{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,21]],"date-time":"2025-11-21T12:40:39Z","timestamp":1763728839805,"version":"build-2065373602"},"reference-count":48,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,10,8]],"date-time":"2024-10-08T00:00:00Z","timestamp":1728345600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"VAJRA","award":["VJR\/2019\/000034"],"award-info":[{"award-number":["VJR\/2019\/000034"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In stock markets, nonlinear interdependencies between various companies result in nontrivial time-varying patterns in stock prices. A network representation of these interdependencies has been successful in identifying and understanding hidden signals before major events like stock market crashes. However, these studies have revolved around the assumption that correlations are mediated in a pairwise manner, whereas, in a system as intricate as this, the interactions need not be limited to pairwise only. Here, we introduce a general methodology using information-theoretic tools to construct a higher-order representation of the stock market data, which we call functional hypergraphs. This framework enables us to examine stock market events by analyzing the following functional hypergraph quantities: Forman\u2013Ricci curvature, von Neumann entropy, and eigenvector centrality. We compare the corresponding quantities of networks and hypergraphs to analyze the evolution of both structures and observe features like robustness towards events like crashes during the course of a time period.<\/jats:p>","DOI":"10.3390\/e26100848","type":"journal-article","created":{"date-parts":[[2024,10,8]],"date-time":"2024-10-08T12:03:49Z","timestamp":1728389029000},"page":"848","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Functional Hypergraphs of Stock Markets"],"prefix":"10.3390","volume":"26","author":[{"given":"Jerry Jones","family":"David","sequence":"first","affiliation":[{"name":"Complex Systems Lab, Department of Physics, Indian Institute of Technology Indore, Khandwa Road, Indore 453552, India"}]},{"given":"Narayan G.","family":"Sabhahit","sequence":"additional","affiliation":[{"name":"Network Science Institute, Northeastern University, Boston, MA 02115, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5873-8564","authenticated-orcid":false,"given":"Sebastiano","family":"Stramaglia","sequence":"additional","affiliation":[{"name":"Dipartimento Interateneo di Fisica, Universit\u00e1 degli Studi di Bari Aldo Moro and INFN, 70125 Bari, Italy"}]},{"given":"T. Di","family":"Matteo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King\u2019s College London, Strand, London WC2R 2LS, UK"},{"name":"Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi, Via Panisperna 89 A, 00184 Rome, Italy"},{"name":"Complexity Science Hub Vienna, Josefst\u00e4dter Stra\u00dfe 39, 1080 Vienna, Austria"}]},{"given":"Stefano","family":"Boccaletti","sequence":"additional","affiliation":[{"name":"CNR\u2014Institute of Complex Systems, Via Madonna del Piano 10, 50019 Sesto Fiorentino, Italy"}]},{"given":"Sarika","family":"Jalan","sequence":"additional","affiliation":[{"name":"Complex Systems Lab, Department of Physics, Indian Institute of Technology Indore, Khandwa Road, Indore 453552, India"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1137\/20M1355896","article-title":"The why, how, and when of representations for complex systems","volume":"63","author":"Torres","year":"2021","journal-title":"SIAM Rev."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.physrep.2005.10.009","article-title":"Complex networks: Structure and dynamics","volume":"424","author":"Boccaletti","year":"2006","journal-title":"Phys. Rep."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"512","DOI":"10.1177\/1073858406293182","article-title":"Small-world brain networks","volume":"12","author":"Bassett","year":"2006","journal-title":"Neuroscientist"},{"key":"ref_4","first-page":"722","article-title":"Public transportation in Great Britain viewed as a complex network","volume":"15","author":"Holovatch","year":"2019","journal-title":"Transp. A Transp. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"046116","DOI":"10.1103\/PhysRevE.76.046116","article-title":"Collective behavior of stock price movements in an emerging market","volume":"76","author":"Pan","year":"2007","journal-title":"Phys. Rev. E\u2014Stat. Nonlinear Soft Matter Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1126\/science.286.5439.509","article-title":"Emergence of scaling in random networks","volume":"286","author":"Albert","year":"1999","journal-title":"Science"},{"key":"ref_7","first-page":"17","article-title":"On the evolution of random graphs","volume":"5","author":"Erdos","year":"1960","journal-title":"Publ. Math. Inst. Hung. Acad. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"10421","DOI":"10.1073\/pnas.0500298102","article-title":"A tool for filtering information in complex systems","volume":"102","author":"Tumminello","year":"2005","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1016\/j.physa.2004.08.045","article-title":"Complex networks on hyperbolic surfaces","volume":"346","author":"Aste","year":"2005","journal-title":"Physica A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"036109","DOI":"10.1103\/PhysRevE.86.036109","article-title":"Exploring complex networks via topological embedding on surfaces","volume":"86","author":"Aste","year":"2012","journal-title":"Phys. Rev. E"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.physrep.2008.09.002","article-title":"Synchronization in complex networks","volume":"469","author":"Arenas","year":"2008","journal-title":"Phys. Rep."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10827-016-0608-6","article-title":"Two\u2019s company, three (or more) is a simplex: Algebraic-topological tools for understanding higher-order structure in neural data","volume":"41","author":"Giusti","year":"2016","journal-title":"J. Comput. Neurosci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1038\/s42005-020-00485-0","article-title":"Higher order interactions in complex networks of phase oscillators promote abrupt synchronization switching","volume":"3","author":"Skardal","year":"2020","journal-title":"Commun. Phys."},{"key":"ref_14","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_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2023.04.002","article-title":"The structure and dynamics of networks with higher order interactions","volume":"1018","author":"Boccaletti","year":"2023","journal-title":"Phys. Rep."},{"key":"ref_16","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_17","doi-asserted-by":"crossref","unstructured":"Sharma, C., and Habib, A. (2019). Mutual information based stock networks and portfolio selection for intraday traders using high frequency data: An Indian market case study. PLoS ONE, 14.","DOI":"10.1371\/journal.pone.0221910"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1471","DOI":"10.1103\/PhysRevLett.83.1471","article-title":"Universal and nonuniversal properties of cross correlations in financial time series","volume":"83","author":"Plerou","year":"1999","journal-title":"Phys. Rev. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"701","DOI":"10.1007\/s11403-023-00389-6","article-title":"A look at financial dependencies by means of econophysics and financial economics","volume":"18","author":"Raddant","year":"2023","journal-title":"J. Econ. Interact. Coord."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"201734","DOI":"10.1098\/rsos.201734","article-title":"Network geometry and market instability","volume":"8","author":"Samal","year":"2021","journal-title":"R. Soc. Open Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjds\/s13688-017-0114-8","article-title":"The shape of collaborations","volume":"6","author":"Patania","year":"2017","journal-title":"EPJ Data Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"20140873","DOI":"10.1098\/rsif.2014.0873","article-title":"Homological scaffolds of brain functional networks","volume":"11","author":"Petri","year":"2014","journal-title":"J. R. Soc. Interface"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Murgas, K.A., Saucan, E., and Sandhu, R. (2022). Hypergraph geometry reflects higher-order dynamics in protein interaction networks. Sci. Rep., 12.","DOI":"10.1038\/s41598-022-24584-w"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"12285","DOI":"10.1038\/ncomms12285","article-title":"High-order species interactions shape ecosystem diversity","volume":"7","author":"Bairey","year":"2016","journal-title":"Nat. Commun."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Herzog, R., Rosas, F.E., Whelan, R., Fittipaldi, S., Santamaria-Garcia, H., Cruzat, J., Birba, A., Moguilner, S., Tagliazucchi, E., and Prado, P. (2022). Genuine high-order interactions in brain networks and neurodegeneration. Neurobiol. Dis., 175.","DOI":"10.1016\/j.nbd.2022.105918"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3954","DOI":"10.1016\/j.neuron.2021.09.042","article-title":"High-order interactions explain the collective behavior of cortical populations in executive but not sensory areas","volume":"109","author":"Chelaru","year":"2021","journal-title":"Neuron"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Santoro, A., Battiston, F., Petri, G., and Amico, E. (2022). Unveiling the higher-order organization of multivariate time series. arXiv.","DOI":"10.1038\/s41567-022-01852-0"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"5766","DOI":"10.1109\/TSP.2022.3221892","article-title":"A new framework for the time-and frequency-domain assessment of high-order interactions in networks of random processes","volume":"70","author":"Faes","year":"2022","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Sawhney, R., Agarwal, S., Wadhwa, A., Derr, T., and Shah, R.R. (2021, January 2\u20139). Stock selection viaspatiotemporal hypergraph attention network: A learning to rank approach. Proceedings of the AAAI Conference on Artificial Intelligence, Online.","DOI":"10.1609\/aaai.v35i1.16127"},{"key":"ref_30","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_31","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1147\/rd.41.0066","article-title":"Information theoretical analysis of multivariate correlation","volume":"4","author":"Watanabe","year":"1960","journal-title":"IBM J. Res. Dev."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1109\/TIT.1954.1057469","article-title":"Multivariate information transmission","volume":"4","author":"McGill","year":"1954","journal-title":"Trans. IRE Prof. Group Inf. Theory"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/S0019-9958(78)90026-8","article-title":"A definition of conditional mutual information for arbitrary ensembles","volume":"38","author":"Wyner","year":"1978","journal-title":"Inf. Control"},{"key":"ref_34","unstructured":"Williams, P.L., and Beer, R.D. (2010). Nonnegative decomposition of multivariate information. arXiv."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Lizier, J.T., Bertschinger, N., Jost, J., and Wibral, M. (2018). Information decomposition of target effects from multi-source interactions: Perspectives on previous, current and future work. Entropy, 20.","DOI":"10.3390\/e20040307"},{"key":"ref_36","unstructured":"Marinazzo, D., Van Roozendaal, J., Rosas, F.E., Stella, M., Comolatti, R., Colenbier, N., Stramaglia, S., and Rosseel, Y. (2022). An information-theoretic approach to hypergraph psychometrics. arXiv."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"032305","DOI":"10.1103\/PhysRevE.100.032305","article-title":"Quantifying high-order interdependencies via multivariate extensions of the mutual information","volume":"100","author":"Rosas","year":"2019","journal-title":"Phys. Rev. E"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Gelfand, I.M., and IAglom, A.M. (1959). Calculation of the Amount of Information about a Random Function Contained Inanother Such Function, American Mathematical Society.","DOI":"10.1090\/trans2\/012\/09"},{"key":"ref_39","unstructured":"Thorne, K.S., Misner, C.W., and Wheeler, J.A. (2000). Gravitation, Freeman."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/s00454-002-0743-x","article-title":"Bochner\u2019s method for cell complexes and combinatorial Ricci curvature","volume":"29","author":"Forman","year":"2003","journal-title":"Discret. Comput. Geom."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"063206","DOI":"10.1088\/1742-5468\/2016\/06\/063206","article-title":"Forman curvature for complex networks","volume":"2016","author":"Sreejith","year":"2016","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"2150003","DOI":"10.1142\/S021952592150003X","article-title":"Forman\u2013Ricci curvature for hypergraphs","volume":"24","author":"Leal","year":"2021","journal-title":"Adv. Complex Syst."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Bengtsson, I., and \u017byczkowski, K. (2017). Geometry of Quantum States: An Introduction to Quantum Entanglement, Cambridge University Press.","DOI":"10.1017\/9781139207010"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1007\/s00026-006-0289-3","article-title":"The Laplacian of a graph as a density matrix: A basic combinatorial approach to separability of mixed states","volume":"10","author":"Braunstein","year":"2006","journal-title":"Ann. Comb."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2889","DOI":"10.1109\/TNSE.2020.3002963","article-title":"Tensor entropy for uniform hypergraphs","volume":"7","author":"Chen","year":"2020","journal-title":"IEEE Trans. Netw. Sci. Eng."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1541","DOI":"10.1002\/hbm.23471","article-title":"A statistical framework for neuroimaging data analysis based on mutual information estimated via a gaussian copula","volume":"38","author":"Ince","year":"2017","journal-title":"Hum. Brain Mapp."},{"key":"ref_47","unstructured":"Ouvrard, X. (2020). Hypergraphs: An introduction and review. arXiv."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"013025","DOI":"10.1103\/PhysRevResearch.5.013025","article-title":"Gradients of O-information: Low-order descriptors of high-order dependencies","volume":"5","author":"Scagliarini","year":"2023","journal-title":"Phys. Rev. Res."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/10\/848\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:09:12Z","timestamp":1760112552000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/10\/848"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,8]]},"references-count":48,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2024,10]]}},"alternative-id":["e26100848"],"URL":"https:\/\/doi.org\/10.3390\/e26100848","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2024,10,8]]}}}