Multimodal Clustering for Community Detection

  • Dmitry I. IgnatovEmail author
  • Alexander Semenov
  • Daria Komissarova
  • Dmitry V. Gnatyshak
Part of the Lecture Notes in Social Networks book series (LNSN)


Multimodal clustering is an unsupervised technique for mining interesting patterns in n-adic binary relations or n-mode networks. Among different types of such generalised patterns one can find biclusters and formal concepts (maximal bicliques) for two-mode case, triclusters and triconcepts for three-mode case, closed n-sets for n-mode case, etc. Object-attribute biclustering (OA-biclustering) for mining large binary datatables (formal contexts or two-mode networks) arose by the end of the last decade due to intractability of computation problems related to formal concepts; this type of patterns was proposed as a meaningful and scalable approximation of formal concepts. In this paper, our aim is to present recent advance in OA-biclustering and its extensions to mining multi-mode communities in SNA setting. We also discuss connection between clustering coefficients known in SNA community for one-mode and two-mode networks and OA-bicluster density, the main quality measure of an OA-bicluster. Our experiments with two-, three-, and four-mode large real-world networks show that this type of patterns is suitable for community detection in multi-mode cases within reasonable time even though the number of corresponding n-cliques is still unknown due to computation difficulties. An interpretation of OA-biclusters for one-mode networks is provided as well.


Two-mode networks Multi-mode networks Formal concept analysis Biclustering Triclustering Social and complex networks Community detection Biclique relaxation 



We would like to thank our colleagues Rakesh Agrawal, Loïc Cerf, Vincent Duquenne, Santo Fortunato, Bernhard Ganter, Jean-François Boulicaut, Mehdi Kaytoue, Boris Mirkin, Amedeo Napoli, Lhouri Nourine, Engelbert Mephu-Nguifo, Sergei Kuznetsov, Rokia Missaoui, Sergei Obiedkov, Camille Roth, Takeaki Uno, Stanley Wasserman, and Leonid Zhukov for their inspirational discussions or a piece of advice, which directly or implicitly influenced this study. We are grateful to our colleagues from the Laboratory for Internet Studies for their piece of advice as well. The study was implemented in the framework of the Basic Research Program at the National Research University Higher School of Economics in 2016 and 2017 and in the Laboratory of Intelligent Systems and Structural Analysis. The first author has also been supported by Russian Foundation for Basic Research.


  1. 1.
    Agrawal, R., Gehrke, J., Gunopulos, D., Raghavan, P.: Automatic subspace clustering of high dimensional data. Data Min. Knowl. Discov. 11(1), 5–33 (2005). DOI  10.1007/s10618-005-1396-1. MathSciNetCrossRefGoogle Scholar
  2. 2.
    Akhmatnurov, M., Ignatov, D.I.: Context-aware recommender system based on boolean matrix factorisation. In: Proceedings of the Twelfth International Conference on Concept Lattices and Their Applications, pp. 99–110, Clermont-Ferrand, 13–16 October 2015.
  3. 3.
    Barabási, A.: Network Science. Cambridge University Press, Cambridge (2016)zbMATHGoogle Scholar
  4. 4.
    Barber, M.J.: Modularity and community detection in bipartite networks. Phys. Rev. E 76, 066102 (2007). DOI 10.1103/PhysRevE.76.066102. MathSciNetCrossRefGoogle Scholar
  5. 5.
    Belohlávek, R., Glodeanu, C.V., Vychodil, V.: Optimal factorization of three-way binary data using triadic concepts. Order 30(2), 437–454 (2013). DOI 10.1007/s11083-012-9254-4. MathSciNetCrossRefGoogle Scholar
  6. 6.
    Belohlávek, R., Trnecka, M.: From-below approximations in boolean matrix factorization: Geometry and new algorithm. J. Comput. Syst. Sci. 81(8), 1678 – 1697 (2015). DOI 06.002. MathSciNetCrossRefGoogle Scholar
  7. 7.
    Belohlávek, R., Vychodil, V.: Discovery of optimal factors in binary data via a novel method of matrix decomposition. J. Comput. Syst. Sci. 76(1), 3–20 (2010). DOI 10.1016/j.jcss.2009.05.002. MathSciNetCrossRefGoogle Scholar
  8. 8.
    Berlingerio, M., Coscia, M., Giannotti, F., Monreale, A., Pedreschi, D.: Multidimensional networks: foundations of structural analysis. World Wide Web 16(5), 567–593 (2013). DOI 10.1007/s11280-012-0190-4. CrossRefGoogle Scholar
  9. 9.
    Berlingerio, M., Pinelli, F., Calabrese, F.: Abacus: frequent pattern mining-based community discovery in multidimensional networks. Data Min. Knowl. Discov. 27(3), 294–320 (2013). DOI 10.1007/s10618-013-0331-0. MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bohman, L.: Bringing the owners back in: an analysis of a 3-mode interlock network. Soc. Netw. 34(2), 275 – 287 (2012). DOI . CrossRefGoogle Scholar
  11. 11.
    Borgatti, S.P., Everett, M.G.: Network analysis of 2-mode data. Soc. Netw. 19(3), 243 – 269 (1997). DOI CrossRefGoogle Scholar
  12. 12.
    Cerf, L., Besson, J., Robardet, C., Boulicaut, J.: Closed patterns meet n-ary relations. TKDD 3(1), 3:1–3:36 (2009). DOI 10.1145/1497577.1497580.
  13. 13.
    Cerf, L., Besson, J., Nguyen, K., Boulicaut, J.: Closed and noise-tolerant patterns in n-ary relations. Data Min. Knowl. Discov. 26(3), 574–619 (2013). DOI 10.1007/s10618-012-0284-8. MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chatterjee, S., Bhattacharyya, M.: Judgment analysis of crowdsourced opinions using biclustering. Inf. Sci. 375, 138–154 (2017). DOI 10.1016/j.ins.2016.09.036. CrossRefGoogle Scholar
  15. 15.
    Cichocki, A., Lee, N., Oseledets, I.V., Phan, A.H., Zhao, Q., Mandic, D.P.: Tensor networks for dimensionality reduction and large-scale optimization: Part 1 low-rank tensor decompositions. Found. Trends Mach. Learn. 9(4–5), 249–429 (2016). DOI 10.1561/2200000059. CrossRefGoogle Scholar
  16. 16.
    Codocedo, V., Napoli, A.: Lattice-based biclustering using partition pattern structures. In: ECAI 2014 - 21st European Conference on Artificial Intelligence, 18–22 August 2014, Prague - Including Prestigious Applications of Intelligent Systems (PAIS 2014), pp. 213–218 (2014). DOI 10.3233/978-1-61499-419-0-213.
  17. 17.
    Davis A., B.B.G., Gardner, M.R.: Deep South. The University of Chicago Press, Chicago (1941)Google Scholar
  18. 18.
    Doreian, P., Batagelj, V., Ferligoj, A.: Generalized blockmodeling of two-mode network data. Soc. Netw. 26(1), 29–53 (2004). DOI CrossRefGoogle Scholar
  19. 19.
    Duquenne, V.: Lattice analysis and the representation of handicap associations. Soc. Netw. 18(3), 217–230 (1996). DOI 10.1016/ 0378-8733(95)00274-X. CrossRefGoogle Scholar
  20. 20.
    Fararo, T.J., Doreian, P.: Tripartite structural analysis: generalizing the Breiger-Wilson formalism. Soc. Netw. 6(2), 141–175 (1984). DOI http: // MathSciNetCrossRefGoogle Scholar
  21. 21.
    Fortunato, S.: Community detection in graphs. Phys. Rep. 486(3–5), 75–174 (2010). DOI MathSciNetCrossRefGoogle Scholar
  22. 22.
    Freeman, L.: Finding social groups: a meta-analysis of the southern women data. In: Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers, pp. 39–97. National Academy Press, Washington, DC (2003)Google Scholar
  23. 23.
    Freeman, L.C., White, D.R.: Using galois lattices to represent network data. Sociol. Methodol. 23, 127–146 (1993)CrossRefGoogle Scholar
  24. 24.
    Freeman, L.C.: Cliques, galois lattices, and the structure of human social groups. Soc. Netw. 18, 173–187 (1996)CrossRefGoogle Scholar
  25. 25.
    Ganter, B., Kuznetsov, S.O.: Pattern structures and their projections. In: Conceptual Structures: Broadening the Base, Proceedings of the 9th International Conference on Conceptual Structures, ICCS 2001, pp. 129–142, Stanford, CA, 30 July–3 August 2001. DOI 10.1007/3-540-44583-8∖_10.
  26. 26.
    Ganter, B., Obiedkov, S.A.: Conceptual Exploration. Springer, Heidelberg (2016). DOI 10.1007/978-3-662-49291-8.
  27. 27.
    Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundations, 1st edn. Springer, New York (1999)CrossRefGoogle Scholar
  28. 28.
    Gnatyshak, D., Ignatov, D.I., Semenov, A., Poelmans, J.: Gaining insight in social networks with biclustering and triclustering. In: BIR, pp. 162–171 (2012)Google Scholar
  29. 29.
    Gnatyshak, D., Ignatov, D.I., Kuznetsov, S.O., Nourine, L.: A one-pass triclustering approach: Is there any room for big data? In: Proceedings of the Eleventh International Conference on Concept Lattices and Their Applications, pp. 231–242, Košice, 7–10 October 2014.
  30. 30.
    Hacene, M.R., Huchard, M., Napoli, A., Valtchev, P.: Relational concept analysis: mining concept lattices from multi-relational data. Ann. Math. Artif. Intell. 67(1), 81–108 (2013). DOI 10.1007/s10472-012-9329-3.
  31. 31.
    Hartigan, J.A.: Direct clustering of a data matrix. J. Am. Stat. Assoc. 67(337), 123–129 (1972). DOI 10.2307/2284710. CrossRefGoogle Scholar
  32. 32.
    Ignatov, D.I.: Introduction to formal concept analysis and its applications in information retrieval and related fields. In: Information Retrieval - 8th Russian Summer School, RuSSIR 2014, pp. 42–141, Nizhniy, Novgorod, 18–22 August 2014. Revised Selected Papers (2014). DOI 10.1007/978-3-319-25485-2∖_3.
  33. 33.
    Ignatov, D.I.: Towards a closure operator for enumeration of maximal tricliques in tripartite hypergraphs. CoRR abs/1602.07267 (2016).
  34. 34.
    Ignatov, D.I., Kornilov, D.: RAPS: a recommender algorithm based on pattern structures. In: Proceedings of the 4th International Workshop “What Can FCA Do for Artificial Intelligence?”, FCA4AI 2015, co-located with the International Joint Conference on Artificial Intelligence (IJCAI 2015), pp. 87–98, Buenos Aires, 25 July 2015.
  35. 35.
    Ignatov, D.I., Kuznetsov, S.O.: Concept-based recommendations for internet advertisement. In: Belohlavek, R., Kuznetsov, S.O. (eds.) Proceedings of the CLA 2008, CEUR WS, vol. 433, pp. 157–166. Palacký University, Olomouc (2008)Google Scholar
  36. 36.
    Ignatov, D.I., Watson, B.W.: Towards a unified taxonomy of biclustering methods. In: Kuznetsov, S.O., Watson, B.W. (eds.) Proceedings of Russian and South African Workshop on Knowledge Discovery Techniques Based on Formal Concept Analysis (RuZA 2015). CEUR Workshop Proceedings, vol. 1552, pp. 23–39 (2015)Google Scholar
  37. 37.
    Ignatov, D., Kaminskaya, A., Kuznetsov, S., Magizov, R.: A concept-based biclustering algorithm. In: Proceedings of the Eight International Conference on Intelligent Information Processing (IIP-8), pp. 140–143. MAKS Press, Moscow (2010) [in Russian]Google Scholar
  38. 38.
    Ignatov, D.I., Kuznetsov, S.O., Magizov, R.A., Zhukov, L.E.: From triconcepts to triclusters. In: Rough Sets, Fuzzy Sets, Data Mining and Granular Computing - Proceedings of the 13th International Conference, RSFDGrC 2011, pp. 257–264, Moscow, 25–27 June 2011. DOI 10.1007/978-3-642-21881-1∖_41.
  39. 39.
    Ignatov, D.I., Kuznetsov, S.O., Poelmans, J.: Concept-based biclustering for internet advertisement. In: ICDM Workshops, pp. 123–130. IEEE Computer Society, Brussels (2012)Google Scholar
  40. 40.
    Ignatov, D.I., Kuznetsov, S.O., Poelmans, J., Zhukov, L.E.: Can triconcepts become triclusters? Int. J. Gen. Syst. 42(6), 572–593 (2013). DOI 10.1080/03081079.2013.798899. MathSciNetCrossRefGoogle Scholar
  41. 41.
    Ignatov, D.I., Mikhailova, M., Zakirova, A.Y., Malioukov, A.: Recommendation of ideas and antagonists for crowdsourcing platform witology. In: Information Retrieval - 8th Russian Summer School, RuSSIR 2014, pp. 276–296, Nizhniy, Novgorod, 18–22 August 2014, Revised Selected Papers (2014). DOI 10.1007/978-3-319-25485-2∖_9. Google Scholar
  42. 42.
    Ignatov, D.I., Nenova, E., Konstantinova, N., Konstantinov, A.V.: Boolean matrix factorisation for collaborative filtering: an fca-based approach. In: Artificial Intelligence: Methodology, Systems, and Applications - Proceedings of the 16th International Conference, AIMSA 2014, pp. 47–58, Varna, 11–13 September 2014. DOI 10.1007/978-3-319-10554-3∖_5.
  43. 43.
    Ignatov, D.I., Kaminskaya, A.Y., Konstantinova, N., Konstantinov, A.V.: Recommender system for crowdsourcing platform witology. In: 2014 IEEE/WIC/ACM International Joint Conferences on Web Intelligence (WI) and Intelligent Agent Technologies (IAT), vol. II, pp. 327–335, Warsaw, 11–14 August 2014. DOI 10.1109/WI-IAT.2014.52.
  44. 44.
    Ignatov, D.I., Kaminskaya, A.Y., Konstantinova, N., Malioukov, A., Poelmans, J.: Fca-based recommender models and data analysis for crowdsourcing platform witology. In: Graph-Based Representation and Reasoning - Proceedings of the 21st International Conference on Conceptual Structures, ICCS 2014, pp. 287–292, Iaşi, 27–30 July 2014. DOI 10.1007/978-3-319-08389-6∖_24. Google Scholar
  45. 45.
    Ignatov, D.I., Gnatyshak, D.V., Kuznetsov, S.O., Mirkin, B.G.: Triadic formal concept analysis and triclustering: searching for optimal patterns. Mach. Learn. 101(1–3), 271–302 (2015). DOI 10.1007/s10994-015-5487-y. MathSciNetCrossRefGoogle Scholar
  46. 46.
    Jäschke, R., Hotho, A., Schmitz, C., Ganter, B., Stumme, G.: TRIAS–an algorithm for mining iceberg tri-lattices. In: Proceedings of the Sixth International Conference on Data Mining, ICDM ’06, pp. 907–911. IEEE Computer Society, Washington, DC (2006). DOI
  47. 47.
    Jelassi, M.N., Yahia, S.B., Nguifo, E.M.: Towards more targeted recommendations in folksonomies. Soc. Netw. Anal. Min. 5(1), 68:1–68:18 (2015). DOI 10.1007/s13278-015-0307-8.
  48. 48.
    Jones, I., Tang, L., Liu, H.: Community discovery in multi-mode networks. In: Paliouras, G., Papadopoulos, S., Vogiatzis, D., Kompatsiaris, Y. (eds.) User Community Discovery, pp. 55–74. Springer, Cham (2015). DOI 10.1007/978-3-319-23835-7∖_3. Google Scholar
  49. 49.
    Kaytoue, M., Kuznetsov, S.O., Napoli, A., Duplessis, S.: Mining gene expression data with pattern structures in formal concept analysis. Inf. Sci. 181(10), 1989–2001 (2011)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Kaytoue, M., Kuznetsov, S.O., Macko, J., Napoli, A.: Biclustering meets triadic concept analysis. Ann. Math. Artif. Intell. 70, 55–79 (2014). DOI 10.1007/s10472-013-9379-1. MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Krasnov, F., Vlasova, E., Yavorskiy, R.: Connectivity analysis of computer science centers based on scientific publications data for major Russian cities. In: Proceedings of the Second International Conference on Information Technology and Quantitative Management, ITQM 2014, pp. 892–899, National Research University Higher School of Economics (HSE), Moscow, 3–5 June 2014. DOI 10.1016/j.procs.2014.05.341. CrossRefGoogle Scholar
  52. 52.
    Krolak-Schwerdt, S., Orlik, P., Ganter, B.: Tripat: a model for analyzing three-mode binary data. In: Bock, H.H., Lenski, W., Richter, M. (eds.) Information Systems and Data Analysis, Studies in Classification, Data Analysis, and Knowledge Organization, pp. 298–307. Springer, Berlin/Heidelberg (1994). DOI 10.1007/978-3-642-46808-7∖_27. zbMATHGoogle Scholar
  53. 53.
    Kuznetsov, S.O.: Stability as an estimate of the degree of substantiation of hypotheses derived on the basis of operational similarity. Nauchn. Tekh. Inf., Ser.2 (Autom. Doc. Math. Ling.) 12, 21 – 29 (1990)Google Scholar
  54. 54.
    Kuznetsov, S.O.: On stability of a formal concept. Ann. Math. Artif. Intell. 49(1–4), 101–115 (2007)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Kuznetsov, S.O., Ignatov, D.: Concept stability for constructing taxonomies of web-site users,. In: Obiedkov, S., Roth, C. (eds.) Proceedings of ICFCA 2007 Satellite Workshop on Social Network Analysis and Conceptual Structures: Exploring Opportunities, pp. 19–24. Clermont-Ferrand (2007)Google Scholar
  56. 56.
    Kuznetsov, S.O., Obiedkov, S.A., Roth, C.: Reducing the representation complexity of lattice-based taxonomies. In: Conceptual Structures: Knowledge Architectures for Smart Applications, Proceedings of the 15th International Conference on Conceptual Structures, ICCS 2007, pp. 241–254, Sheffield, 22–27 July 2007. DOI 10.1007/978-3-540-73681-3∖_18.
  57. 57.
    Latapy, M., Magnien, C., Vecchio, N.D.: Basic notions for the analysis of large two-mode networks. Soc. Netw. 30(1), 31 – 48 (2008). DOI 10. 1016/j.socnet.2007.04.006. CrossRefGoogle Scholar
  58. 58.
    Lehmann, F., Wille, R.: A triadic approach to formal concept analysis. In: Proceedings of the Third International Conference on Conceptual Structures: Applications, Implementation and Theory, pp. 32–43. Springer, London (1995). CrossRefGoogle Scholar
  59. 59.
    Lijffijt, J., Spyropoulou, E., Kang, B., Bie, T.D.: P-n-rminer: a generic framework for mining interesting structured relational patterns. Int. J. Data Sci. Anal. 1(1), 61–76 (2016). DOI 10.1007/s41060-016-0004-3. CrossRefGoogle Scholar
  60. 60.
    Liu, X., Murata, T.: Evaluating community structure in bipartite networks. In: Elmagarmid, A.K., Agrawal, D. (eds.) SocialCom/PASSAT, pp. 576–581. IEEE Computer Society, Washington, DC (2010)Google Scholar
  61. 61.
    Metzler, S., Miettinen, P.: Clustering boolean tensors. Data Min. Knowl. Discov. 29(5), 1343–1373 (2015). DOI 10.1007/s10618-015-0420-3. MathSciNetCrossRefGoogle Scholar
  62. 62.
    Miettinen, P.: Boolean tensor factorizations. In: 11th IEEE International Conference on Data Mining, ICDM 2011, pp. 447–456, Vancouver, BC, 11–14 December 2011. DOI 10.1109/ICDM.2011.28.
  63. 63.
    Mirkin, B.: Mathematical Classification and Clustering. Kluwer, Dordrecht (1996)CrossRefGoogle Scholar
  64. 64.
    Mirkin, B.G., Kramarenko, A.V.: Approximate bicluster and tricluster boxes in the analysis of binary data. In: Proceedings of the 13th International Conference on Rough Sets, Fuzzy Sets, Data Mining and Granular Computing, RSFDGrC’11, pp. 248–256. Springer, Berlin/Heidelberg (2011). Google Scholar
  65. 65.
    Mohr, J.W., Duquenne, V.: The Duality of Culture and Practice: Poverty Relief in New York City, 1888–1917. Theory Soc. 26(2/3), 305–356 (1997). Special Double Issue on New Directions in Formalization and Historical AnalysisGoogle Scholar
  66. 66.
    Murata, T.: Detecting communities from tripartite networks. In: Rappa, M., Jones, P., Freire, J., Chakrabarti, S. (eds.) WWW, pp. 1159–1160. ACM, New York (2010)Google Scholar
  67. 67.
    Newman, M.E.J.: Scientific collaboration networks. II. shortest paths, weighted networks, and centrality. Phys. Rev. E 64, 016,132 (2001). DOI 10.1103/PhysRevE.64.016132.
  68. 68.
    Nussbaum, D., Pu, S., Sack, J., Uno, T., Zarrabi-Zadeh, H.: Finding maximum edge bicliques in convex bipartite graphs. Algorithmica 64(2), 311–325 (2012). DOI 10.1007/s00453-010-9486-x. MathSciNetCrossRefGoogle Scholar
  69. 69.
    Opsahl, T.: Triadic closure in two-mode networks: redefining the global and local clustering coefficients. Soc. Netw. 34 (2011). DOI 10.1016/j. socnet.2011.07.001. (in press)
  70. 70.
    Padilha, V.A., Campello, R.J.G.B.: A systematic comparative evaluation of biclustering techniques. BMC Bioinf. 18(1), 55:1–55:25 (2017). DOI 10.1186/s12859-017-1487-1.
  71. 71.
    Papalexakis, E.E., Faloutsos, C., Sidiropoulos, N.D.: Tensors for data mining and data fusion: Models, applications, and scalable algorithms. ACM Trans. Intell. Syst. Technol. 8(2), 16:1–16:44 (2016). DOI 10.1145/2915921. CrossRefGoogle Scholar
  72. 72.
    Poelmans, J., Elzinga, P., Ignatov, D.I., Kuznetsov, S.O.: Semi-automated knowledge discovery: identifying and profiling human trafficking. Int. J. Gen. Syst. 41(8), 774–804 (2012). DOI 10.1080/03081079.2012.721662. MathSciNetCrossRefGoogle Scholar
  73. 73.
    Poelmans, J., Ignatov, D.I., Kuznetsov, S.O., Dedene, G.: Formal concept analysis in knowledge processing: a survey on applications. Expert Syst. Appl. 40(16), 6538–6560 (2013). DOI 10.1016/j.eswa.2013.05.009. CrossRefGoogle Scholar
  74. 74.
    Poelmans, J., Kuznetsov, S.O., Ignatov, D.I., Dedene, G.: Formal concept analysis in knowledge processing: a survey on models and techniques. Expert Syst. Appl. 40(16), 6601–6623 (2013). DOI 10.1016/j.eswa.2013.05.007. CrossRefGoogle Scholar
  75. 75.
    Roth, C.: Generalized preferential attachment: towards realistic socio-semantic network models. In: ISWC 4th Intl Semantic Web Conference, Workshop on Semantic Network Analysis, Galway, CEUR-WS Series (ISSN 1613-0073), vol. 171, pp. 29–42 (2005)Google Scholar
  76. 76.
    Roth, C., Cointet, J.P.: Social and semantic coevolution in knowledge networks. Soc. Netw. 32, 16–29 (2010)CrossRefGoogle Scholar
  77. 77.
    Roth, C., Obiedkov, S.A., Kourie, D.G.: Towards concise representation for taxonomies of epistemic communities. In: Yahia, S.B., Nguifo, E.M., Belohlávek, R. (eds.) CLA. Lecture Notes in Computer Science, vol. 4923, pp. 240–255. Springer, Heidelberg (2006)Google Scholar
  78. 78.
    Roth, C., Obiedkov, S.A., Kourie, D.G.: On succinct representation of knowledge community taxonomies with formal concept analysis. Int. J. Found. Comput. Sci. 19(2), 383–404 (2008). DOI 10.1142/S0129054108005735. MathSciNetCrossRefGoogle Scholar
  79. 79.
    Shin, K., Hooi, B., Faloutsos, C.: M-zoom: Fast dense-block detection in tensors with quality guarantees. In: Machine Learning and Knowledge Discovery in Databases - Proceedings of the European Conference, ECML PKDD 2016, Part I, pp. 264–280, Riva del Garda, 19–23 September 2016. DOI 10.1007/978-3-319-46128-1∖_17. CrossRefGoogle Scholar
  80. 80.
    Spyropoulou, E., Bie, T.D., Boley, M.: Interesting pattern mining in multi-relational data. Data Min. Knowl. Discov. 28(3), 808–849 (2014). DOI 10.1007/s10618-013-0319-9. MathSciNetCrossRefGoogle Scholar
  81. 81.
    Tang, L., Liu, H., Zhang, J., Nazeri, Z.: Community evolution in dynamic multi-mode networks. In: Proceedings of the 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 677–685, Las Vegas, NV, 24–27 August 2008. DOI 10.1145/1401890.1401972.
  82. 82.
    Tsymbal, A., Pechenizkiy, M., Cunningham, P.: Diversity in search strategies for ensemble feature selection. Inf. Fusion 6(1), 83–98 (2005)CrossRefGoogle Scholar
  83. 83.
    Vander Wal, T.: Folksonomy coinage and definition. URL (2007). Accessed 12 Mar 2012
  84. 84.
    Veremyev, A., Prokopyev, O.A., Butenko, S., Pasiliao, E.L.: Exact mip-based approaches for finding maximum quasi-cliques and dense subgraphs. Comput. Optim. Appl. 64(1), 177–214 (2016). DOI 10.1007/s10589-015-9804-y. MathSciNetCrossRefGoogle Scholar
  85. 85.
    Voutsadakis, G.: Polyadic concept analysis. Order 19(3), 295–304 (2002)MathSciNetCrossRefGoogle Scholar
  86. 86.
    White, D.R.: Statistical entailments and the galois lattice. Soc. Netw. 18(3), 201–215 (1996). DOI 10.1016/0378-8733(95)00273-1. CrossRefGoogle Scholar
  87. 87.
    Wille, R.: The basic theorem of triadic concept analysis. Order 12, 149–158 (1995)MathSciNetCrossRefGoogle Scholar
  88. 88.
    Wu, Z., Bu, Z., Cao, J., Zhuang, Y.: Discovering communities in multi-relational networks. In: Paliouras, G., Papadopoulos, S., Vogiatzis, D., Kompatsiaris, Y. (eds.) User Community Discovery, pp. 75–95. Springer, Cham (2015). DOI 10.1007/978-3-319-23835-7∖_4. Google Scholar
  89. 89.
    Yavorsky, R.: Research challenges of dynamic socio-semantic networks. In: Ignatov, D., Poelmans, J., Kuznetsov, S. (eds.) CEUR Workshop Proceedings, CDUD’11 - Concept Discovery in Unstructured Data, vol. 757, pp. 119–122 (2011)Google Scholar
  90. 90.
    Zachary, W.W.: An information flow model for conflict and fission in small groups. J. Anthropol. Res. 33(4), 452–473 (1977). CrossRefGoogle Scholar
  91. 91.
    Zakhlebin, I., Semenov, A., Tolmach, A., Nikolenko, S.I.: Detecting opinion polarisation on twitter by constructing pseudo-bimodal networks of mentions and retweets. In: Information Retrieval - 9th Russian Summer School, RuSSIR 2015, pp. 169–178, Saint Petersburg, 24–28 August 2015, Revised Selected Papers (2015). DOI 10.1007/978-3-319-41718-9∖_10. Google Scholar
  92. 92.
    Zhao, L., Zaki, M.J.: Tricluster: An effective algorithm for mining coherent clusters in 3d microarray data. In: Proceedings of the ACM SIGMOD International Conference on Management of Data, pp. 694–705, Baltimore, Maryland, 14–16 June 2005. DOI 10.1145/1066157.1066236.
  93. 93.
    Zhuk, R., Ignatov, D.I., Konstantinova, N.: Concept learning from triadic data. In: Proceedings of the Second International Conference on Information Technology and Quantitative Management, ITQM 2014, pp. 928–938, National Research University Higher School of Economics (HSE), Moscow, 3–5 June 2014. DOI 10.1016/j.procs.2014.05.345.
  94. 94.
    Zudin, S., Gnatyshak, D.V., Ignatov, D.I.: Putting oac-triclustering on mapreduce. In: Proceedings of the Twelfth International Conference on Concept Lattices and Their Applications, pp. 47–58, Clermont-Ferrand, 13–16 October 2015.

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© Springer International Publishing AG 2017

Authors and Affiliations

  • Dmitry I. Ignatov
    • 1
    Email author
  • Alexander Semenov
    • 1
    • 2
  • Daria Komissarova
    • 1
  • Dmitry V. Gnatyshak
    • 1
  1. 1.National Research University Higher School of EconomicsMoscowRussia
  2. 2.Mobile TeleSystems PJSCMoscowRussia

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