{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T03:01:01Z","timestamp":1768273261414,"version":"3.49.0"},"reference-count":53,"publisher":"Association for Computing Machinery (ACM)","issue":"8","license":[{"start":{"date-parts":[[2024,7,31]],"date-time":"2024-07-31T00:00:00Z","timestamp":1722384000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"name":"Key Research and Development Plan of Shaanxi Province","award":["2024GX-YBXM-021"],"award-info":[{"award-number":["2024GX-YBXM-021"]}]},{"name":"National NaturalScience Foundation of China","award":["U22B2025"],"award-info":[{"award-number":["U22B2025"]}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"crossref","award":["G2023KY05105"],"award-info":[{"award-number":["G2023KY05105"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"crossref"}]},{"name":"National Key Research and Development Plan","award":["2020YFB1805400"],"award-info":[{"award-number":["2020YFB1805400"]}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Knowl. Discov. Data"],"published-print":{"date-parts":[[2024,9,30]]},"abstract":"\n Scale-free networks are prevalently observed in a great variety of complex systems, which triggers various researches relevant to networked models of such type. In this work, we propose a family of growth tree networks\n \n \\(\\mathcal{T}_{t}\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n , which turn out to be scale-free, in an iterative manner. As opposed to most of published tree models with scale-free feature, our tree networks have the power-law exponent\n \n \\(\\gamma=1{ + }\\ln 5\/\\ln 2\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n that is obviously larger than\n \n \\(3\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n . At the same time, \u201csmall-world\u201d property can not be found particularly because models\n \n \\(\\mathcal{T}_{t}\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n have an ultra-large diameter\n \n \\(D_{t}\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n (i.e.,\n \n \\(D_{t}\\sim|\\mathcal{T}_{t}|^{\\ln 3\/\\ln 5}\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n ) and a greater average shortest path length\n \n \\(\\langle\\mathcal{W}_{t}\\rangle\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n (namely,\n \n \\(\\langle\\mathcal{W}_{t}\\rangle\\sim|\\mathcal{T}_{t}|^{\\ln 3\/\\ln 5}\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n ) where\n \n \\(|\\mathcal{T}_{t}|\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n represents vertex number. Next, we determine Pearson correlation coefficient and verify that networks\n \n \\(\\mathcal{T}_{t}\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n display disassortative mixing structure. In addition, we study random walks on tree networks\n \n \\(\\mathcal{T}_{t}\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n and derive exact solution to mean hitting time\n \n \\(\\langle\\mathcal{H}_{t}\\rangle\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n . The results suggest that the analytic formula for quantity\n \n \\(\\langle\\mathcal{H}_{t}\\rangle\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n as a function of vertex number\n \n \\(|\\mathcal{T}_{t}|\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n shows a power-law form, i.e.,\n \n \\(\\langle\\mathcal{H}_{t}\\rangle\\sim|\\mathcal{T}_{t}|^{1+\\ln 3\/\\ln 5}\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n . Accordingly, we execute extensive experimental simulations, and demonstrate that empirical analysis is in strong agreement with theoretical results. Lastly, we provide a guide to extend the proposed iterative manner in order to generate more general scale-free tree networks with large diameter.\n <\/jats:p>","DOI":"10.1145\/3674146","type":"journal-article","created":{"date-parts":[[2024,6,20]],"date-time":"2024-06-20T12:31:29Z","timestamp":1718886689000},"page":"1-26","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":4,"title":["Structural Properties on Scale-Free Tree Network with an Ultra-Large Diameter"],"prefix":"10.1145","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4269-2565","authenticated-orcid":false,"given":"Fei","family":"Ma","sequence":"first","affiliation":[{"name":"School of Computer Science, Northwestern Polytechnical University, Xi\u2019an, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8854-2079","authenticated-orcid":false,"given":"Ping","family":"Wang","sequence":"additional","affiliation":[{"name":"National Engineering Research Center for Software Engineering, Peking University, Beijing, China, School of Software and Microelectronics, Peking University, Beijing, China, and Key Laboratory of High Confidence Software Technologies (PKU), Ministry of Education, Peking University, Beijing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2024,7,31]]},"reference":[{"key":"e_1_3_3_2_1","volume-title":"Network Science","author":"Barab\u00e1si A. -L.","year":"2016","unstructured":"A. -L. Barab\u00e1si. 2016. Network Science. Cambridge University Press."},{"key":"e_1_3_3_3_1","doi-asserted-by":"publisher","DOI":"10.1137\/S003614450342480"},{"key":"e_1_3_3_4_1","volume-title":"Network: An Introduction","author":"Newman M. E. J.","year":"2020","unstructured":"M. E. J. Newman. 2020. Network: An Introduction. Oxford University Press."},{"key":"e_1_3_3_5_1","doi-asserted-by":"publisher","DOI":"10.1038\/30918"},{"key":"e_1_3_3_6_1","doi-asserted-by":"publisher","DOI":"10.1126\/science.286.5439.509"},{"key":"e_1_3_3_7_1","doi-asserted-by":"publisher","DOI":"10.1140\/epjb\/e2005-00120-9"},{"key":"e_1_3_3_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2007.11.005"},{"key":"e_1_3_3_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2017.11.047"},{"key":"e_1_3_3_10_1","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.200327197"},{"key":"e_1_3_3_11_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.103.022318"},{"key":"e_1_3_3_12_1","doi-asserted-by":"publisher","DOI":"10.1089\/brain.2011.0038"},{"key":"e_1_3_3_13_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.90.058701"},{"key":"e_1_3_3_14_1","doi-asserted-by":"publisher","DOI":"10.1140\/epjb\/e2007-00107-6"},{"key":"e_1_3_3_15_1","doi-asserted-by":"publisher","DOI":"10.1063\/1.5105354"},{"key":"e_1_3_3_16_1","doi-asserted-by":"publisher","DOI":"10.1063\/1.5123594"},{"key":"e_1_3_3_17_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.94.018702"},{"key":"e_1_3_3_18_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.65.066122"},{"key":"e_1_3_3_19_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.67.026112"},{"key":"e_1_3_3_20_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.101.052315"},{"key":"e_1_3_3_21_1","first-page":"625","article-title":"Invariance principle for variable speed random walks on trees","volume":"45","author":"Athreya S.","year":"2014","unstructured":"S. Athreya, W. L\u00f6hr, and A. Winter. 2014. Invariance principle for variable speed random walks on trees. Annals of Probability 45, 625\u2013667.","journal-title":"Annals of Probability"},{"key":"e_1_3_3_22_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.78.011114"},{"key":"e_1_3_3_23_1","doi-asserted-by":"publisher","DOI":"10.1140\/epjb\/e2007-00310-5"},{"key":"e_1_3_3_24_1","doi-asserted-by":"crossref","first-page":"P04009","DOI":"10.1088\/1742-5468\/2014\/04\/P04009","article-title":"Highly dispersed networks generated by enhanced redirection","volume":"4","author":"Gabel A.","year":"2014","unstructured":"A. Gabel, P. L. Krapivsky, and S. Redner. 2014. Highly dispersed networks generated by enhanced redirection. Journal of Statistical Mechanics: Theory and Experiment 4, P04009.","journal-title":"Journal of Statistical Mechanics: Theory and Experiment"},{"key":"e_1_3_3_25_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.65.056101"},{"key":"e_1_3_3_26_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2011.08.002"},{"key":"e_1_3_3_27_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.81.016114"},{"key":"e_1_3_3_28_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2021.126199"},{"key":"e_1_3_3_29_1","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/16\/17\/003"},{"key":"e_1_3_3_30_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.69.1955"},{"key":"e_1_3_3_31_1","doi-asserted-by":"publisher","DOI":"10.1209\/0295-5075\/133\/40004"},{"key":"e_1_3_3_32_1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511606014"},{"key":"e_1_3_3_33_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.77.011128"},{"key":"e_1_3_3_34_1","doi-asserted-by":"publisher","DOI":"10.1109\/TKDE.2020.3014191"},{"key":"e_1_3_3_35_1","doi-asserted-by":"publisher","DOI":"10.1145\/3382208"},{"key":"e_1_3_3_36_1","doi-asserted-by":"publisher","DOI":"10.1145\/3561388"},{"key":"e_1_3_3_37_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.92.118701"},{"key":"e_1_3_3_38_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00373-012-1175-x"},{"key":"e_1_3_3_39_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.102.022305"},{"issue":"7","key":"e_1_3_3_40_1","first-page":"1014","article-title":"Commute times of random walks on trees","volume":"161","author":"Mokhtar K.","year":"2013","unstructured":"K. Mokhtar, A. A. Fahimah, and T. Andr\u00e1s. 2013. Commute times of random walks on trees. Discrete Applied Mathematics 161, 7 (2013), 1014\u20131021.","journal-title":"Discrete Applied Mathematics"},{"key":"e_1_3_3_41_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-84628-970-5"},{"key":"e_1_3_3_42_1","volume-title":"Elements of Random Walk and Diffusion Processes","author":"Oliver C. I.","year":"2013","unstructured":"C. I. Oliver. 2013. Elements of Random Walk and Diffusion Processes. Wiley."},{"key":"e_1_3_3_43_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.89.208701"},{"key":"e_1_3_3_44_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.70.046126"},{"key":"e_1_3_3_45_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-35653-8_16"},{"key":"e_1_3_3_46_1","doi-asserted-by":"publisher","DOI":"10.1109\/CDC.2015.7402000"},{"key":"e_1_3_3_47_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.101.052317"},{"key":"e_1_3_3_48_1","doi-asserted-by":"publisher","DOI":"10.1088\/1367-2630\/9\/6\/175"},{"key":"e_1_3_3_49_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.109.088701"},{"key":"e_1_3_3_50_1","doi-asserted-by":"publisher","DOI":"10.1038\/ncomms6121"},{"key":"e_1_3_3_51_1","doi-asserted-by":"publisher","DOI":"10.1145\/3406241"},{"key":"e_1_3_3_52_1","doi-asserted-by":"publisher","DOI":"10.1021\/ci950116s"},{"key":"e_1_3_3_53_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01270385"},{"key":"e_1_3_3_54_1","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.22029"}],"container-title":["ACM Transactions on Knowledge Discovery from Data"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3674146","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3674146","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T23:57:26Z","timestamp":1750291046000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3674146"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,31]]},"references-count":53,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2024,9,30]]}},"alternative-id":["10.1145\/3674146"],"URL":"https:\/\/doi.org\/10.1145\/3674146","relation":{},"ISSN":["1556-4681","1556-472X"],"issn-type":[{"value":"1556-4681","type":"print"},{"value":"1556-472X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,7,31]]},"assertion":[{"value":"2023-03-10","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2024-06-17","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2024-07-31","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}