{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T23:00:59Z","timestamp":1777590059069,"version":"3.51.4"},"reference-count":50,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100002703","name":"Jiangsu University","doi-asserted-by":"publisher","award":["23JDG010"],"award-info":[{"award-number":["23JDG010"]}],"id":[{"id":"10.13039\/501100002703","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62303198"],"award-info":[{"award-number":["62303198"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012216","name":"Jiangsu University of Science and Technology","doi-asserted-by":"publisher","award":["1052932204"],"award-info":[{"award-number":["1052932204"]}],"id":[{"id":"10.13039\/501100012216","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Communications in Nonlinear Science and Numerical Simulation"],"published-print":{"date-parts":[[2026,1]]},"DOI":"10.1016\/j.cnsns.2025.109369","type":"journal-article","created":{"date-parts":[[2025,9,27]],"date-time":"2025-09-27T04:52:22Z","timestamp":1758948742000},"page":"109369","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":6,"special_numbering":"PD","title":["Local entropy and nonextensivity of networks ensemble"],"prefix":"10.1016","volume":"152","author":[{"given":"Meizhu","family":"Li","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6692-0737","authenticated-orcid":false,"given":"Qi","family":"Zhang","sequence":"additional","affiliation":[]}],"member":"78","reference":[{"key":"10.1016\/j.cnsns.2025.109369_bib0001","series-title":"Networks","author":"Newman","year":"2018"},{"issue":"1","key":"10.1016\/j.cnsns.2025.109369_bib0002","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41467-017-01076-4","article-title":"Dynamics of lineage commitment revealed by single-cell transcriptomics of differentiating embryonic stem cells","volume":"8","author":"Semrau","year":"2017","journal-title":"Nat Commun"},{"issue":"290\u2013297","key":"10.1016\/j.cnsns.2025.109369_bib0003","first-page":"18","article-title":"On random graphs i","volume":"6","author":"ERDdS","year":"1959","journal-title":"Publ Math Debrecen"},{"key":"10.1016\/j.cnsns.2025.109369_bib0004","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.plrev.2017.11.003","article-title":"Network science of biological systems at different scales: a review","volume":"24","author":"Gosak","year":"2018","journal-title":"Phys Life Rev"},{"issue":"2","key":"10.1016\/j.cnsns.2025.109369_bib0005","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/s11299-015-0167-y","article-title":"Network science: a useful tool in economics and finance","volume":"14","author":"Kenett","year":"2015","journal-title":"Mind Soci"},{"key":"10.1016\/j.cnsns.2025.109369_bib0006","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"},{"issue":"10","key":"10.1016\/j.cnsns.2025.109369_bib0007","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":"10.1016\/j.cnsns.2025.109369_bib0008","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2020.08.003","article-title":"Sequential dynamics of complex networks in mind: consciousness and creativity","volume":"883","author":"Rabinovich","year":"2020","journal-title":"Phys Rep"},{"key":"10.1016\/j.cnsns.2025.109369_bib0009","doi-asserted-by":"crossref","DOI":"10.1016\/j.physletb.2022.137147","article-title":"Nonextensive statistical field theory","volume":"830","author":"Carvalho","year":"2022","journal-title":"Phys Lett B"},{"issue":"5","key":"10.1016\/j.cnsns.2025.109369_bib0010","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.72.056138","article-title":"Phase transition in the link weight structure of networks","volume":"72","author":"Van Mieghem","year":"2005","journal-title":"Phys Rev E"},{"issue":"3","key":"10.1016\/j.cnsns.2025.109369_bib0011","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.72.036117","article-title":"Phase transition in random catalytic networks","volume":"72","author":"Hanel","year":"2005","journal-title":"Phys Rev E"},{"issue":"3","key":"10.1016\/j.cnsns.2025.109369_bib0012","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"},{"issue":"4","key":"10.1016\/j.cnsns.2025.109369_bib0013","article-title":"Double percolation phase transition in clustered complex networks","volume":"4","author":"Colomer-de Sim\u00f3n","year":"2014","journal-title":"Phys Rev X"},{"issue":"1","key":"10.1016\/j.cnsns.2025.109369_bib0014","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevLett.105.018101","article-title":"Post-transcriptional regulation of noise in protein distributions during gene expression","volume":"105","author":"Jia","year":"2010","journal-title":"Phys Rev Lett"},{"issue":"24","key":"10.1016\/j.cnsns.2025.109369_bib0015","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevLett.103.240503","article-title":"Entanglement percolation in quantum complex networks","volume":"103","author":"Cuquet","year":"2009","journal-title":"Phys Rev Lett"},{"issue":"24","key":"10.1016\/j.cnsns.2025.109369_bib0016","doi-asserted-by":"crossref","first-page":"5632","DOI":"10.1103\/PhysRevLett.86.5632","article-title":"Bose-Einstein condensation in complex networks","volume":"86","author":"Bianconi","year":"2001","journal-title":"Phys Rev Lett"},{"issue":"2","key":"10.1016\/j.cnsns.2025.109369_bib0017","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevResearch.1.023022","article-title":"Condensation versus ordering: from the spherical models to Bose-Einstein condensation in the canonical and grand canonical ensemble","volume":"1","author":"Crisanti","year":"2019","journal-title":"Phys Rev Res"},{"issue":"3","key":"10.1016\/j.cnsns.2025.109369_bib0018","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevLett.102.038701","article-title":"Generalized Bose-Fermi statistics and structural correlations in weighted networks","volume":"102","author":"Garlaschelli","year":"2009","journal-title":"Phys Rev Lett"},{"key":"10.1016\/j.cnsns.2025.109369_bib0019","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2023.113546","article-title":"Ensemble nonequivalence and Bose\u2013Einstein condensation in weighted networks","volume":"172","author":"Zhang","year":"2023","journal-title":"Chaos Solitons Fractals"},{"issue":"1","key":"10.1016\/j.cnsns.2025.109369_bib0020","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":"Nature Rev Phys"},{"issue":"6","key":"10.1016\/j.cnsns.2025.109369_bib0021","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.87.062806","article-title":"Statistical mechanics of multiplex networks: entropy and overlap","volume":"87","author":"Bianconi","year":"2013","journal-title":"Phys Rev E"},{"issue":"6","key":"10.1016\/j.cnsns.2025.109369_bib0022","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.70.066117","article-title":"Statistical mechanics of networks","volume":"70","author":"Park","year":"2004","journal-title":"Phys Rev E"},{"issue":"4","key":"10.1016\/j.cnsns.2025.109369_bib0023","doi-asserted-by":"crossref","DOI":"10.1088\/1367-2630\/ac5e89","article-title":"Strong ensemble nonequivalence in systems with local constraints","volume":"24","author":"Zhang","year":"2022","journal-title":"New J Phys"},{"key":"10.1016\/j.cnsns.2025.109369_bib0024","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2022.112264","article-title":"A betweenness structural entropy of complex networks","volume":"161","author":"Zhang","year":"2022","journal-title":"Chaos Solitons Fractals"},{"issue":"1","key":"10.1016\/j.cnsns.2025.109369_bib0025","doi-asserted-by":"crossref","first-page":"1130","DOI":"10.1038\/s41598-020-80939-1","article-title":"Connecting complex networks to nonadditive entropies","volume":"11","author":"de Oliveira","year":"2021","journal-title":"Sci Rep"},{"key":"10.1016\/j.cnsns.2025.109369_bib0026","unstructured":"Hollander F.d., Mandjes M., Roccaverde A., Starreveld N.J.. Breaking of ensemble equivalence for perturbed erd\u2216h {o} sr\u2216\u2019enyi random graphs. arXiv preprint arXiv: 1807077502018."},{"issue":"2","key":"10.1016\/j.cnsns.2025.109369_bib0027","doi-asserted-by":"crossref","DOI":"10.1209\/0295-5075\/81\/28005","article-title":"The entropy of randomized network ensembles","volume":"81","author":"Bianconi","year":"2007","journal-title":"EPL (Europhys Lett)"},{"issue":"4","key":"10.1016\/j.cnsns.2025.109369_bib0028","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1103\/PhysRev.106.620","article-title":"Information theory and statistical mechanics","volume":"106","author":"Jaynes","year":"1957","journal-title":"Phys Rev."},{"issue":"5","key":"10.1016\/j.cnsns.2025.109369_bib0029","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.69.056111","article-title":"Entropy and information in neural spike trains: progress on the sampling problem","volume":"69","author":"Nemenman","year":"2004","journal-title":"Phys Rev E"},{"issue":"3","key":"10.1016\/j.cnsns.2025.109369_bib0030","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.79.036114","article-title":"Entropy of network ensembles","volume":"79","author":"Bianconi","year":"2009","journal-title":"Phys Rev E"},{"issue":"5","key":"10.1016\/j.cnsns.2025.109369_bib0031","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.70.056131","article-title":"Analysis of weighted networks","volume":"70","author":"Newman","year":"2004","journal-title":"Phys Rev E"},{"issue":"5","key":"10.1016\/j.cnsns.2025.109369_bib0032","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.74.055101","article-title":"Correlations in weighted networks","volume":"74","author":"Serrano","year":"2006","journal-title":"Phys Rev E"},{"issue":"1","key":"10.1016\/j.cnsns.2025.109369_bib0033","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1038\/s42005-025-02182-2","article-title":"Entropy-based models to randomise real-world hypergraphs","volume":"8","author":"Saracco","year":"2025","journal-title":"Commun Phys"},{"issue":"5","key":"10.1016\/j.cnsns.2025.109369_bib0034","doi-asserted-by":"crossref","DOI":"10.1063\/5.0090864","article-title":"Complex network growth model: possible isomorphism between nonextensive statistical mechanics and random geometry","volume":"32","author":"Tsallis","year":"2022","journal-title":"Chaos Interdiscip J Nonlinear Sci"},{"issue":"6","key":"10.1016\/j.cnsns.2025.109369_bib0035","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1051\/epn:2005612","article-title":"Nonextensive statistical mechanics and complex scale-free networks","volume":"36","author":"Thurner","year":"2005","journal-title":"Europhys News"},{"issue":"3\u20136","key":"10.1016\/j.cnsns.2025.109369_bib0036","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.physrep.2009.07.001","article-title":"Statistical mechanics and dynamics of solvable models with long-range interactions","volume":"480","author":"Campa","year":"2009","journal-title":"Phys Rep"},{"issue":"5","key":"10.1016\/j.cnsns.2025.109369_bib0037","doi-asserted-by":"crossref","first-page":"999","DOI":"10.1023\/A:1026446225804","article-title":"Large deviation principles and complete equivalence and nonequivalence results for pure and mixed ensembles","volume":"101","author":"Ellis","year":"2000","journal-title":"J Stat Phys"},{"issue":"3","key":"10.1016\/j.cnsns.2025.109369_bib0038","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1016\/S0960-0779(01)00019-4","article-title":"Entropic nonextensivity: a possible measure of complexity","volume":"13","author":"Tsallis","year":"2002","journal-title":"Chaos Solitons Fractals"},{"issue":"5","key":"10.1016\/j.cnsns.2025.109369_bib0039","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevLett.124.050601","article-title":"Thermodynamic cost and benefit of memory","volume":"124","author":"Still","year":"2020","journal-title":"Phys Rev Lett"},{"issue":"10","key":"10.1016\/j.cnsns.2025.109369_bib0040","doi-asserted-by":"crossref","first-page":"744","DOI":"10.1038\/nphys1803","article-title":"Emergent complex neural dynamics","volume":"6","author":"Chialvo","year":"2010","journal-title":"Nat Phys"},{"issue":"3","key":"10.1016\/j.cnsns.2025.109369_bib0041","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1007\/s003329900051","article-title":"Complex chaotic systems and emergent phenomena","volume":"8","author":"Singh","year":"1998","journal-title":"J Nonlinear Sci"},{"issue":"7","key":"10.1016\/j.cnsns.2025.109369_bib0042","doi-asserted-by":"crossref","first-page":"1033","DOI":"10.1038\/s41567-023-02020-8","article-title":"Emergent stability in complex network dynamics","volume":"19","author":"Meena","year":"2023","journal-title":"Nat Phys"},{"issue":"27","key":"10.1016\/j.cnsns.2025.109369_bib0043","doi-asserted-by":"crossref","DOI":"10.1073\/pnas.2322939121","article-title":"Unraveling emergent network indeterminacy in complex ecosystems: a random matrix approach","volume":"121","author":"Kawatsu","year":"2024","journal-title":"Proc Natl Acad Sci"},{"key":"10.1016\/j.cnsns.2025.109369_bib0044","series-title":"Elementary principles in statistical mechanics: developed with especial reference to the rational foundations of thermodynamics","author":"Gibbs","year":"1902"},{"issue":"1\u20134","key":"10.1016\/j.cnsns.2025.109369_bib0045","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/S0378-4371(01)00444-7","article-title":"Complexity through nonextensivity","volume":"302","author":"Bialek","year":"2001","journal-title":"Physica A Stat Mech Appl"},{"issue":"5","key":"10.1016\/j.cnsns.2025.109369_bib0046","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevB.74.054432","article-title":"Nonextensivity in magnetic nanoparticle ensembles","volume":"74","author":"Binek","year":"2006","journal-title":"Phys Rev B"},{"issue":"2","key":"10.1016\/j.cnsns.2025.109369_bib0047","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.95.022102","article-title":"Extensivity and additivity of the Kolmogorov-Sinai entropy for simple fluids","volume":"95","author":"Das","year":"2017","journal-title":"Phys Rev E"},{"issue":"26","key":"10.1016\/j.cnsns.2025.109369_bib0048","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevLett.115.268701","article-title":"Breaking of ensemble equivalence in networks","volume":"115","author":"Squartini","year":"2015","journal-title":"Phys Rev Lett"},{"issue":"3","key":"10.1016\/j.cnsns.2025.109369_bib0049","doi-asserted-by":"crossref","first-page":"644","DOI":"10.1007\/s10955-018-2114-x","article-title":"Covariance structure behind breaking of ensemble equivalence in random graphs","volume":"173","author":"Garlaschelli","year":"2018","journal-title":"J Stat Phys"},{"key":"10.1016\/j.cnsns.2025.109369_bib0050","unstructured":"Squartini T., Garlaschelli D.. Reconnecting statistical physics and combinatorics beyond ensemble equivalence. arXiv preprint arXiv: 1710114222017."}],"container-title":["Communications in Nonlinear Science and Numerical Simulation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1007570425007786?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1007570425007786?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T14:44:08Z","timestamp":1773153848000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S1007570425007786"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1]]},"references-count":50,"alternative-id":["S1007570425007786"],"URL":"https:\/\/doi.org\/10.1016\/j.cnsns.2025.109369","relation":{},"ISSN":["1007-5704"],"issn-type":[{"value":"1007-5704","type":"print"}],"subject":[],"published":{"date-parts":[[2026,1]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Local entropy and nonextensivity of networks ensemble","name":"articletitle","label":"Article Title"},{"value":"Communications in Nonlinear Science and Numerical Simulation","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.cnsns.2025.109369","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"109369"}}