{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:26:40Z","timestamp":1760239600810,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2020,12,13]],"date-time":"2020-12-13T00:00:00Z","timestamp":1607817600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Information"],"abstract":"<jats:p>Frequency analysis is often used to investigate the structure of systems representing multi-scale real-world phenomena. In many different environments, functional relationships characterized by a power law have been recognized, but, in many cases this simple model has turned out to be absolutely inadequate and other models have been proposed. In this paper, we propose a general abstract model which constitutes a unifying framework, including many models found in literature, like the mixed model, the exponential cut-off and the log-normal. It is based on a discrete-time stochastic process, which leads to a recurrence relation describing the temporal evolution of the system. The steady state solution of the system highlights the probability distribution, which underlies the frequency behavior. A particular instance of the general model, called cubic-cut-off, was analyzed and tested in a number of experiments, producing good answers in difficult cases, even in the presence of peculiar behaviors.<\/jats:p>","DOI":"10.3390\/info11120580","type":"journal-article","created":{"date-parts":[[2020,12,13]],"date-time":"2020-12-13T20:56:57Z","timestamp":1607893017000},"page":"580","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Model for the Frequency Distribution of Multi-Scale Phenomena"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0382-3195","authenticated-orcid":false,"given":"Paola","family":"Favati","sequence":"first","affiliation":[{"name":"Istituto di Informatica e Telematica (IIT\u2013CNR), Via G. Moruzzi 1, 56124 Pisa, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Grazia","family":"Lotti","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica, University of Parma, Parco Area delle Scienze 53\/A, 43124 Parma, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ornella","family":"Menchi","sequence":"additional","affiliation":[{"name":"Dipartimento di Informatica, University of Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8890-4490","authenticated-orcid":false,"given":"Francesco","family":"Romani","sequence":"additional","affiliation":[{"name":"Dipartimento di Informatica, University of Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,13]]},"reference":[{"key":"ref_1","unstructured":"Barabasi, A.L. (2016). Network Science, Cambridge University Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1080\/15427951.2004.10129088","article-title":"A Brief History of Generative Models for Power Law and Lognormal Distributions","volume":"1","author":"Mitzenmacher","year":"2003","journal-title":"Internet Math."},{"key":"ref_3","unstructured":"Adamic, L.A. (2012). Zipf, Power-Laws, and Pareto\u2014A Ranking Tutorial, Information Dynamics Lab, HP Labs. Available online: http:\/\/www.hpl.hp.com\/research\/idl\/papers\/ranking\/ranking.html."},{"key":"ref_4","unstructured":"Caldarelli, G., De Los Rios, P., Laura, L., Leonardi, S., and Millozzi, S. (2003). A Study of Stochastic Models for the Web Graph, Dip. di Informatica e Sistemistica, Universita\u2019 di Roma \u201cLa Sapienza\u201d. Technical Report 04-03."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1093\/biomet\/42.3-4.425","article-title":"On a Class of Skew Distribution Functions","volume":"42","author":"Simon","year":"1955","journal-title":"Biometrika"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Cooper, C., and Frieze, A.M. (2001). A general model of undirected web graphs. European Symposium on Algorithms, Springer.","DOI":"10.1007\/3-540-44676-1_42"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"4633","DOI":"10.1103\/PhysRevLett.85.4633","article-title":"Structure of Growing Networks: Exact Solution of the Barabasi-Albert\u2019s model","volume":"85","author":"Dorogovtsev","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"5207","DOI":"10.1073\/pnas.032085699","article-title":"Winners don\u2019t take all: Characterizing the competition for links on the web","volume":"99","author":"Pennock","year":"2002","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1222","DOI":"10.1214\/EJP.v14-647","article-title":"Random Networks with Sublinear Preferential Attachment: Degree Evolutions","volume":"14","author":"Dereich","year":"2009","journal-title":"Electron. J. Probab."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Kunegis, J., Blattner, M., and Moser, C. (2013). Preferential Attachment in Online Networks: Measurement and Explanations. arXiv.","DOI":"10.1145\/2464464.2464514"},{"key":"ref_11","unstructured":"Abramowitz, M., and Stegun, I. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, 10th Printing."},{"key":"ref_12","unstructured":"(2020, December 02). Beta Function. Available online: https:\/\/en.wikipedia.org\/wiki\/Beta_function."},{"key":"ref_13","unstructured":"(2020, December 09). Supplementary Data for This Paper. Available online: http:\/\/pages.di.unipi.it\/romani\/MDPIdata.zip."},{"key":"ref_14","unstructured":"(2020, December 04). Wolfram Language & System Documentation Center. Available online: https:\/\/reference.wolfram.com\/language\/ref\/CityData.html."},{"key":"ref_15","unstructured":"(2020, December 04). Project Gutenberg Literary Archive. Available online: ftp:\/\/mirrors.pglaf.org\/mirrors\/gutenberg-iso\/pgdvd042010.iso."},{"key":"ref_16","unstructured":"(2020, December 04). Laboratory for Web Algorithmics. Available online: http:\/\/law.di.unimi.it\/datasets.php."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1145\/988672.988752","article-title":"The Web Graph Framework I: Compression Techniques","volume":"Volume 99","author":"Boldi","year":"2004","journal-title":"Proceedings of the 13th International Conference on World Wide Web"},{"key":"ref_18","unstructured":"Srinivasan, S., Ramamritham, K., Kumar, A., Ravindra, M.P., Bertino, E., and Kumar, R. (April, January 28). Layered Label Propagation: A MultiResolution Coordinate-Free Ordering for Compressing Social Networks. Proceedings of the 20th International Conference on World Wide Web, Hyderabad, India."},{"key":"ref_19","unstructured":"(2020, December 04). ClueWeb12 Web Graph. Available online: http:\/\/www.lemurproject.org\/clueweb12\/webgraph.php\/."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Boldi, P., Marino, A., Santini, M., and Vigna, S. (2014, January 7\u201311). BUbiNG: Massive Crawling for the Masses. Proceedings of the Companion Publication of the 23rd International Conference on World Wide Web, Seoul, Korea.","DOI":"10.1145\/2567948.2577304"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1002\/spe.587","article-title":"Ubicrawler: A Scalable Fully Distributed Web Crawler","volume":"34","author":"Boldi","year":"2004","journal-title":"Software Pract. Exp."},{"key":"ref_22","unstructured":"Guidi, B., Michienzi, A., and Ricci, L. A graph-based socio-economic analysis of Steemit. IEEE Trans. Comput. Soc. Syst., Submitted to."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Kwak, H., Lee, C., Park, H., and Moon, S. (2010, January 26\u201330). What is Twitter, a Social Network or a News Media?. Proceedings of the 19th International World Wide Web (WWW) Conference, Raleigh, NC, USA.","DOI":"10.1145\/1772690.1772751"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Ziegler, C.N., McNee, S.M., Konstan, J.A., and Lausen, G. (2005, January 10\u201314). Improving Recommendation Lists Through Topic Diversification. Proceedings of the 14th International World Wide Web Conference (WWW \u201905), Chiba, Japan.","DOI":"10.1145\/1060745.1060754"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"McAuley, J., and Leskovec, J. (2020, December 04). From Amateurs to Connoisseurs: Modeling the Evolution of User Expertise through Online Reviews. WWW 2013. Available online: https:\/\/snap.stanford.edu\/data\/web-FineFoods.html.","DOI":"10.1145\/2488388.2488466"},{"key":"ref_26","unstructured":"Bradan, A.L. (2020). Forecast Emerging Artists Success on Last.fm Music Service: A Data-Driven Study. [Master\u2019s Thesis, University of Pisa]."},{"key":"ref_27","unstructured":"(2020, December 04). Movie Lens. Available online: https:\/\/grouplens.org\/datasets\/movielens\/."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Pennacchioli, D., Coscia, M., Rinzivillo, S., Pedreschi, D., and Giannotti, F. (2013, January 6\u20139). Explaining the product range effect in purchase data. Proceedings of the 2013 IEEE International Conference on Big Data, Big Data 2013, Santa Clara, CA, USA.","DOI":"10.1109\/BigData.2013.6691634"},{"key":"ref_29","unstructured":"(2020, December 11). Yahoo! Webscope Dataset ydata-ymusic-user-artist-ratings-v1_0 Yahoo! Music User Ratings of Musical Artists, Version 1.0. Available online: http:\/\/webscope.sandbox.yahoo.com\/catalog.php?datatype=r."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1295","DOI":"10.1111\/rssb.12233","article-title":"Sparse Graphs using Exchangeable Random Measures","volume":"79","author":"Caron","year":"2017","journal-title":"J. R. Stat. Soc. Ser. B"}],"container-title":["Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2078-2489\/11\/12\/580\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:44:33Z","timestamp":1760179473000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2078-2489\/11\/12\/580"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,13]]},"references-count":30,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2020,12]]}},"alternative-id":["info11120580"],"URL":"https:\/\/doi.org\/10.3390\/info11120580","relation":{},"ISSN":["2078-2489"],"issn-type":[{"type":"electronic","value":"2078-2489"}],"subject":[],"published":{"date-parts":[[2020,12,13]]}}}