{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T18:19:25Z","timestamp":1780078765676,"version":"3.54.0"},"reference-count":42,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,2,23]],"date-time":"2024-02-23T00:00:00Z","timestamp":1708646400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"CUFE Postgraduate Students Support Program for the Integration of Research and Teaching","award":["202209"],"award-info":[{"award-number":["202209"]}]},{"name":"CUFE Postgraduate Students Support Program for the Integration of Research and Teaching","award":["22JZD011"],"award-info":[{"award-number":["22JZD011"]}]},{"name":"Key Project of Major Research Projects of Philosophy and Social Sciences of the Ministry of Education","award":["202209"],"award-info":[{"award-number":["202209"]}]},{"name":"Key Project of Major Research Projects of Philosophy and Social Sciences of the Ministry of Education","award":["22JZD011"],"award-info":[{"award-number":["22JZD011"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>With the deepening of the diversification and openness of financial systems, financial vulnerability, as an endogenous attribute of financial systems, becomes an important measurement of financial security. Based on a network analysis, we introduce a network curvature indicator improved by Copula entropy as an innovative metric of financial vulnerability. Compared with the previous network curvature analysis method, the CE-based curvature proposed in this paper can measure market vulnerability and systematic risk with significant advantages.<\/jats:p>","DOI":"10.3390\/e26030192","type":"journal-article","created":{"date-parts":[[2024,2,23]],"date-time":"2024-02-23T06:07:39Z","timestamp":1708668459000},"page":"192","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Vulnerability Analysis Method Based on Network and Copula Entropy"],"prefix":"10.3390","volume":"26","author":[{"given":"Mengyuan","family":"Chen","sequence":"first","affiliation":[{"name":"School of Finance, Central University of Finance and Economics, Beijing 102206, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jilan","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Finance, Central University of Finance and Economics, Beijing 102206, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ning","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Finance, Central University of Finance and Economics, Beijing 102206, China"},{"name":"China Fintech Research Center, Central University of Finance and Economics, Beijing 102206, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yichao","family":"Zheng","sequence":"additional","affiliation":[{"name":"School of Finance, Central University of Finance and Economics, Beijing 102206, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,23]]},"reference":[{"key":"ref_1","unstructured":"Minsky, H.P. (1982). The Financial-Instability Hypothesis: Capitalist Processes and the Behavior of the Economy, Cambridge University Press."},{"key":"ref_2","first-page":"41","article-title":"On Financial Fragility","volume":"3","author":"Huang","year":"2001","journal-title":"Financ. Res."},{"key":"ref_3","first-page":"77","article-title":"Theoretical Analysis of Banking Crisis Transmission Mechanism","volume":"4","author":"Yang","year":"2008","journal-title":"China Circ. Econ."},{"key":"ref_4","first-page":"69","article-title":"The Vulnerability Test of my country\u2019s Stock Market in the Process of Emerging Market Change","volume":"5","author":"Liu","year":"2013","journal-title":"Shanghai Econ. Res."},{"key":"ref_5","first-page":"4","article-title":"Research on the Relationship between Financial Fragility, Risk Management System and Financial Futures Market","volume":"3","author":"Chang","year":"2019","journal-title":"China Secur. Futures"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2307\/3867328","article-title":"Leading indicators of currency crises","volume":"45","author":"Kaminsky","year":"1998","journal-title":"Staff Pap."},{"key":"ref_7","first-page":"18","article-title":"Research on dynamic risk measurement of financial markets under asymmetric structure","volume":"24","author":"Lin","year":"2012","journal-title":"Manag. Rev."},{"key":"ref_8","first-page":"299","article-title":"Construction and analysis of my country\u2019s financial vulnerability indicators since the subprime mortgage crisis","volume":"1","author":"Wang","year":"2017","journal-title":"Mod. Econ. Inf."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"736","DOI":"10.1016\/j.jbusres.2019.10.043","article-title":"Financial crises: Uncovering self-organized patterns and predicting stock markets instability","volume":"129","author":"Spelta","year":"2021","journal-title":"J. Bus. Res."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Cerqueti, R., Rotundo, G., and Ausloos, M. (2018). Investigating the configurations in cross-shareholding: A joint copula-entropy approach. Entropy, 20.","DOI":"10.3390\/e20020134"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/s11403-022-00364-7","article-title":"The weighted cross-shareholding complex network: A copula approach to concentration and control in financial markets","volume":"18","author":"Cerqueti","year":"2023","journal-title":"J. Econ. Interact. Coord."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/s10479-016-2362-6","article-title":"Financial contagion behavior analysis based on complex network approach","volume":"268","author":"Zhu","year":"2018","journal-title":"Ann. Oper. Res."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Chen, X., Hao, A., and Li, Y. (2020). The impact of financial contagion on real economy-An empirical research based on combination of complex network technology and spatial econometrics model. PLoS ONE, 15.","DOI":"10.1371\/journal.pone.0229913"},{"key":"ref_14","first-page":"1","article-title":"Debtrank: Too central to fail?","volume":"2","author":"Battiston","year":"2012","journal-title":"financial networks, the fed and systemic risk. Sci. Rep."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1038\/nature09659","article-title":"Systemic risk in banking ecosystems","volume":"469","author":"Haldane","year":"2011","journal-title":"Nature"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/s100510050929","article-title":"Hierarchical structure in financial markets","volume":"11","author":"Mantegna","year":"1999","journal-title":"Eur. Phys. J. B-Condens. Matter Complex Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"056110","DOI":"10.1103\/PhysRevE.68.056110","article-title":"Dynamics of market correlations: Taxonomy and portfolio analysis","volume":"68","author":"Onnela","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_18","first-page":"42","article-title":"Review of Financial Market Interconnection and Risk Communication: From Time Series to Complex Network","volume":"37","author":"Niu","year":"2018","journal-title":"Invest. Res."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"784","DOI":"10.1016\/j.jbusres.2019.09.046","article-title":"Complexity, interconnectedness and stability: New perspectives applied to the European banking system","volume":"129","author":"Chabot","year":"2021","journal-title":"J. Bus. Res."},{"key":"ref_20","unstructured":"Sandhu, R., Georgiou, T., and Tannenbaum, A. (2015). Market fragility, systemic risk, and Ricci curvature. arXiv."},{"key":"ref_21","unstructured":"Ma, J., and Sun, Z. (2008). Dependence Structure Estimation via Copula. arXiv."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1755","DOI":"10.1007\/s00477-013-0838-3","article-title":"Copula entropy coupled with artificial neural network for rainfall\u2013runoff simulation","volume":"28","author":"Chen","year":"2014","journal-title":"Stoch. Environ. Res. Risk Assess."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2253","DOI":"10.3390\/e17042253","article-title":"Integrating Entropy and Copula Theories for Hydrologic Modeling and Analysis","volume":"17","author":"Hao","year":"2015","journal-title":"Entropy"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1016\/j.jhydrol.2017.09.046","article-title":"A two-phase copula entropy-based multiobjective optimization approach to hydrometeorological gauge network design","volume":"555","author":"Xu","year":"2017","journal-title":"J. Hydrol."},{"key":"ref_25","first-page":"229","article-title":"Fonctions de repartition an dimensions et leurs marges","volume":"8","author":"Sklar","year":"1959","journal-title":"Publ. Inst. Statist. Univ. Paris"},{"key":"ref_26","first-page":"112","article-title":"Correlation study of portfolio risk in financial markets","volume":"27","author":"Li","year":"2007","journal-title":"Syst. Eng. Theory Pract."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"827","DOI":"10.1016\/j.jimonfin.2006.04.007","article-title":"The copula-garch model of conditional dependencies: An international stock market application","volume":"25","author":"Jondeau","year":"2006","journal-title":"J. Int. Money Financ."},{"key":"ref_28","first-page":"5","article-title":"Empirical study on the volatility asymmetry of China\u2019s stock market under the financial crisis","volume":"30","author":"Wen","year":"2010","journal-title":"J. Daqing Norm. Univ."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"e00173","DOI":"10.1016\/j.jeca.2020.e00173","article-title":"Bayesian sequential stock return prediction through copulas","volume":"22","author":"Frey","year":"2020","journal-title":"J. Econ. Asymmetries"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/S1007-0214(11)70008-6","article-title":"Mutual information is copula entropy","volume":"16","author":"Ma","year":"2011","journal-title":"Tsinghua Sci. Technol."},{"key":"ref_31","first-page":"01093","article-title":"Research on LSTM drought prediction model based on drive analysis","volume":"360","author":"Li","year":"2022","journal-title":"J. Math. Pract. Theory"},{"key":"ref_32","first-page":"88","article-title":"The american journal of psychology","volume":"15","author":"Spearman","year":"1904","journal-title":"Am. J. Psychol."},{"key":"ref_33","unstructured":"Ma, J. (2021). Copula Entropy: Theory and Applications. ChinaXiv, Available online: https:\/\/chinaxiv.org\/abs\/202105.00070."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"066138","DOI":"10.1103\/PhysRevE.69.066138","article-title":"Estimating mutual information","volume":"69","author":"Kraskov","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_35","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_36","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1111\/j.1540-6261.1990.tb03695.x","article-title":"Positive feedback investment strategies and destabilizing rational speculation","volume":"45","author":"Shleifer","year":"1990","journal-title":"J. Financ."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1903","DOI":"10.1111\/jofi.12286","article-title":"Arbitrage asymmetry and the idiosyncratic volatility puzzle","volume":"70","author":"Stambaugh","year":"2015","journal-title":"J. Financ."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"e1501495","DOI":"10.1126\/sciadv.1501495","article-title":"Ricci curvature: An economic indicator for market fragility and systemic risk","volume":"2","author":"Sandhu","year":"2016","journal-title":"Sci. Adv."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1086\/260061","article-title":"Risk, Return, and Equilibrium: Empirical Tests","volume":"81","author":"Fama","year":"1973","journal-title":"J. Political Econ."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"063206","DOI":"10.1088\/1742-5468\/2016\/06\/063206","article-title":"Forman curvature for complex networks","volume":"6","author":"Sreejith","year":"2016","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"S106","DOI":"10.1017\/nws.2020.42","article-title":"A simple differential geometry for complex networks","volume":"9","author":"Saucan","year":"2021","journal-title":"Netw. Sci."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Saucan, E., Samal, A., and Jost, J. (2019, January 10\u201312). A simple differential geometry for networks and its generalizations. Proceedings of the Eighth International Conference on Complex Networks and Their Applications, Lisbon, Portugal.","DOI":"10.1007\/978-3-030-36687-2_78"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/3\/192\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:03:50Z","timestamp":1760105030000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/3\/192"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,23]]},"references-count":42,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["e26030192"],"URL":"https:\/\/doi.org\/10.3390\/e26030192","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,2,23]]}}}