A Formal Concept Analysis Look at the Analysis of Affiliation Networks

  • Francisco J. Valverde-AlbaceteEmail author
  • Carmen Peláez-Moreno
Part of the Lecture Notes in Social Networks book series (LNSN)


In this paper we try to analyse the dual-projection approach to weighted 2-mode networks using the tools of\(\mathcal{K}\)-Formal Concept Analysis (\(\mathcal{K}\)-FCA), an extension of FCA for incidences with values in a particular kind of semiring.

For this purpose, we first revisit the isomorphisms between 2-mode networks with formal contexts. In the quest for similar relations when the networks have non-Boolean weights, we relate the dual-projection method to both the Singular Value Decomposition and the Eigenvalue Problem of matrices with values in such algebras, as embodied in Kleinberg’s Hubs and Authorities (HITS) algorithm.

To recover a relation with multi-valued extensions of FCA, we introduce extensions of the HITS algorithm to calculate the influence of nodes in a network whose adjacency matrix takes values over dioids, zerosumfree semirings with a natural order. In this way, we show the original HITS algorithm to be a particular instance of the generic construction, but also the advantages of working in idempotent semifields, instances of dioids.

Subsequently, we also make some connections with extended\(\mathcal{K}\)-FCA, where the particular kind of dioid is an idempotent semifield, and provide theoretical reasoning and evidence that the type of knowledge extracted from a matrix by one procedure and the other are different.


Weighted two-mode networks Dual-projection analysis HITS SVD Idempotent semifield Idempotent Formal Concept Analysis Idempotent HITS 



The authors have been partially supported by the Spanish Government-MinECo projects TEC2014-53390-P and TEC2014-61729-EXP for this work.

We would like to thank the reviewers of earlier versions of this paper for their help in improving it.


  1. 1.
    Bang-Jensen, J., Gutin, G.: Digraphs. Theory, Algorithms, and Applications, 3rd edn. Springer, Heidelberg (2001)Google Scholar
  2. 2.
    Agneessens, F., Everett, M.G.: Introduction to the special issue on advances in two-mode social networks. Soc. Netw. 35, 145–147 (2013)CrossRefGoogle Scholar
  3. 3.
    Latapy, M., Magnien, C., Vecchio, N.D.: Basic notions for the analysis of large two-mode networks. Soc. Netw. 30, 31–48 (2008)CrossRefGoogle Scholar
  4. 4.
    Everett, M.G., Borgatti, S.P.: The dual-projection approach for two-mode networks. Soc. Netw. 35, 204–210 (2013)CrossRefGoogle Scholar
  5. 5.
    Strang, G.: The fundamental theorem of linear algebra. Am. Math. Mon. 100, 848–855 (1993)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. JHU Press, Baltimore (2012)zbMATHGoogle Scholar
  7. 7.
    Landauer, T.K., McNamara, D.S., Dennis, S., Kintsch, W.: Handbook of Latent Semantic Analysis. Lawrence Erlbaum Associates, Mahwah (2007)Google Scholar
  8. 8.
    Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundations. Springer, Berlin/Heidelberg (1999)CrossRefGoogle Scholar
  9. 9.
    Wille, R.: Restructuring lattice theory: an approach based on hierarchies of concepts. In: Ordered Sets (Banff, Alta., 1981), pp. 445–470. Reidel, Boston (1982)CrossRefGoogle Scholar
  10. 10.
    Davey, B., Priestley, H.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002)CrossRefGoogle Scholar
  11. 11.
    Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society, Providence (1967)zbMATHGoogle Scholar
  12. 12.
    Rota, G.C.: Indiscrete Thoughts. Springer, Boston, MA (2009)Google Scholar
  13. 13.
    Freeman, L.C., White, D.R.: Using Galois lattices to represent network data. Sociol. Methodol. 23, 127–146 (1993)CrossRefGoogle Scholar
  14. 14.
    Domenach, F.: CryptoLat - a pedagogical software on lattice cryptomorphisms and lattice properties. In: Ojeda-Aciego, M., Outrata, J. (eds.) 10th International Conference on Concept Lattices and Their Applications (2013)Google Scholar
  15. 15.
    Gaume, B., Navarro, E., Prade, H.: A parallel between extended formal concept analysis and bipartite graphs analysis. In: IPMU’10: Proceedings of the Computational Intelligence for Knowledge-Based Systems Design, and 13th International Conference on Information Processing and Management of Uncertainty, Universite Paul Sabatier Toulouse III. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  16. 16.
    Kuznetsov, S.O.: Interpretation on graphs and complexity characteristics of a search for specific patterns. Nauchno-Tekhnicheskaya Informatsiya Seriya - Informationnye i sistemy 1, 23–27 (1989)Google Scholar
  17. 17.
    Falzon, L.: Determining groups from the clique structure in large social networks. Soc. Netw. 22, 159–172 (2000)CrossRefGoogle Scholar
  18. 18.
    Ali, S.S., Bentayeb, F., Missaoui, R., Boussaid, O.: An efficient method for community detection based on formal concept analysis. In: Foundations of Intelligent Systems, pp. 61–72. Springer, New York (2014)Google Scholar
  19. 19.
    Roth, C., Bourgine, P.: Epistemic communities: description and hierarchic categorization. Math. Popul. Stud. 12 107–130 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Freeman, L.C.: Methods of social network visualization. In: Meyers, R.A. (ed.) Encyclopedia of Complexity and Systems Science, Entry 25, pp. 1–19. Springer, New York (2008)Google Scholar
  21. 21.
    Duquenne, V.: On lattice approximations: syntactic aspects. Soc. Netw. 18, 189–199 (1996)CrossRefGoogle Scholar
  22. 22.
    Bělohlávek, R., Vychodil, V.: Formal concepts as optimal factors in Boolean factor analysis: implications and experiments. In: Proceedings of the 5th International Conference on Concept Lattices and Their Applications, (CLA07), Montpellier, 24–26 October 2007Google Scholar
  23. 23.
    Bělohlávek, R.: Fuzzy Relational Systems. Foundations and Principles. IFSR International Series on Systems Science and Engineering, vol. 20. Kluwer Academic, Norwell (2002)CrossRefGoogle Scholar
  24. 24.
    Valverde-Albacete, F.J., Peláez-Moreno, C.: Extending conceptualisation modes for generalised Formal Concept Analysis. Inf. Sci. 181, 1888–1909 (2011)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Bělohlávek, R.: Optimal decompositions of matrices with entries from residuated lattices. J. Log. Comput. 22 (2012) 1405–1425MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kleinberg, J.M.: Authoritative sources in a hyperlinked environment. J. ACM 46 (1999) 604–632MathSciNetCrossRefGoogle Scholar
  27. 27.
    Easley, D.A., Kleinberg, J.M.: Networks, Crowds, and Markets - Reasoning About a Highly Connected World. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  28. 28.
    Kolaczyk, E.D., Csárdi, G.: Statistical Analysis of Network Data with R. Use R!, vol. 65. Springer, New York, NY (2014)zbMATHGoogle Scholar
  29. 29.
    Lanczos, C.: Linear Differential Operators. Dover, New York (1997)zbMATHGoogle Scholar
  30. 30.
    Golan, J.S.: Power Algebras over Semirings. With Applications in Mathematics and Computer Science. Mathematics and Its applications, vol. 488. Kluwer Academic, Dordrecht (1999)CrossRefGoogle Scholar
  31. 31.
    Pouly, M., Kohlas, J.: Generic Inference. A Unifying Theory for Automated Reasoning. Wiley, Hoboken (2012)zbMATHGoogle Scholar
  32. 32.
    Golan, J.S.: Semirings and Their Applications. Kluwer Academic, Dordrecht (1999)CrossRefGoogle Scholar
  33. 33.
    Gondran, M., Minoux, M.: Graphs, Dioids and Semirings. New Models and Algorithms. Operations Research/Computer Science Interfaces. Springer, New York (2008)zbMATHGoogle Scholar
  34. 34.
    Gondran, M., Minoux, M.: Valeurs propres et vecteurs propres dans les dioïdes et leur interprétation en théorie des graphes. EDF, Bulletin de la Direction des Etudes et Recherches, Serie C, Mathématiques Informatique 2, 25–41 (1977)Google Scholar
  35. 35.
    Valverde-Albacete, F.J., Peláez-Moreno, C.: The spectra of irreducible matrices over completed idempotent semifields. Fuzzy Sets Syst. 271, 46–69 (2015)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Valverde-Albacete, F.J., Peláez-Moreno, C.: The spectra of reducible matrices over complete commutative idempotent semifields and their spectral lattices. Int. J. Gen. Syst. 45, 86–115 (2016)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Valverde-Albacete, F.J., Peláez-Moreno, C.: Spectral lattices of reducible matrices over completed idempotent semifields. In: Ojeda-Aciego, M., Outrata, J., (eds.) Concept Lattices and Applications (CLA 2013), pp. 211–224. Université de la Rochelle, Laboratory L31, La Rochelle (2013)Google Scholar
  38. 38.
    Cohen, G., Gaubert, S., Quadrat, J.P.: Duality and separation theorems in idempotent semimodules. Linear Algebra Appl. 379, 395–422 (2004)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Peláez-Moreno, C., García-Moral, A.I., Valverde-Albacete, F.J.: Analyzing phonetic confusions using Formal Concept Analysis. J. Acoust. Soc. Am. 128, 1377–1390 (2010)CrossRefGoogle Scholar
  40. 40.
    Valverde-Albacete, F.J., González-Calabozo, J.M., Peñas, A., Peláez-Moreno, C.: Supporting scientific knowledge discovery with extended, generalized formal concept analysis. Expert Syst. Appl. 44, 198–216 (2016)CrossRefGoogle Scholar
  41. 41.
    González-Calabozo, J.M., Valverde-Albacete, F.J., Peláez-Moreno, C.: Interactive knowledge discovery and data mining on genomic expression data with numeric formal concept analysis. BMC Bioinf. 17, 374 (2016)CrossRefGoogle Scholar
  42. 42.
    Valverde-Albacete, F.J., Peláez-Moreno, C.: The linear algebra in extended formal concept analysis over idempotent semifields. In: Bertet, K., Borchmann, D. (eds.) Formal Concept Analysis, Springer Berlin Heidelberg, 211–227 (2017)CrossRefGoogle Scholar
  43. 43.
    Mirkin, B.: Mathematical Classification and Clustering. Nonconvex Optimization and Its Applications, vol. 11. Kluwer Academic, Dordrecht (1996)CrossRefGoogle Scholar
  44. 44.
    Akian, M., Gaubert, S., Ninove, L.: Multiple equilibria of nonhomogeneous Markov chains and self-validating web rankings. arXiv:0712.0469 (2007)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Francisco J. Valverde-Albacete
    • 1
    Email author
  • Carmen Peláez-Moreno
    • 1
  1. 1.Department of Signal Theory and CommunicationsUniversidad Carlos III de MadridLeganésSpain

Personalised recommendations