{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,8]],"date-time":"2026-08-08T04:44:21Z","timestamp":1786164261125,"version":"build-2736575974"},"reference-count":58,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2026,8,6]],"date-time":"2026-08-06T00:00:00Z","timestamp":1785974400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12301602"],"award-info":[{"award-number":["12301602"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"award":["12301602"],"award-info":[{"award-number":["12301602"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012456","name":"National Social Science Foundation of China","doi-asserted-by":"crossref","award":["22FJLB33"],"award-info":[{"award-number":["22FJLB33"]}],"id":[{"id":"10.13039\/501100012456","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Quantifying the direction, strength and temporal clustering of cross-border financial risk contagion calls for methods that go beyond linear correlation. We suggest a two-layer framework that brings together transfer entropy and a multivariate Hawkes self-exciting point process on a time-varying, directed, weighted network. In the first layer, one-to-one transfer entropies of sovereign credit default swap spreads are estimated with a bias-corrected k nearest neighbour estimator, and this step detects nonlinear and directional information transfer between spreads. The second layer is a multivariate Hawkes process that models how extreme loss events arrive and mutually excite one another across countries, and it gives an excitation intensity matrix, encoding the way a tail event in one country raises the likelihood of an instantaneous hazard occurring in another. By merging these two layers, we obtain a composite, directed, weighted adjacency matrix in which the weights of the edges reflect both information flow and event clustering. We introduce a network-level contagion intensity index and split it into direct, indirect and feedback terms using the graph Laplacian spectrum. Von Neumann graph entropy together with the spectral gap ratio serve as entropy-based measures of the complexity and fragility of the evolving network. We validate the choice of Shannon-type entropy through a Tsallis q-sensitivity analysis, and we verify the nonlinear dependence structure of the data using BDS tests and maximal Lyapunov exponent estimates. Three empirical findings emerge from analysing 20 sovereign CDS markets from January 2015 to December 2025: (i) directional risk spillover signals derived based on transfer entropy are more timely than those derived from variance decomposition; (ii) the Hawkes excitation component amplifies measured contagion intensity by 35 to 58 percent during the COVID-19 shock and the 2022 European energy crisis relative to a transfer-entropy-only baseline; (iii) von Neumann graph entropy reaches historically extreme values 7 to 12 trading days before the peak drawdown in a Global Sovereign Bond Index. These results hold across rolling window lengths, significance thresholds, alternative entropy functionals and alternative Hawkes kernels.<\/jats:p>","DOI":"10.3390\/e28080887","type":"journal-article","created":{"date-parts":[[2026,8,6]],"date-time":"2026-08-06T11:43:02Z","timestamp":1786016582000},"page":"887","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Transfer-Entropy- and Hawkes-Process-Driven Dynamic Measurement of Cross-Border Financial Risk Contagion in Directed, Weighted Networks"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7019-2552","authenticated-orcid":false,"given":"Lei","family":"An","sequence":"first","affiliation":[{"name":"School of Finance, Nanjing Audit University, Nanjing 211815, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jinping","family":"Dai","sequence":"additional","affiliation":[{"name":"School of Economics, Nankai University, Tianjin 300071, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1016\/j.jeconom.2014.04.012","article-title":"On the network topology of variance decompositions: Measuring the connectedness of financial firms","volume":"182","author":"Diebold","year":"2014","journal-title":"J. Econom."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.jeconom.2023.01.021","article-title":"On the past, present, and future of the Diebold\u2013Yilmaz approach to dynamic network connectedness","volume":"234","author":"Diebold","year":"2023","journal-title":"J. Econom."},{"key":"ref_3","first-page":"271","article-title":"Measuring the frequency dynamics of financial connectedness and systemic risk","volume":"16","year":"2018","journal-title":"J. Financ. Econom."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/jae.2585","article-title":"Estimating global bank network connectedness","volume":"33","author":"Demirer","year":"2018","journal-title":"J. Appl. Econom."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"105761","DOI":"10.1016\/j.jbankfin.2020.105761","article-title":"How connected is the global sovereign credit risk network?","volume":"113","author":"Bostanci","year":"2020","journal-title":"J. Bank. Financ."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1198\/073500102288618487","article-title":"Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models","volume":"20","author":"Engle","year":"2002","journal-title":"J. Bus. Econ. Stat."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1080\/17446540500428843","article-title":"Flexible dynamic conditional correlation multivariate GARCH models for asset allocation","volume":"2","author":"Billio","year":"2006","journal-title":"Appl. Financ. Econ. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"100680","DOI":"10.1016\/j.mulfin.2021.100680","article-title":"Dynamic measures of asymmetric & pairwise connectedness within an optimal currency area: Evidence from the ERM I system","volume":"60","author":"Gabauer","year":"2021","journal-title":"J. Multinatl. Financ. Manag."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1016\/j.jeconom.2005.01.013","article-title":"Regime switching for dynamic correlations","volume":"131","author":"Pelletier","year":"2006","journal-title":"J. Econom."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1103\/PhysRevLett.85.461","article-title":"Measuring information transfer","volume":"85","author":"Schreiber","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","first-page":"85","article-title":"Using transfer entropy to measure information flows between financial markets","volume":"17","author":"Dimpfl","year":"2013","journal-title":"Stud. Nonlinear Dyn. Econom."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1016\/j.physa.2018.10.048","article-title":"Group transfer entropy with an application to cryptocurrencies","volume":"516","author":"Dimpfl","year":"2019","journal-title":"Phys. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2971","DOI":"10.1016\/j.physa.2011.12.064","article-title":"R\u00e9nyi\u2019s information transfer between financial time series","volume":"391","author":"Jizba","year":"2012","journal-title":"Phys. A"},{"key":"ref_14","unstructured":"Khalilian, P., Golestani, A.N., Eslamifar, M., Firouzjaee, M.T., and Firouzjaee, J.T. (2025). Mapping crisis-driven market dynamics: A transfer entropy and Kramers\u2013Moyal approach to financial networks. arXiv."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Scagliarini, T., Faes, L., Marinazzo, D., Stramaglia, S., and Mantegna, R.N. (2020). Synergistic information transfer in the global system of financial markets. Entropy, 22.","DOI":"10.3390\/e22091000"},{"key":"ref_16","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_17","doi-asserted-by":"crossref","unstructured":"Bossomaier, T., Barnett, L., Harr\u00e9, M., and Lizier, J.T. (2016). An Introduction to Transfer Entropy: Information Flow in Complex Systems, Springer International Publishing.","DOI":"10.1007\/978-3-319-43222-9"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1093\/biomet\/58.1.83","article-title":"Spectra of some self-exciting and mutually exciting point processes","volume":"58","author":"Hawkes","year":"1971","journal-title":"Biometrika"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1016\/j.jfineco.2015.03.002","article-title":"Modeling financial contagion using mutually exciting jump processes","volume":"117","author":"Laeven","year":"2015","journal-title":"J. Financ. Econ."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1550005","DOI":"10.1142\/S2382626615500057","article-title":"Hawkes processes in finance","volume":"1","author":"Bacry","year":"2015","journal-title":"Mark. Microstruct. Liq."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"056108","DOI":"10.1103\/PhysRevE.85.056108","article-title":"Quantifying reflexivity in financial markets: Toward a prediction of flash crashes","volume":"85","author":"Filimonov","year":"2012","journal-title":"Phys. Rev. E"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"062807","DOI":"10.1103\/PhysRevE.90.062807","article-title":"Branching-ratio approximation for the self-exciting Hawkes process","volume":"90","author":"Hardiman","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1239\/jap\/1318940477","article-title":"Multivariate Hawkes processes: An application to financial data","volume":"48A","author":"Embrechts","year":"2011","journal-title":"J. Appl. Probab."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"999","DOI":"10.1080\/14697688.2016.1260759","article-title":"The role of volume in order book dynamics: A multivariate Hawkes process analysis","volume":"17","author":"Rambaldi","year":"2017","journal-title":"Quant. Financ."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1080\/14697688.2017.1403142","article-title":"High-dimensional Hawkes processes for limit order books: Modelling, empirical analysis, and numerical calibration","volume":"18","author":"Lu","year":"2018","journal-title":"Quant. Financ."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1016\/j.jfs.2017.12.004","article-title":"Measuring sovereign contagion in Europe","volume":"34","author":"Caporin","year":"2018","journal-title":"J. Financ. Stab."},{"key":"ref_27","unstructured":"Zuo, S., Jiang, H., Li, Z., Zhao, T., and Zha, H. (2020, January 13\u201318). Transformer Hawkes process. Proceedings of the 37th International Conference on Machine Learning (ICML), Virtual."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1016\/j.jfineco.2011.12.010","article-title":"Econometric measures of connectedness and systemic risk in the finance and insurance sectors","volume":"104","author":"Billio","year":"2012","journal-title":"J. Financ. Econ."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1038\/srep00541","article-title":"DebtRank: Too central to fail? Financial networks, the FED and systemic risk","volume":"2","author":"Battiston","year":"2012","journal-title":"Sci. Rep."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1705","DOI":"10.1257\/aer.20120555","article-title":"CoVaR","volume":"106","author":"Adrian","year":"2016","journal-title":"Am. Econ. Rev."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1093\/rfs\/hhw060","article-title":"SRISK: A conditional capital shortfall measure of systemic risk","volume":"30","author":"Brownlees","year":"2017","journal-title":"Rev. Financ. Stud."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"564","DOI":"10.1257\/aer.20130456","article-title":"Systemic risk and stability in financial networks","volume":"105","author":"Acemoglu","year":"2015","journal-title":"Am. Econ. Rev."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/j.jfs.2015.08.001","article-title":"The multi-layer network nature of systemic risk and its implications for the costs of financial crises","volume":"20","author":"Poledna","year":"2015","journal-title":"J. Financ. Stab."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1417","DOI":"10.1080\/14697688.2016.1272762","article-title":"Extreme risk spillover network: Application to financial institutions","volume":"17","author":"Wang","year":"2017","journal-title":"Quant. Financ."},{"key":"ref_35","unstructured":"Gonon, L., Meyer-Brandis, T., and Weber, N. (2024). Computing systemic risk measures with graph neural networks. arXiv."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1016\/j.jeconom.2016.02.013","article-title":"TENET: Tail-Event driven NETwork risk","volume":"192","author":"Wang","year":"2016","journal-title":"J. Econom."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"888","DOI":"10.1038\/srep00888","article-title":"Changes in cross-correlations as an indicator for systemic risk","volume":"2","author":"Zheng","year":"2012","journal-title":"Sci. Rep."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ememar.2017.12.001","article-title":"Interconnectedness and systemic risk of China\u2019s financial institutions","volume":"35","author":"Wang","year":"2018","journal-title":"Emerg. Mark. Rev."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Giudici, P., Pagnottoni, P., and Polinesi, G. (2020). Network models to enhance automated cryptocurrency portfolio management. Front. Artif. Intell., 3.","DOI":"10.3389\/frai.2020.00022"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/j.pacfin.2018.05.009","article-title":"Measuring systemic risk of the banking industry in China: A DCC-MIDAS-t approach","volume":"51","author":"Xu","year":"2018","journal-title":"Pac.-Basin Financ. J."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"733","DOI":"10.1016\/j.ijforecast.2018.05.004","article-title":"Forecasting risk with Markov-switching GARCH models: A large-scale performance study","volume":"34","author":"Ardia","year":"2018","journal-title":"Int. J. Forecast."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1111\/j.1468-2354.2006.00387.x","article-title":"Modelling asymmetric exchange rate dependence","volume":"47","author":"Patton","year":"2006","journal-title":"Int. Econ. Rev."},{"key":"ref_43","first-page":"041062","article-title":"Spectral entropies as information-theoretic tools for complex network comparison","volume":"6","author":"Biamonte","year":"2016","journal-title":"Phys. Rev. X"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"032810","DOI":"10.1103\/PhysRevE.92.032810","article-title":"Thermodynamic characterization of networks using graph polynomials","volume":"92","author":"Ye","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Pele, D.T., Lazar, E., and Dufour, A. (2017). Information entropy and measures of market risk. Entropy, 19.","DOI":"10.3390\/e19050226"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"101646","DOI":"10.1016\/j.irfa.2020.101646","article-title":"Return connectedness across asset classes around the COVID-19 outbreak","volume":"73","author":"Bouri","year":"2021","journal-title":"Int. Rev. Financ. Anal."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann\u2013Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. Stat. Phys."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1081","DOI":"10.21105\/joss.01081","article-title":"IDTxl: The information dynamics toolkit xl: A Python package for the efficient analysis of multivariate information dynamics in networks","volume":"4","author":"Wollstadt","year":"2019","journal-title":"J. Open Source Softw."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"614","DOI":"10.1198\/016214508000000148","article-title":"Estimation of space-time branching process models in seismology using an EM-type algorithm","volume":"103","author":"Veen","year":"2008","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"046117","DOI":"10.1103\/PhysRevE.84.046117","article-title":"Randomizing world trade. I. A binary network analysis","volume":"84","author":"Squartini","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"101528","DOI":"10.1016\/j.frl.2020.101528","article-title":"Financial markets under the global pandemic of COVID-19","volume":"36","author":"Zhang","year":"2020","journal-title":"Financ. Res. Lett."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"125988","DOI":"10.1016\/j.physa.2021.125988","article-title":"Analysis of stock market efficiency during crisis periods in the US stock market: Differences between the global financial crisis and COVID-19 pandemic","volume":"574","author":"Choi","year":"2021","journal-title":"Phys. A"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1080\/07474939608800353","article-title":"A test for independence based on the correlation dimension","volume":"15","author":"Broock","year":"1996","journal-title":"Econom. Rev."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/S0304-4076(02)00202-6","article-title":"Testing for a unit root in the nonlinear STAR framework","volume":"112","author":"Kapetanios","year":"2003","journal-title":"J. Econom."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/0304-4076(92)90104-Y","article-title":"Testing the null hypothesis of stationarity against the alternative of a unit root","volume":"54","author":"Kwiatkowski","year":"1992","journal-title":"J. Econom."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1111\/j.1467-9892.1983.tb00371.x","article-title":"The estimation and application of long memory time series models","volume":"4","author":"Geweke","year":"1983","journal-title":"J. Time Ser. Anal."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"1890","DOI":"10.1214\/009053605000000309","article-title":"Exact local Whittle estimation of fractional integration","volume":"33","author":"Shimotsu","year":"2005","journal-title":"Ann. Stat."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/0167-2789(93)90009-P","article-title":"A practical method for calculating largest Lyapunov exponents from small data sets","volume":"65","author":"Rosenstein","year":"1993","journal-title":"Physica D"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/8\/887\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,8]],"date-time":"2026-08-08T04:27:29Z","timestamp":1786163249000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/8\/887"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,6]]},"references-count":58,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2026,8]]}},"alternative-id":["e28080887"],"URL":"https:\/\/doi.org\/10.3390\/e28080887","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,6]]}}}