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Our model is based on the fuzzy Partition Around Medoids, and it combines it with techniques for community detection in bipartite complex networks based on Barber modularity maximization. The goal is to produce a partition of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {N}\\cup \\mathcal {M}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>N<\/mml:mi>\n                    <mml:mo>\u222a<\/mml:mo>\n                    <mml:mi>M<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> into clusters, each of which is also identified by two medoids, one in <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {N}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>N<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and one in <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {M}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>M<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, which represent the typical units in the cluster for each set. Such clusters are optimized so that units in the same cluster both have similar values on their attributes and are likely to be adjacent. We test the algorithm on both simulated and real data, to show how it is able to capture a wide range of different interactions between the distribution of the attributes and the network structure.\n<\/jats:p>","DOI":"10.1007\/s11222-024-10495-5","type":"journal-article","created":{"date-parts":[[2024,11,8]],"date-time":"2024-11-08T03:31:22Z","timestamp":1731036682000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Fuzzy clustering with Barber modularity regularization"],"prefix":"10.1007","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7406-6411","authenticated-orcid":false,"given":"Pierpaolo","family":"D\u2019Urso","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7806-2170","authenticated-orcid":false,"given":"Livia","family":"De Giovanni","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3231-3901","authenticated-orcid":false,"given":"Lorenzo","family":"Federico","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9512-3609","authenticated-orcid":false,"given":"Vincenzina","family":"Vitale","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,11,8]]},"reference":[{"key":"10495_CR1","first-page":"1981","volume":"9","author":"EM Airoldi","year":"2008","unstructured":"Airoldi, E.M., Blei, D.M., Fienberg, S.E., et al.: Mixed membership stochastic blockmodels. 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