{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,22]],"date-time":"2025-05-22T04:12:33Z","timestamp":1747887153066,"version":"3.41.0"},"publisher-location":"Cham","reference-count":42,"publisher":"Springer Nature Switzerland","isbn-type":[{"value":"9783031928970","type":"print"},{"value":"9783031928987","type":"electronic"}],"license":[{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025]]},"DOI":"10.1007\/978-3-031-92898-7_3","type":"book-chapter","created":{"date-parts":[[2025,5,21]],"date-time":"2025-05-21T13:13:33Z","timestamp":1747833213000},"page":"30-44","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Modularity of\u00a0Random Intersection Graphs"],"prefix":"10.1007","author":[{"given":"Katarzyna","family":"Rybarczyk","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,5,22]]},"reference":[{"key":"3_CR1","doi-asserted-by":"publisher","first-page":"066118","DOI":"10.1103\/PhysRevE.85.066118","volume":"85","author":"JP Bagrow","year":"2012","unstructured":"Bagrow, J.P.: Communities and bottlenecks: trees and treelike networks have high modularity. Phys. Rev. E 85, 066118 (2012)","journal-title":"Phys. Rev. E"},{"issue":"1","key":"3_CR2","doi-asserted-by":"publisher","first-page":"R17","DOI":"10.37236\/935","volume":"14","author":"M Behrisch","year":"2007","unstructured":"Behrisch, M.: Component evolution in random intersection graphs. Electron. J. Comb. 14(1), R17 (2007)","journal-title":"Electron. J. Comb."},{"issue":"12","key":"3_CR3","doi-asserted-by":"publisher","first-page":"113071","DOI":"10.1016\/j.disc.2022.113071","volume":"345","author":"P Bennett","year":"2022","unstructured":"Bennett, P., Dudek, A.: A gentle introduction to the differential equation method and dynamic concentration. Discret. Math. 345(12), 113071 (2022)","journal-title":"Discret. Math."},{"issue":"10","key":"3_CR4","doi-asserted-by":"publisher","first-page":"P10008","DOI":"10.1088\/1742-5468\/2008\/10\/P10008","volume":"2008","author":"VD Blondel","year":"2008","unstructured":"Blondel, V.D., Guillaume, J.L., Lambiotte, R., Lefebvre, E.: Fast unfolding of communities in large networks. J. Stat. Mech: Theory Exp. 2008(10), P10008 (2008)","journal-title":"J. Stat. Mech: Theory Exp."},{"key":"3_CR5","doi-asserted-by":"crossref","unstructured":"Bloznelis, M., Godehardt, E., Jaworski, J., Kurauskas, V., Rybarczyk, K.: Recent Progress in Complex Network Analysis: Models of Random Intersection Graphs. In: Data Science, Learning by Latent Structures, and Knowledge Discovery, pp. 69\u201378. Springer, Heidelberg (2015)","DOI":"10.1007\/978-3-662-44983-7_6"},{"key":"3_CR6","doi-asserted-by":"crossref","unstructured":"Bloznelis, M., Godehardt, E., Jaworski, J., Kurauskas, V., Rybarczyk, K.: Recent Progress in Complex Network Analysis: Properties of Random Intersection Graphs. In: Data Science, Learning by Latent Structures, and Knowledge Discovery, pp. 79\u201388. Springer, Heidelberg (2015)","DOI":"10.1007\/978-3-662-44983-7_7"},{"issue":"4","key":"3_CR7","doi-asserted-by":"publisher","first-page":"1188","DOI":"10.1017\/S0269964821000310","volume":"36","author":"M Bloznelis","year":"2021","unstructured":"Bloznelis, M., Karjalainen, J., Leskel\u00e4, L.: Assortativity and bidegree distributions on Bernoulli random graph superpositions. Probab. Eng. Inf. Sci. 36(4), 1188\u20131213 (2021)","journal-title":"Probab. Eng. Inf. Sci."},{"issue":"4\u20135","key":"3_CR8","first-page":"385","volume":"11","author":"M Bloznelis","year":"2014","unstructured":"Bloznelis, M., Karo\u0144ski, M.: Random intersection graph process. Internet Math. 11(4\u20135), 385\u2013402 (2014)","journal-title":"Internet Math."},{"issue":"2","key":"3_CR9","doi-asserted-by":"publisher","first-page":"283","DOI":"10.1002\/rsa.21140","volume":"63","author":"M Bloznelis","year":"2023","unstructured":"Bloznelis, M., Leskel\u00e4, L.: Clustering and percolation on superpositions of Bernoulli random graphs. Random Struct. Algorithms 63(2), 283\u2013342 (2023)","journal-title":"Random Struct. Algorithms"},{"issue":"2","key":"3_CR10","doi-asserted-by":"publisher","first-page":"172","DOI":"10.1109\/TKDE.2007.190689","volume":"20","author":"U Brandes","year":"2008","unstructured":"Brandes, U., et al.: On modularity clustering. IEEE Trans. Knowl. Data Eng. 20(2), 172\u2013188 (2008)","journal-title":"IEEE Trans. Knowl. Data Eng."},{"key":"3_CR11","doi-asserted-by":"crossref","unstructured":"Brennan, M., Bresler, G., Nagaraj, D.: Phase transitions for detecting latent geometry in random graphs. Probab. Theory Relat. Fields 1215\u20131289 (2020)","DOI":"10.1007\/s00440-020-00998-3"},{"issue":"1","key":"3_CR12","doi-asserted-by":"publisher","first-page":"cnab051","DOI":"10.1093\/comnet\/cnab051","volume":"10","author":"J Chellig","year":"2021","unstructured":"Chellig, J., Fountoulakis, N., Skerman, F.: The modularity of random graphs on the hyperbolic plane. J. Complex Netw. 10(1), cnab051 (2021)","journal-title":"J. Complex Netw."},{"key":"3_CR13","doi-asserted-by":"crossref","unstructured":"Dinh, T.N., Thai, M.T.: Finding community structure with performance guarantees in scale-free networks. In: 2011 IEEE Third International Conference on Privacy, Security, Risk and Trust and 2011 IEEE Third International Conference on Social Computing, pp. 888\u2013891 (2011)","DOI":"10.1109\/PASSAT\/SocialCom.2011.185"},{"key":"3_CR14","doi-asserted-by":"publisher","first-page":"156","DOI":"10.1002\/(SICI)1098-2418(200003)16:2<156::AID-RSA3>3.0.CO;2-H","volume":"16","author":"JA Fill","year":"2000","unstructured":"Fill, J.A., Scheinerman, E.R., Singer-Cohen, K.B.: Random intersection graphs when $$m=\\omega (n)$$: an equivalence theorem relating the evolution of the G(n, m, p) and G(n, p) models. Random Struct. Algorithms 16, 156\u2013176 (2000)","journal-title":"Random Struct. Algorithms"},{"issue":"6","key":"3_CR15","doi-asserted-by":"publisher","first-page":"pgad180","DOI":"10.1093\/pnasnexus\/pgad180","volume":"2","author":"G Gilad","year":"2023","unstructured":"Gilad, G., Sharan, R.: From Leiden to Tel-Aviv University (TAU): exploring clustering solutions via a genetic algorithm. PNAS Nexus 2(6), pgad180 (2023)","journal-title":"PNAS Nexus"},{"key":"3_CR16","doi-asserted-by":"crossref","unstructured":"Godehardt, E., Jaworski, J.: Two models of random intersection graphs for classification. In: Opitz, O., Schwaiger, M. (eds.) Studies in Classifcation, Data Analysis and Knowledge Organization, vol. 22, pp. 67\u201381. Springer (2003)","DOI":"10.1007\/978-3-642-55721-7_8"},{"issue":"5","key":"3_CR17","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1016\/j.ipl.2004.03.007","volume":"90","author":"J-L Guillaume","year":"2004","unstructured":"Guillaume, J.-L., Latapy, M.: Bipartite structure of all complex networks. Inf. Process. Lett. 90(5), 215\u2013221 (2004)","journal-title":"Inf. Process. Lett."},{"key":"3_CR18","doi-asserted-by":"publisher","DOI":"10.1002\/9781118032718","volume-title":"Random Graphs","author":"S Janson","year":"2000","unstructured":"Janson, S., \u0141uczak, T., Ruci\u0144ski, A.: Random Graphs. Wiley, Hoboken (2000)"},{"issue":"5","key":"3_CR19","first-page":"26","volume":"12","author":"B Kami\u0144ski","year":"2024","unstructured":"Kami\u0144ski, B., Misiorek, P., Pra\u0142at, P., Th\u00e9berge, F.: Modularity based community detection in hypergraphs. J. Complex Netw. 12(5), 26 (2024). Paper No. cnae041","journal-title":"J. Complex Netw."},{"key":"3_CR20","doi-asserted-by":"crossref","unstructured":"Kami\u0144ski, B., Pankratz, B., Pra\u0142at, P., Th\u00e9berge, F.: Modularity of the abcd random graph model with community structure. J. Complex Netw. 10(6) (2022)","DOI":"10.1093\/comnet\/cnac050"},{"key":"3_CR21","doi-asserted-by":"publisher","first-page":"e0224307","DOI":"10.1371\/journal.pone.0224307","volume":"14","author":"B Kami\u0144ski","year":"2019","unstructured":"Kami\u0144ski, B., Poulin, V., Pra\u0142at, P., Szufel, P., Th\u00e9berge, F.: Clustering via hypergraph modularity. PLoS ONE 14, e0224307 (2019)","journal-title":"PLoS ONE"},{"key":"3_CR22","doi-asserted-by":"crossref","unstructured":"Kami\u0144ski, B., Pra\u0142at, P., Th\u00e9berge, F.: Mining Complex Networks. Chapman and Hall\/CRC (2021)","DOI":"10.1201\/9781003218869"},{"key":"3_CR23","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1017\/S0963548398003459","volume":"8","author":"M Karo\u0144ski","year":"1999","unstructured":"Karo\u0144ski, M., Scheinerman, E.R., Singer-Cohen, K.B.: On random intersection graphs: the subgraph problem. Comb. Probab. Comput. 8, 131\u2013159 (1999)","journal-title":"Comb. Probab. Comput."},{"issue":"4","key":"3_CR24","doi-asserted-by":"publisher","first-page":"662","DOI":"10.1002\/rsa.20750","volume":"52","author":"JH Kim","year":"2018","unstructured":"Kim, J.H., Lee, S.J., Na, J.: On the total variation distance between the binomial random graph and the random intersection graph. Random Struct. Algorithms 52(4), 662\u2013679 (2018)","journal-title":"Random Struct. Algorithms"},{"issue":"1","key":"3_CR25","doi-asserted-by":"publisher","first-page":"N10","DOI":"10.37236\/885","volume":"15","author":"AN Lager\u00e5s","year":"2008","unstructured":"Lager\u00e5s, A.N., Lindholm, M.: A note on the component structure in random intersection graphs with tunable clustering. Electron. J. Comb. 15(1), N10 (2008)","journal-title":"Electron. J. Comb."},{"issue":"4","key":"3_CR26","doi-asserted-by":"publisher","first-page":"728","DOI":"10.1002\/jgt.22896","volume":"102","author":"M Laso\u0144","year":"2023","unstructured":"Laso\u0144, M., Sulkowska, M.: Modularity of minor-free graphs. J. Graph Theory 102(4), 728\u2013736 (2023)","journal-title":"J. Graph Theory"},{"key":"3_CR27","doi-asserted-by":"crossref","unstructured":"Majstorovi\u0107, S., Stevanovi\u0107, D.: A note on graphs whose largest eigenvalues of the modularity matrix equals zero. Electron. J. Linear Algebra 27 (2014)","DOI":"10.13001\/1081-3810.1921"},{"key":"3_CR28","doi-asserted-by":"publisher","first-page":"431","DOI":"10.1016\/j.endm.2013.07.063","volume":"43","author":"C McDiarmid","year":"2013","unstructured":"McDiarmid, C., Skerman, F.: Modularity in random regular graphs and lattices. Electron. Notes Discrete Math. 43, 431\u2013437 (2013)","journal-title":"Electron. Notes Discrete Math."},{"issue":"4","key":"3_CR29","doi-asserted-by":"publisher","first-page":"596","DOI":"10.1093\/comnet\/cnx046","volume":"6","author":"C McDiarmid","year":"2018","unstructured":"McDiarmid, C., Skerman, F.: Modularity of regular and treelike graphs. J. Complex Netw. 6(4), 596\u2013619 (2018)","journal-title":"J. Complex Netw."},{"issue":"1","key":"3_CR30","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1002\/rsa.20910","volume":"57","author":"C McDiarmid","year":"2020","unstructured":"McDiarmid, C., Skerman, F.: Modularity of Erd\u0151s-R\u00e9nyi random graphs. Random Struct. Algorithms 57(1), 211\u2013243 (2020)","journal-title":"Random Struct. Algorithms"},{"key":"3_CR31","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780199206650.001.0001","volume-title":"Networks: An Introduction","author":"MEJ Newman","year":"2010","unstructured":"Newman, M.E.J.: Networks: An Introduction. Oxford University Press, Oxford (2010)"},{"issue":"2","key":"3_CR32","doi-asserted-by":"publisher","first-page":"026113","DOI":"10.1103\/PhysRevE.69.026113","volume":"69","author":"M Newman","year":"2004","unstructured":"Newman, M., Girvan, M.: Finding and evaluating community structure in networks. Phys. Rev. E 69(2), 026113 (2004)","journal-title":"Phys. Rev. E"},{"key":"3_CR33","doi-asserted-by":"crossref","unstructured":"Ostroumova Prokhorenkova, L., Pra\u0142at, P., Raigorodskii, A.: Modularity of complex networks models. In: Bonato, A., Graham, F.C., Pra\u0142at, P. (eds.) WAW 2016. LNCS, vol. 10088, pp. 115\u2013126. Springer, Cham (2016)","DOI":"10.1007\/978-3-319-49787-7_10"},{"key":"3_CR34","doi-asserted-by":"publisher","first-page":"947","DOI":"10.1016\/j.endm.2017.07.058","volume":"61","author":"L Prokhorenkova","year":"2017","unstructured":"Prokhorenkova, L., Pra\u0142at, P., Raigorodskii, A.M.: Modularity in several random graph models. Electron. Notes Discrete Math. 61, 947\u2013953 (2017)","journal-title":"Electron. Notes Discrete Math."},{"key":"3_CR35","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1002\/rsa.20356","volume":"38","author":"K Rybarczyk","year":"2011","unstructured":"Rybarczyk, K.: Equivalence of the random intersection graph and G(n, p). Random Struct. Algorithms 38, 205\u2013234 (2011)","journal-title":"Random Struct. Algorithms"},{"issue":"1","key":"3_CR36","doi-asserted-by":"publisher","first-page":"P36","DOI":"10.37236\/523","volume":"18","author":"K Rybarczyk","year":"2011","unstructured":"Rybarczyk, K.: Sharp threshold functions for random intersection graphs via a coupling method. Electron. J. Comb. 18(1), P36 (2011)","journal-title":"Electron. J. Comb."},{"issue":"2","key":"3_CR37","doi-asserted-by":"publisher","first-page":"P2.10","DOI":"10.37236\/5186","volume":"24","author":"K Rybarczyk","year":"2017","unstructured":"Rybarczyk, K.: The coupling method for inhomogeneous random intersection graphs. Electron. J. Combin. 24(2), P2.10 (2017)","journal-title":"Electron. J. Combin."},{"key":"3_CR38","unstructured":"Rybarczyk, K., Sulkowska, M.: Modularity of preferential attachment graphs. Preprint (2025). arxiv.org\/abs\/2501.06771"},{"key":"3_CR39","doi-asserted-by":"crossref","unstructured":"Spirakis, P.G., Nikoletseas, S., Raptopoulos, C.: A guided tour in random intersection graphs. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013. LNCS, vol. 7966, pp. 29\u201335. Springer, Heidelberg (2013)","DOI":"10.1007\/978-3-642-39212-2_5"},{"key":"3_CR40","doi-asserted-by":"crossref","unstructured":"Traag, V.A., Waltman, L., van Eck, N.J.: From Louvain to Leiden: guaranteeing well-connected communities. Sci. Rep. 9(5233) (2019)","DOI":"10.1038\/s41598-019-41695-z"},{"issue":"3","key":"3_CR41","doi-asserted-by":"publisher","first-page":"406","DOI":"10.1002\/rsa.21063","volume":"60","author":"R van der Hofstad","year":"2021","unstructured":"van der Hofstad, R., Komj\u00e1thy, J., Vadon, V.: Phase transition in random intersection graphs with communities. Random Struct. Algorithms 60(3), 406\u2013461 (2021)","journal-title":"Random Struct. Algorithms"},{"key":"3_CR42","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1017\/S0963548315000103","volume":"25","author":"L Warnke","year":"2022","unstructured":"Warnke, L.: On the method of typical bounded differences. Combin. Probab. Comput. 25, 269\u2013299 (2022)","journal-title":"Combin. Probab. Comput."}],"container-title":["Lecture Notes in Computer Science","Modelling and Mining Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-031-92898-7_3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,21]],"date-time":"2025-05-21T13:13:43Z","timestamp":1747833223000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-031-92898-7_3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025]]},"ISBN":["9783031928970","9783031928987"],"references-count":42,"URL":"https:\/\/doi.org\/10.1007\/978-3-031-92898-7_3","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025]]},"assertion":[{"value":"22 May 2025","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"WAW","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"International Workshop on Modelling and Mining Networks","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Vilnius","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Lithuania","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2025","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"30 June 2025","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"3 July 2025","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"20","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"waw2025","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/math.torontomu.ca\/waw2025\/","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}}]}}