{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T06:13:07Z","timestamp":1778220787519,"version":"3.51.4"},"reference-count":48,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,7,15]],"date-time":"2020-07-15T00:00:00Z","timestamp":1594771200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012325","name":"National Office for Philosophy and Social Sciences","doi-asserted-by":"publisher","award":["17BTJ017"],"award-info":[{"award-number":["17BTJ017"]}],"id":[{"id":"10.13039\/501100012325","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11571368"],"award-info":[{"award-number":["11571368"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Wuhan Institute of Technology supporting fund","award":["1234"],"award-info":[{"award-number":["1234"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Complex network is a powerful tool to discover important information from various types of big data. Although substantial studies have been conducted for the development of stock relation networks, correlation coefficient is dominantly used to measure the relationship between stock pairs. Information theory is much less discussed for this important topic, though mutual information is able to measure nonlinear pairwise relationship. In this work we propose to use part mutual information for developing stock networks. The path-consistency algorithm is used to filter out redundant relationships. Using the Australian stock market data, we develop four stock relation networks using different orders of part mutual information. Compared with the widely used planar maximally filtered graph (PMFG), we can generate networks with cliques of large size. In addition, the large cliques show consistency with the structure of industrial sectors. We also analyze the connectivity and degree distributions of the generated networks. Analysis results suggest that the proposed method is an effective approach to develop stock relation networks using information theory.<\/jats:p>","DOI":"10.3390\/e22070773","type":"journal-article","created":{"date-parts":[[2020,7,15]],"date-time":"2020-07-15T10:35:18Z","timestamp":1594809318000},"page":"773","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":34,"title":["Development of Stock Networks Using Part Mutual Information and Australian Stock Market Data"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5833-4607","authenticated-orcid":false,"given":"Yan","family":"Yan","sequence":"first","affiliation":[{"name":"School of Mathematics and Physics, Wuhan Institute of Technology, Wuhan 430205, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Boyao","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Mathematics, Monash University, Melbourne, VIC 3800, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6191-0209","authenticated-orcid":false,"given":"Tianhai","family":"Tian","sequence":"additional","affiliation":[{"name":"School of Mathematics, Monash University, Melbourne, VIC 3800, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hu","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1103\/RevModPhys.74.47","article-title":"Statistical mechanics of complex networks","volume":"74","author":"Albert","year":"2001","journal-title":"Rev. Mod. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1126\/science.aah3449","article-title":"Network analytics in the age of big data","volume":"353","author":"Przulj","year":"2016","journal-title":"Science"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1243089","DOI":"10.1126\/science.1243089","article-title":"Economics in the age of big data","volume":"346","author":"Einav","year":"2014","journal-title":"Science"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1126\/science.1215842","article-title":"Identifying influential and susceptible members of social networks","volume":"337","author":"Aral","year":"2012","journal-title":"Science"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1581","DOI":"10.1016\/j.cell.2018.05.015","article-title":"Next-generation machine learning for biological networks","volume":"173","author":"Camacho","year":"2018","journal-title":"Cell"},{"key":"ref_6","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_7","doi-asserted-by":"crossref","first-page":"198301","DOI":"10.1103\/PhysRevLett.119.198301","article-title":"Closed-loop control of complex networks: A trade-off between time and energy","volume":"119","author":"Sun","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1038\/s42254-018-0002-6","article-title":"The statistical physics of real-world networks","volume":"1","author":"Cimini","year":"2019","journal-title":"Nat. Rev. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1016\/j.jebo.2010.01.004","article-title":"Correlation, hierarchies and networks in financial markets","volume":"75","author":"Tumminello","year":"2008","journal-title":"J. Econ. Behav. Organ."},{"key":"ref_10","first-page":"187","article-title":"Correlation of financial markets in times of crisis","volume":"391","author":"Junior","year":"2011","journal-title":"Physica A"},{"key":"ref_11","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"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"352","DOI":"10.1103\/PhysRevE.68.046130","article-title":"Topology of correlation based minimal spanning trees in real and model markets","volume":"68","author":"Bonanno","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_13","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_14","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1140\/epjb\/e2006-00414-4","article-title":"Correlation based networks of equity returns sampled at different time horizons","volume":"55","author":"Tumminello","year":"2007","journal-title":"Eur. Phys. J. B"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1016\/j.jempfin.2010.04.008","article-title":"A network perspective of the stock market","volume":"17","author":"Chi","year":"2010","journal-title":"J. Empir. Financ."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1007\/s10614-015-9481-z","article-title":"Analysis of Correlation Based Networks Representing DAX 30 Stock Price Returns","volume":"47","author":"Birch","year":"2016","journal-title":"Comput. Econ."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1016\/j.physa.2014.07.067","article-title":"Dynamic spanning trees in stock market networks: The case of Asia-Pacific","volume":"414","author":"Sensoy","year":"2014","journal-title":"Physica A"},{"key":"ref_18","first-page":"161","article-title":"Network Filtering for Big Data: Triangulated Maximally Filtered Graph","volume":"5","author":"Massara","year":"2017","journal-title":"J. Complex Netw."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"41379","DOI":"10.1038\/srep41379","article-title":"Topological characteristics of the hong kong stock market: A test-based p-threshold approach to understanding network complexity","volume":"7","author":"Xu","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/j.physa.2014.01.011","article-title":"Cointegration analysis and influence rank?a network approach to global stock markets","volume":"400","author":"Yang","year":"2014","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_21","first-page":"26","article-title":"Sector identification in a set of stock return time series traded at the London Stock Exchange","volume":"36","author":"Coronnello","year":"2005","journal-title":"Acta Phys. Pol. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"026108","DOI":"10.1103\/PhysRevE.84.026108","article-title":"Evolution of worldwide stock markets, correlation structure, and correlation-based graphs","volume":"84","author":"Song","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"654","DOI":"10.1016\/j.physleta.2015.11.015","article-title":"Structure and dynamics of stock market in times of crisis","volume":"380","author":"Zhao","year":"2016","journal-title":"Phys. Lett. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1016\/j.physa.2013.08.053","article-title":"Stock network stability in times of crisis","volume":"393","author":"Heiberger","year":"2014","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_25","unstructured":"Marti, G., Nielsen, F., Binkowski, M., and Donnat, P. (2017). A review of two decades of correlations, hierarchies, networks and clustering in financial markets. arXiv."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1581","DOI":"10.1080\/01605682.2019.1595193","article-title":"Computational approaches and data analytics in financial services: A literature review","volume":"70","author":"Andriosopoulos","year":"2019","journal-title":"J. Oper. Res. Soc."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Kenett, D.Y., Tumminello, M., Madi, A., Gur-Gershgoren, G., Mantegna, R.N., and Ben-Jacob, E. (2010). Dominating clasp of the financial sector revealed by partial correlation analysis of the stock market. PLoS ONE, 5.","DOI":"10.1371\/journal.pone.0015032"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1080\/14697688.2014.946660","article-title":"Partial correlation analysis: Applications for financial markets","volume":"15","author":"Kenett","year":"2015","journal-title":"Quant. Financ."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"S241","DOI":"10.1016\/j.jbankfin.2015.08.034","article-title":"Connecting the dots: Econometric methods for uncovering networks with an application to the australian financial institutions","volume":"61","author":"Anufriev","year":"2015","journal-title":"J. Bank. Financ."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Corso, G., Ferreira, G.M.F., and Levinsohn, T.M. (2020). Mutual information as a general measure of structure in interaction networks. Entropy, 22.","DOI":"10.3390\/e22050528"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Si, S., Wang, B., Liu, X., Yu, C., Ding, C., and Zhao, H. (2019). Brain network modeling based on mutual information and graph theory for predicting the connection mechanism in the progression of Alzheimer?s disease. Entropy, 21.","DOI":"10.3390\/e21030300"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Guo, X., Zhang, H., and Tian, T. (2018). Development of stock correlation networks using mutual information and financial big data. PLoS ONE, 13.","DOI":"10.1371\/journal.pone.0195941"},{"key":"ref_33","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_34","doi-asserted-by":"crossref","first-page":"266","DOI":"10.3390\/jrfm8020266","article-title":"Network analysis of the Shanghai stock exchange based on partial mutual information","volume":"8","author":"Tao","year":"2015","journal-title":"J. Risk Financ. Manag."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"052801","DOI":"10.1103\/PhysRevE.89.052801","article-title":"Networks in financial markets based on the mutual information rate","volume":"89","author":"Fiedor","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2050018","DOI":"10.1142\/S0219477520500182","article-title":"An effective stock classification method via MDS based on modified mutual information distance","volume":"19","author":"Jiang","year":"2020","journal-title":"Fluct. Noise Lett."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1093\/bioinformatics\/btr626","article-title":"Inferring gene regulatory networks from gene expression data by path consistency algorithm based on conditional mutual information","volume":"28","author":"Zhang","year":"2012","journal-title":"Bioinformatics"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"E31","DOI":"10.1093\/nar\/gku1315","article-title":"Conditional mutual inclusive information enables accurate quantification of associations in gene regulatory networks","volume":"43","author":"Zhang","year":"2015","journal-title":"Nucleic Acids Res."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Villaverde, A.F., Ross, J., Moran, F., and Banga, J.R. (2014). MIDER: Network inference with mutual information distance and entropy reduction. PLoS ONE, 9.","DOI":"10.1371\/journal.pone.0096732"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"5130","DOI":"10.1073\/pnas.1522586113","article-title":"Part mutual information for quantifying direct associations in networks","volume":"113","author":"Zhao","year":"2016","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_41","first-page":"613","article-title":"Estimating high-dimensional directed acyclic graphs with the PC-algorithm","volume":"8","author":"Kalisch","year":"2007","journal-title":"J. Mach. Learn. Res."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"773","DOI":"10.1198\/106186008X381927","article-title":"Robustification of the PC-algorithm for directed acyclicgraphs","volume":"17","author":"Kalisch","year":"2008","journal-title":"J. Comput. Graph. Stat."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"S231","DOI":"10.1093\/bioinformatics\/18.suppl_2.S231","article-title":"The mutual information: Detecting and evaluating dependencies between variables","volume":"18","author":"Steuer","year":"2002","journal-title":"Bioinformatics"},{"key":"ref_44","unstructured":"Aste, T. (2020, June 19). MATLAB Central File Exchange. Available online: https:\/\/www.mathworks.com\/matlabcentral\/fileexchange\/27360-pmfg."},{"key":"ref_45","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":"Reka","year":"1999","journal-title":"Science"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1016","DOI":"10.1038\/s41467-019-09038-8","article-title":"Rare and everywhere: Perspectives on scale-free networks","volume":"10","author":"Petter","year":"2019","journal-title":"Nat. Commun."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1038\/30918","article-title":"Collective dynamics of \u2018small-world\u2019 networks","volume":"393","author":"Watts","year":"1998","journal-title":"Nature"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1017\/nws.2017.5","article-title":"How small is it? Comparing indices of small worldliness","volume":"5","author":"Neal","year":"2017","journal-title":"Netw. Sci."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/7\/773\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:48:48Z","timestamp":1760176128000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/7\/773"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,15]]},"references-count":48,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2020,7]]}},"alternative-id":["e22070773"],"URL":"https:\/\/doi.org\/10.3390\/e22070773","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,15]]}}}