{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:56:42Z","timestamp":1760151402352,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,3,15]],"date-time":"2022-03-15T00:00:00Z","timestamp":1647302400000},"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":"publisher","award":["No.61872166, No.61662066"],"award-info":[{"award-number":["No.61872166, No.61662066"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Characterizing the topology and random walk of a random network is difficult because the connections in the network are uncertain. We propose a class of the generalized weighted Koch network by replacing the triangles in the traditional Koch network with a graph Rs according to probability 0\u2264p\u22641 and assign weight to the network. Then, we determine the range of several indicators that can characterize the topological properties of generalized weighted Koch networks by examining the two models under extreme conditions, p=0 and p=1, including average degree, degree distribution, clustering coefficient, diameter, and average weighted shortest path. In addition, we give a lower bound on the average trapping time (ATT) in the trapping problem of generalized weighted Koch networks and also reveal the linear, super-linear, and sub-linear relationships between ATT and the number of nodes in the network.<\/jats:p>","DOI":"10.3390\/e24030409","type":"journal-article","created":{"date-parts":[[2022,3,16]],"date-time":"2022-03-16T03:34:13Z","timestamp":1647401653000},"page":"409","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Structure and First-Passage Properties of Generalized Weighted Koch Networks"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9071-5725","authenticated-orcid":false,"given":"Jing","family":"Su","sequence":"first","affiliation":[{"name":"School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China"},{"name":"Key Laboratory of High Confidence Software Technologies, Peking University, Beijing 100871, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mingjun","family":"Zhang","sequence":"additional","affiliation":[{"name":"China Northwest Center of Financial Research, Lanzhou University of Finance and Economics, Lanzhou 730020, China"},{"name":"School of Information Engineering, Lanzhou University of Finance and Economics, Lanzhou 730020, China"},{"name":"Key Laboratory of E-Business Technology and Application, Lanzhou 730020, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bing","family":"Yao","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"12321","DOI":"10.3233\/JIFS-210458","article-title":"Efficient detection of hacker community based on twitter data using complex networks and machine learning algorithm","volume":"40","year":"2021","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"013130","DOI":"10.1063\/5.0033335","article-title":"Evolutionary dynamics of cooperation with the celebrity effect in complex networks","volume":"31","author":"Fu","year":"2021","journal-title":"Chaos"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"125001","DOI":"10.1016\/j.physa.2020.125001","article-title":"Scale-free and small-world properties of a multiple-hub network with fractal structure","volume":"558","author":"Huang","year":"2020","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"125897","DOI":"10.1016\/j.physa.2021.125897","article-title":"A directed weighted scale-free network model with an adaptive evolution mechanism","volume":"572","author":"Pi","year":"2021","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Rak, R., and Rak, E. (2020). The Fractional Preferential Attachment Scale-Free Network Model. Entropy, 22.","DOI":"10.3390\/e22050509"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.tre.2018.11.008","article-title":"Efficiency and robustness of weighted air transport networks","volume":"122","author":"Zhou","year":"2019","journal-title":"Transp. Res. Part E Logist. Transp. Rev."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"107481","DOI":"10.1016\/j.patcog.2020.107481","article-title":"Neural Network With Multiple Connection Weights","volume":"107","author":"Zhang","year":"2020","journal-title":"Pattern Recognit."},{"key":"ref_8","first-page":"3286","article-title":"On Random Graphs I","volume":"4","author":"Erdos","year":"1959","journal-title":"Publ. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1038\/30918","article-title":"Collective dynamics of small-world networks","volume":"393","author":"Watts","year":"1998","journal-title":"Nature"},{"key":"ref_10","first-page":"509","article-title":"Emergence of scaling in random networks","volume":"5439","author":"Albert","year":"1999","journal-title":"Science"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"088701","DOI":"10.1103\/PhysRevLett.109.088701","article-title":"First passage time for random walks in heterogeneous networks","volume":"109","author":"Hwang","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","first-page":"45","article-title":"Community Detection Algorithm Based on Random Walk of Signal Propagation with Bias","volume":"46","author":"Yin","year":"2019","journal-title":"Comput. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1016\/j.ces.2018.09.055","article-title":"The dual phase moisture conductivity of fibrous materials using random walk techniques in X-ray microcomputed tomographic structures","volume":"195","author":"Defrenne","year":"2018","journal-title":"Chem. Eng. Sci."},{"key":"ref_14","first-page":"691","article-title":"Mean first-passage time for random walks on undirected networks","volume":"84","author":"Zhang","year":"2011","journal-title":"Eur. Phys. J. B\u2014Condens. Matter"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"062140","DOI":"10.1103\/PhysRevE.87.062140","article-title":"Random walks in weighted networks with a perfect trap: An application of laplacian spectra","volume":"87","author":"Lin","year":"2013","journal-title":"Phys. Rev. E Stat. Nonlinear Soft Matter Phys."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Chelali, M., Kurtz, C., Puissant, A., and Vincent, N. (2020, January 19\u201323). From pixels to random walk based segments for image time series deep classification. Proceedings of the International Conference on Pattern Recognition and Artificial Intelligence, Zhongshan, China.","DOI":"10.1007\/978-3-030-59830-3_30"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1700","DOI":"10.1093\/icb\/icz117","article-title":"High-throughput segmentation of tiled biological structures using random walk distance transforms","volume":"59","author":"Baum","year":"2019","journal-title":"Integr. Comp. Biol."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1108\/IR-05-2018-0096","article-title":"Matching for navigation map building for automated guided robot based on laser navigation without a reflector","volume":"46","author":"Zhang","year":"2019","journal-title":"Ind. Robot"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1063\/1.4922265","article-title":"Spectrum of walk matrix for Koch network and its application","volume":"142","author":"Xie","year":"2015","journal-title":"J. Chem. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1063\/1.3493406","article-title":"Impact of degree heterogeneity on the behavior of trapping in Koch networks","volume":"20","author":"Zhang","year":"2010","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"6165","DOI":"10.1016\/j.physa.2012.06.066","article-title":"Scaling of average receiving time and average weighted shortest path on weighted Koch networks","volume":"391","author":"Dai","year":"2012","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1140\/epjb\/e2013-30905-x","article-title":"Expanded Koch networks: Structure and trapping time of random walks","volume":"64","author":"Hou","year":"2013","journal-title":"Eur. Phys. J. B"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physa.2016.03.097","article-title":"Average receiving scaling of the weighted polygon Koch networks with the weight-dependent walk","volume":"458","author":"Ye","year":"2016","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"178701","DOI":"10.1103\/PhysRevLett.107.178701","article-title":"All scale-free network are sparse","volume":"107","author":"Gross","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1038\/scientificamerican0503-60","article-title":"Scale-free network","volume":"288","author":"Barabasi","year":"2003","journal-title":"Sci. Am."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"083042","DOI":"10.1088\/1367-2630\/10\/8\/083042","article-title":"Mean clustering coefficient-on clustering measures for small-world networks","volume":"10","author":"Kaiser","year":"2008","journal-title":"New J. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2134","DOI":"10.1016\/j.physa.2010.01.019","article-title":"Weighted Fractal Networks","volume":"389","author":"Carletti","year":"2009","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1016\/j.physa.2014.03.088","article-title":"Scaling of average weighted shortest path and average receiving time on weighted hierarchical networks","volume":"407","author":"Sun","year":"2014","journal-title":"Phys. A Stat. Mech. Its Appl."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/3\/409\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:36:37Z","timestamp":1760135797000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/3\/409"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,15]]},"references-count":28,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,3]]}},"alternative-id":["e24030409"],"URL":"https:\/\/doi.org\/10.3390\/e24030409","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,3,15]]}}}