{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:03:31Z","timestamp":1760238211677,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,7,29]],"date-time":"2022-07-29T00:00:00Z","timestamp":1659052800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["71991485","71991480","72088101"],"award-info":[{"award-number":["71991485","71991480","72088101"]}]},{"name":"Basic Science Center Project for National Natural Science Foundation of China","award":["71991485","71991480","72088101"],"award-info":[{"award-number":["71991485","71991480","72088101"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The input\u2013output (IO) network is the quantitative description of an IO-based economy in which nodes represent industries and edges connecting nodes represent the economic connection between industries. Robustness refers to the ability of tolerating perturbations that might affect the system\u2019s functional body. There is both practical and theoretical significance to explore the robustness of the IO network for economic development. In this paper, we probe the robustness of the Chinese IO network based on the relative entropy of the probability distribution of network parameters (node degree, strongest path betweenness, downstream closeness and upstream closeness) under random node or edge failure and intentional node or edge attack. It is found that the Chinese IO network shows relatively weak robustness when it is under intentional attack, but relatively strong robustness when it is under random failure. Our experiment also verifies the applicability and effectiveness of the relative entropy model in measuring the robustness of the IO network.<\/jats:p>","DOI":"10.3390\/e24081043","type":"journal-article","created":{"date-parts":[[2022,7,31]],"date-time":"2022-07-31T23:37:29Z","timestamp":1659310649000},"page":"1043","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Research on the Robustness of the Chinese Input\u2013Output Network Based on Relative Entropy Theory"],"prefix":"10.3390","volume":"24","author":[{"given":"Weidong","family":"Li","sequence":"first","affiliation":[{"name":"Institute of Mineral Resources, Chinese Academy of Geological Sciences, Beijing 100037, China"},{"name":"Research Center for Strategy of Global Mineral Resources, Chinese Academy of Geological Sciences, Beijing 100037, China"}]},{"given":"Anjian","family":"Wang","sequence":"additional","affiliation":[{"name":"Institute of Mineral Resources, Chinese Academy of Geological Sciences, Beijing 100037, China"},{"name":"Research Center for Strategy of Global Mineral Resources, Chinese Academy of Geological Sciences, Beijing 100037, China"}]},{"given":"Wanli","family":"Xing","sequence":"additional","affiliation":[{"name":"Development and Research Center, Chinese Geological Survey, Beijing 100037, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"105","DOI":"10.2307\/1927837","article-title":"Quantitative Input and Output Relations in the Economic Systems of the United States","volume":"18","author":"Leontief","year":"1936","journal-title":"Rev. Econ. Stat."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"6427","DOI":"10.1016\/j.physa.2013.07.063","article-title":"Network structure of inter-industry flows","volume":"392","author":"James","year":"2013","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"046127","DOI":"10.1103\/PhysRevE.83.046127","article-title":"Vertex centralities in input\u2013output networks reveal the structure of modern economies","volume":"83","author":"Blochl","year":"2011","journal-title":"Phys. Rev. E Stat. Nonlinear Soft Matter Phys."},{"key":"ref_4","unstructured":"McNerney, J. (2009). Network Properties of Economic-Input Output Networks, lnternational lnstitute for Applied Systems Analysis."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"121178","DOI":"10.1016\/j.physa.2019.121178","article-title":"Input\u2013output networks offer new insights of economic structure","volume":"527","author":"Xu","year":"2019","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"604","DOI":"10.1108\/JES-05-2020-0225","article-title":"The asymmetric Brazilian input\u2013output network","volume":"48","author":"Matsushita","year":"2021","journal-title":"J. Econ. Stud."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Cerina, F., Zhu, Z., Chessa, A., and Riccaboni, M. (2015). World Input-Output Network. PLoS ONE, 10.","DOI":"10.1371\/journal.pone.0134025"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1007\/s11403-015-0171-7","article-title":"Network analysis of inter-sectoral relationships and key sectors in the Greek economy","volume":"12","author":"Theodore","year":"2017","journal-title":"J. Econ. Interact. Coord."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.physa.2017.02.068","article-title":"Structure analysis and core community detection of embodied resources networks among regional industries","volume":"479","author":"He","year":"2017","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"102590","DOI":"10.1016\/j.resourpol.2022.102590","article-title":"The role of mineral-related industries in Chinese industrial pattern","volume":"76","author":"Li","year":"2022","journal-title":"Resour. Policy"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1977","DOI":"10.3982\/ECTA9623","article-title":"The Network Origins of Aggregate Fluctuations","volume":"80","author":"Acemoglu","year":"2012","journal-title":"Econometrica"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"126993","DOI":"10.1016\/j.physa.2022.126993","article-title":"An empirical analysis of the global input\u2013output network and its evolution","volume":"594","author":"Grazzini","year":"2022","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1038\/35019019","article-title":"Error and attack tolerance of complex networks","volume":"406","author":"Albert","year":"2000","journal-title":"Nature"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"056109","DOI":"10.1103\/PhysRevE.65.056109","article-title":"Attack vulnerability of complex networks","volume":"65","author":"Holme","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3838","DOI":"10.1073\/pnas.1009440108","article-title":"Mitigation of malicious attacks on networks","volume":"108","author":"Schneider","year":"2011","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Sen, A., Mazumder, A., Banerjee, J., Das, A., and Compton, R. (May, January 27). Identification of K most vulnerable nodes in multi-layered network using a new model of interdependency. Proceedings of the 2014 IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS), Toronto, ON, Canada.","DOI":"10.1109\/INFCOMW.2014.6849338"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"6043","DOI":"10.1038\/s41467-020-19841-3","article-title":"Robustness and lethality in multilayer biological molecular networks","volume":"11","author":"Liu","year":"2020","journal-title":"Nat. Commun."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1889","DOI":"10.1016\/j.jedc.2013.04.009","article-title":"Stability of the World Trade Web over time\u2014An extinction analysis","volume":"37","author":"Foti","year":"2013","journal-title":"J. Econ. Dyn. Control"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1138","DOI":"10.1073\/pnas.1523412113","article-title":"Breakdown of interdependent directed networks","volume":"113","author":"Liu","year":"2016","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"025101","DOI":"10.1103\/PhysRevE.64.025101","article-title":"Giant strongly connected component of directed networks","volume":"64","author":"Dorogovtsev","year":"2001","journal-title":"Phys. Rev. E Stat. Nonlin. Soft Matter Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.physd.2013.03.002","article-title":"A quantitative method for determining the robustness of complex networks","volume":"253","author":"Qin","year":"2013","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1214\/aoms\/1177729694","article-title":"On Information and Sufficiency","volume":"22","author":"Kullback","year":"1951","journal-title":"Ann. Math. Stat."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"107025","DOI":"10.1016\/j.knosys.2021.107025","article-title":"Layered Relative Entropy Policy Search","volume":"223","author":"Izadi","year":"2021","journal-title":"Knowl.-Based Syst."},{"key":"ref_24","first-page":"1567","article-title":"An improved harmony search algorithm for solving optimization problems","volume":"188","author":"Mahdavi","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/j.chaos.2017.08.010","article-title":"A new method to identify influential nodes based on relative entropy","volume":"104","author":"Fei","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1219","DOI":"10.21629\/JSEE.2016.06.10","article-title":"Integrated evaluation approach for node importance of complex networks based on relative entropy","volume":"27","author":"Chen","year":"2016","journal-title":"J. Syst. Eng. Electron."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"104867","DOI":"10.1016\/j.cnsns.2019.104867","article-title":"Node similarity measuring in complex networks with relative entropy","volume":"78","author":"Wen","year":"2019","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1016\/j.physa.2017.09.042","article-title":"Measure the structure similarity of nodes in complex networks based on relative entropy","volume":"491","author":"Zhang","year":"2018","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/j.physa.2017.02.054","article-title":"Efficient calculation of the robustness measure R for complex networks","volume":"478","author":"Hong","year":"2017","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"111","DOI":"10.2307\/2232220","article-title":"Structural path analysis and multiplier decomposition within a social accounting matrix framework","volume":"94","author":"Defourny","year":"1984","journal-title":"Econ. J."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/8\/1043\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:58:57Z","timestamp":1760140737000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/8\/1043"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,29]]},"references-count":30,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["e24081043"],"URL":"https:\/\/doi.org\/10.3390\/e24081043","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,7,29]]}}}