{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,20]],"date-time":"2026-05-20T22:16:20Z","timestamp":1779315380387,"version":"3.51.4"},"reference-count":35,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,9,28]],"date-time":"2023-09-28T00:00:00Z","timestamp":1695859200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"STI 2030\u2014Major Projects","award":["2022ZD0211400"],"award-info":[{"award-number":["2022ZD0211400"]}]},{"name":"STI 2030\u2014Major Projects","award":["T2293771"],"award-info":[{"award-number":["T2293771"]}]},{"name":"STI 2030\u2014Major Projects","award":["2023NSFSC1919"],"award-info":[{"award-number":["2023NSFSC1919"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2022ZD0211400"],"award-info":[{"award-number":["2022ZD0211400"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["T2293771"],"award-info":[{"award-number":["T2293771"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2023NSFSC1919"],"award-info":[{"award-number":["2023NSFSC1919"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Sichuan Science and Technology Program","award":["2022ZD0211400"],"award-info":[{"award-number":["2022ZD0211400"]}]},{"name":"Sichuan Science and Technology Program","award":["T2293771"],"award-info":[{"award-number":["T2293771"]}]},{"name":"Sichuan Science and Technology Program","award":["2023NSFSC1919"],"award-info":[{"award-number":["2023NSFSC1919"]}]},{"name":"New Cornerstone Science Foundation","award":["2022ZD0211400"],"award-info":[{"award-number":["2022ZD0211400"]}]},{"name":"New Cornerstone Science Foundation","award":["T2293771"],"award-info":[{"award-number":["T2293771"]}]},{"name":"New Cornerstone Science Foundation","award":["2023NSFSC1919"],"award-info":[{"award-number":["2023NSFSC1919"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Null models are crucial tools for investigating network topological structures. However, research on null models for higher-order networks is still relatively scarce. In this study, we introduce an innovative method to construct null models for hypergraphs, namely the hyperedge swapping-based method. By preserving certain network properties while altering others, we generate six hyper-null models with various orders and analyze their interrelationships. To validate our approach, we first employ hypergraph entropy to assess the randomness of these null models across four datasets. Furthermore, we examine the differences in important statistical properties between the various null models and the original networks. Lastly, we investigate the impact of hypergraph randomness on network dynamics using the proposed hyper-null models, focusing on dismantling and epidemic contagion. The findings show that our proposed hyper-null models are applicable to various scenarios. By introducing a comprehensive framework for generating and analyzing hyper-null models, this research opens up avenues for further exploration of the intricacies of network structures and their real-world implications.<\/jats:p>","DOI":"10.3390\/e25101390","type":"journal-article","created":{"date-parts":[[2023,9,28]],"date-time":"2023-09-28T08:24:44Z","timestamp":1695889484000},"page":"1390","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Hyper-Null Models and Their Applications"],"prefix":"10.3390","volume":"25","author":[{"given":"Yujie","family":"Zeng","sequence":"first","affiliation":[{"name":"Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China"},{"name":"Yangtze Delta Region Institute (Huzhou), University of Electronic Science and Technology of China, Huzhou 313001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2802-4405","authenticated-orcid":false,"given":"Bo","family":"Liu","sequence":"additional","affiliation":[{"name":"Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China"},{"name":"Yangtze Delta Region Institute (Huzhou), University of Electronic Science and Technology of China, Huzhou 313001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fang","family":"Zhou","sequence":"additional","affiliation":[{"name":"Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China"},{"name":"Yangtze Delta Region Institute (Huzhou), University of Electronic Science and Technology of China, Huzhou 313001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Linyuan","family":"L\u00fc","sequence":"additional","affiliation":[{"name":"Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China"},{"name":"Yangtze Delta Region Institute (Huzhou), University of Electronic Science and Technology of China, Huzhou 313001, China"},{"name":"School of Cyber Science and Technology, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,28]]},"reference":[{"key":"ref_1","unstructured":"Barab\u00e1si, A.L. (2016). Network Science, Cambridge University Press."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Newman, M. (2018). Networks, Oxford University Press.","DOI":"10.1093\/oso\/9780198805090.001.0001"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"20220043","DOI":"10.1098\/rsif.2022.0043","article-title":"Dynamics on higher-order networks: A review","volume":"19","author":"Majhi","year":"2022","journal-title":"J. R. Soc. Interface"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2020.05.004","article-title":"Networks beyond pairwise interactions: Structure and dynamics","volume":"874","author":"Battiston","year":"2020","journal-title":"Phys. Rep."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1093","DOI":"10.1038\/s41567-021-01371-4","article-title":"The physics of higher-order interactions in complex systems","volume":"17","author":"Battiston","year":"2021","journal-title":"Nat. Phys."},{"key":"ref_6","unstructured":"Zeng, Y., Huang, Y., Ren, X.L., and L\u00fc, L. (2023). Identifying vital nodes through augmented random walks on higher-order networks. arXiv."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Bianconi, G. (2021). Higher-Order Networks, Cambridge University Press.","DOI":"10.1017\/9781108770996"},{"key":"ref_8","unstructured":"Berge, C. (1984). Hypergraphs: Combinatorics of Finite Sets, Elsevier."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"LaRock, T., and Lambiotte, R. (2023). Encapsulation Structure and Dynamics in Hypergraphs. arXiv.","DOI":"10.1088\/2632-072X\/ad0b39"},{"key":"ref_10","unstructured":"Feng, Y., You, H., Zhang, Z., Ji, R., and Gao, Y. (February, January 27). Hypergraph neural networks. Proceedings of the AAAI Conference on Artificial Intelligence, Hilton, HI, USA."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Liao, X., Xu, Y., and Ling, H. (2021, January 11\u201317). Hypergraph neural networks for hypergraph matching. Proceedings of the IEEE\/CVF International Conference on Computer Vision, Montreal, BC, Canada.","DOI":"10.1109\/ICCV48922.2021.00130"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"824","DOI":"10.1126\/science.298.5594.824","article-title":"Network motifs: Simple building blocks of complex networks","volume":"298","author":"Milo","year":"2002","journal-title":"Science"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1538","DOI":"10.1126\/science.1089167","article-title":"Superfamilies of evolved and designed networks","volume":"303","author":"Milo","year":"2004","journal-title":"Science"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Liu, B., Xu, S., Li, T., Xiao, J., and Xu, X.K. (2018). Quantifying the effects of topology and weight for link prediction in weighted complex networks. Entropy, 20.","DOI":"10.3390\/e20050363"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"8627","DOI":"10.1038\/ncomms9627","article-title":"Quantifying randomness in real networks","volume":"6","author":"Orsini","year":"2015","journal-title":"Nat. Commun."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Gjoka, M., Kurant, M., and Markopoulou, A. (2013, January 14\u201319). 2.5K-graphs: From sampling to generation. Proceedings of the 2013 IEEE INFOCOM, Turin, Italy.","DOI":"10.1109\/INFCOM.2013.6566997"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Mahadevan, P., Hubble, C., Krioukov, D., Huffaker, B., and Vahdat, A. (2007, January 27\u201331). Orbis: Rescaling degree correlations to generate annotated internet topologies. Proceedings of the 2007 Conference on Applications, Technologies, Architectures, and Protocols for Computer Communications, Kyoto, Japan.","DOI":"10.1145\/1282380.1282417"},{"key":"ref_18","unstructured":"Zeng, Y., Huang, Y., Wu, Q., and L\u00fc, L. (2023). Influential Simplices Mining via Simplicial Convolutional Network. arXiv."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1145\/1151659.1159930","article-title":"Systematic topology analysis and generation using degree correlations","volume":"36","author":"Mahadevan","year":"2006","journal-title":"ACM Sigcomm Comput. Commun. Rev."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"cnaa028","DOI":"10.1093\/comnet\/cnaa028","article-title":"Hypergraphs for predicting essential genes using multiprotein complex data","volume":"9","author":"Klimm","year":"2021","journal-title":"J. Complex Netw."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"cnaa018","DOI":"10.1093\/comnet\/cnaa018","article-title":"Configuration models of random hypergraphs","volume":"8","author":"Chodrow","year":"2020","journal-title":"J. Complex Netw."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Miyashita, R., Nakajima, K., Fukuda, M., and Shudo, K. (2023, January 13\u201316). Randomizing Hypergraphs Preserving Two-mode Clustering Coefficient. Proceedings of the 2023 IEEE International Conference on Big Data and Smart Computing (BigComp), Jeju, Republic of Korea.","DOI":"10.1109\/BigComp57234.2023.00064"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1139","DOI":"10.1109\/TNSE.2021.3133380","article-title":"Randomizing hypergraphs preserving degree correlation and local clustering","volume":"9","author":"Nakajima","year":"2021","journal-title":"IEEE Trans. Netw. Sci. Eng."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Omar, Y.M., and Plapper, P. (2020). A survey of information entropy metrics for complex networks. Entropy, 22.","DOI":"10.3390\/e22121417"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1073\/pnas.0605965104","article-title":"Resolution limit in community detection","volume":"104","author":"Fortunato","year":"2007","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"026113","DOI":"10.1103\/PhysRevE.69.026113","article-title":"Finding and evaluating community structure in networks","volume":"69","author":"Newman","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1238","DOI":"10.1007\/s10114-019-8093-2","article-title":"Extremality of graph entropy based on degrees of uniform hypergraphs with few edges","volume":"35","author":"Hu","year":"2019","journal-title":"Acta Math. Sin. Engl. Ser."},{"key":"ref_29","unstructured":"Berge, C. (1973). Graphs and Hypergraphs, North-Holland Publishing Company."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1016\/j.physa.2005.12.002","article-title":"Subgraph centrality and clustering in complex hyper-networks","volume":"364","author":"Estrada","year":"2006","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"208701","DOI":"10.1103\/PhysRevLett.89.208701","article-title":"Assortative Mixing in Networks","volume":"89","author":"Newman","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Amburg, I., Veldt, N., and Benson, A. (2020, January 20\u201324). Clustering in graphs and hypergraphs with categorical edge labels. Proceedings of the Web Conference 2020, Taipei, Taiwan.","DOI":"10.1145\/3366423.3380152"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"D515","DOI":"10.1093\/nar\/gkv1049","article-title":"BiGG Models: A platform for integrating, standardizing and sharing genome-scale models","volume":"44","author":"King","year":"2016","journal-title":"Nucleic Acids Res."},{"key":"ref_34","first-page":"4679","article-title":"Disintegrate hypergraph networks by attacking hyperedge","volume":"34","author":"Peng","year":"2022","journal-title":"J. King Saud-Univ.-Comput. Inf. Sci."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1007\/s11538-016-0158-0","article-title":"SIS epidemic propagation on hypergraphs","volume":"78","author":"Katona","year":"2016","journal-title":"Bull. Math. Biol."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/10\/1390\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:01:04Z","timestamp":1760130064000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/10\/1390"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,28]]},"references-count":35,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2023,10]]}},"alternative-id":["e25101390"],"URL":"https:\/\/doi.org\/10.3390\/e25101390","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,28]]}}}