Science 290 (5500) (2000) 2323-2326.

[32] B. Shaw, T. Jebara, Structure preserving embedding, in: Proceedings of the 26th International Conference on Machine Learning, ACM, 2009, pp. 937-944.

[33] S. Cao, W. Lu, Q. Xu, Grarep: learning graph representations with global structural information,in:Proceedings of the 24th ACM International Conference on Informa-tion and Knowledge Management, ACM, 2015, pp. 891-900.

[34] J. Tang, M. Qu, M. Wang, M. Zhang, J. Yan, Q. Mei, Line: large-scale information network embedding,in:Proceedings of the 24th International Conference on World Wide Web, ACM, 2015, pp. 1067-1077.

[35] D. Wang, P. Cui, W. Zhu, Structural deep network embedding, in: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, ACM, 2016, pp. 1225-1234.

[36] X. Su, et al., A comprehensive survey on community detection with deep learning, IEEE Trans. Neural Netw. Learn. Syst. 33 (2022).

[37] A.A. Amini, A. Chen, P.J. Bickel, E. Levina, et al., Pseudo-likelihood methods for community detection in large sparse networks, Ann. Stat. 41 (4) (2013) 2097-2122.

[38] D. de Lange, M. de Reus, M. van den Heuvel, The Laplacian spectrum of neural networks, Front. Comput. Neurosci. 7 (2014) 189.

[39] W. Holland, K.B. Laskey, S. Leinhardt, Stochastic blockmodels: first steps, Soc. Netw. 5 (2) (1983) 109-137.

[40] B. Karrer, M.E. Newman, Stochastic blockmodels and community structure in net-works, Phys. Rev. E 83 (1) (2011) 016107.

[41] E.M. Airoldi, D.M. Blei, S.E. Fienberg, E.P. Xing, Mixed membership stochastic block-models, J. Mach. Learn. Res. 9 (2008) 1981-2014.

[42] Y. LeCun, Y. Bengio, et al. Convolutional Networks for Images, Speech and Time Series, 1998, p. 3361.

[43] T.N. Kipf,M. Welling, Semi-supervised classification with graph convolutional net-works, arXiv:1609.02907.

[44] I. Goodfellow, et al., Generative adversarial networks, Commun. ACM 63 (11) (2020) 139-144.

[45] P. Veličković, G. Cucurull, A. Casanova, A. Romero, P. Lio,Y. Bengio, Graph attention networks, arXiv:1710.10903.

[46] C. Yang, Z. Liu, D. Zhao, M. Sun, E. Chang, Network representation learning with rich text information, in: Proceedings of the 24th International Joint Conference on Artificial Intelligence, AAAI Press, 2015.

[47] C. Tu, H. Liu, Z. Liu, M. Sun, Cane: context-aware network embedding for relation modeling,in: Proceedings of the 55th Annual Meeting of the Association for Compu-tational Linguistics (Volume 1: Long Papers), vol. 1, Association for Computational Linguistics, 2017, pp. 1722-1731.

[48] C. Tu, Z. Zhang, Z. Liu, M. Sun, Transnet: translation-based network representation learning for social relation extraction,in:Proceedings of the 26th International Joint Conference on Artificial Intelligence, AAAI Press, 2017, pp. 2864-2870.

[49] C. Tu, W. Zhang, Z. Liu, M. Sun, et al., Max-margin deepwalk: discriminative learning of network representation,in:Proceedings of the 25th International Joint Conference on Artificial Intelligence, AAAI Press, 2016, pp. 3889-3895.

[50] J. Li, J. Zhu, B. Zhang,Discriminative deep random walk for network classification, in:Proceedings of the 54th Annual Meeting of the Association for Computational Lin-guistics (Volume 1: Long Papers), vol. 1, Association for Computational Linguistics, 2016, pp. 1004-1013.

[51] A. Grover, J. Leskovec, node2vec: scalable feature learning for networks, in: Proceed-ings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, ACM, 2016, pp. 855-864.

[52] Z. Yang, W. Cohen, R. Salakhudinov, Revisiting semi-supervised learning with graph embeddings, in: Proceedings of the 33rd International Conference on Machine Learn-ing, PMLR, 2016, pp. 40-48.

[53] G. Garcia-Perez, A. Allard, M.A. Serrano, M. Boguñá, Mercator: uncovering faithful hyperbolic embeddings of complex networks, New J. Phys. 21 (2019) 123033.

[54] F. Papadopoulos, C. Psomas, D. Krioukov, Network mapping by replaying hyperbolic growth, IEEE/ACM Trans. Netw. 23 (1) (2014) 198-211.

[55] F. Papadopoulos, R. Aldecoa, D. Krioukov, Network geometry inference using com-mon neighbors, Phys. Rev. E 92 (2) (2015) 022807.

[56] T. Bläsius, T. Friedrich, A. Krohmer, S. Laue, Efficient embedding of scale-free graphs in the hyperbolic plane, IEEE/ACM Trans. Netw. 26 (2) (2018) 920-933.

[57] G. Alanis-Lobato, P. Mier, M.A. Andrade-Navarro, Efficient embedding of complex networks to hyperbolic space via their Laplacian, Sci. Rep. 6 (1) (2016) 1-10.

[58] A. Muscoloni, J.M. Thomas, S. Ciucci, G. Bianconi, C.V. Cannistraci, Machine learn-ing meets complex networks via coalescent embedding in the hyperbolic space, Nat. Commun. 8 (1) (2017) 1-19.

[59] A. Muscoloni,C.V. Cannistraci, Minimum curvilinear automata with similarity at-tachment for network embedding and link prediction in the hyperbolic space, arXiv: 1802.01183.

[60] X. Wang, X. Li, G. Chen, Introduction to Network Science, Higher Education Press, 2012.

[61] X. Wang, Y. Zhang, C. Shi,Hyperbolic heterogeneous information network embed-ding, in:Proceedings of the AAAI Conference on Artificial Intelligence, vol. 33, AAAI Press, 2019, pp. 5337-5344.

[62] B.P. Chamberlain, J. Clough,M.P. Deisenroth, Neural embeddings of graphs in hy-perbolic space, arXiv:1705.10359.

[63] Y. Tay, L.A. Tuan, S.C. Hui, Hyperbolic representation learning for fast and efficient neural question answering, in: Proceedings of the Eleventh ACM International Con-ference on Web Search and Data Mining, ACM, 2018, pp. 583-591.

[64] F. Sala, C. De Sa, A. Gu, C. Ré, Representation tradeoffs for hyperbolic embeddings, in: Proceedings of the 35th International Conference on Machine Learning, PMLR, 2018, pp. 4460-4469.

[65] M. Nickel, D. Kiela, Poincaré embeddings for learning hierarchical representations, in: Advances in Neural Information Processing Systems, vol. 30, Curran Associates, Inc., 2017.

[66] F. Papadopoulos, M. Kitsak, M. Serrano, M. Bogun, D. Krioukov, Popularity versus similarity in growing networks, Nature 489 (7417) (2012) 537-540.

[67] A. Muscoloni, C.V. Cannistraci, A nonuniform popularity-similarity optimization (npso) model to efficiently generate realistic complex networks with communities, New J. Phys. 20 (5) (2018).

[68] K.K. Kleineberg, M. Bogun, M.Á. Serrano, F. Papadopoulos, Hidden geometric cor-relations in real multiplex networks, Nat. Phys. 12 (11) (2016) 1076-1081.

[69] J.L. Contreras, J.H. Reichman, Sharing by design: data and decentralized commons, Science 350 (6266) (2015) 1312-1314.

[70] T. Simas, M. Chavez, P.R. Rodriguez, A. Diaz-Guilera, An algebraic topological method for multimodal brain networks comparisons, Front. Psychol. 6 (2015) 904.

[71] M.E.J. Newman, M. Girvan, Finding and evaluating community structure in net-works, Phys. Rev. E 69 (2) (2004) 026113.

[72] F. Radicchi, C. Castellano, F. Cecconi, V. Loreto, D. Parisi, Defining and identifying communities in networks, Proc. Natl. Acad. Sci. USA 101 (9) (2004) 2658-2663.

[73] D. Krioukov, F. Papadopoulos, M. Bogun, A. Vahdat, et al., Efficient navigation in scale-free networks embedded in hyperbolic metric spaces, arXiv:0805.1266.

[74] E. Stai, V. Karyotis, S. Papavassiliou, A hyperbolic space analytics framework for big network data and their applications, IEEE Netw. 30 (1) (2016) 11-17.

[75] E. Ortiz, M. Starnini, M. Serrano, Navigability of temporal networks in hyperbolic space, Sci. Rep. 7 (1) (2017) 1-9.

[76] M. Kitsak, I. Voitalov, D. Krioukov, Link prediction with hyperbolic geometry, Phys. Rev. Res. 2 (4) (2020) 043113.

[77] C. Gulcehre, et al., Hyperbolic attention networks, arXiv:1805.09786.

[78] H. Cho, B. DeMeo, J. Peng, B. Berger, Large-margin classification in hyperbolic space, in: Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics, PMLR, 2019, pp. 1832-1840.

[79] O. Ganea, G. Bécigneul, T. Hofmann, Hyperbolic neural networks, in: Advances in Neural Information Processing Systems, vol. 31, Curran Associates, Inc., 2018.

[80] I. Chami, Z. Ying, C. Ré, J. Leskovec, Hyperbolic graph convolutional neural net-works, in: Advances in Neural Information Processing Systems, vol. 32, Curran As-sociates, Inc., 2019.