Evaluating the quality of image matrices in blockmodeling

  • Stefan WiesbergEmail author
  • Gerhard Reinelt
Original Paper


One approach for analyzing large networks is to partition its nodes into classes where the nodes in a class have similar characteristics with respect to their connections in the network. A class is represented as a blockmodel (or image matrix). In this context, a specific question is to test whether a presumed blockmodel is well reflected in the network or to select from a choice of possible blockmodels the one fitting best. In this paper, we formulate these problems as combinatorial optimization problems. We show that the evaluation of a blockmodel’s quality is a generalization of well-known optimization problems such as quadratic assignment, minimum \(k\)-cut, traveling salesman, and minimum edge cover. A quadratic integer programming formulation is derived and linearized by making use of properties of these special cases. With a branch-and-cut approach, the resulting formulation is solved up to 10,000 times faster than a comparable formulation from the literature.


Regular equivalence Blockmodeling Quadratic integer programming Linearization techniques 

Mathematics Subject Classification

80-08 90C11 90B10 05C82 


  1. Achterberg T (2009) SCIP: solving constraint integer programs. Math Prog Comp 1(1):1–41CrossRefGoogle Scholar
  2. Balas E (1964) Extension de l’algorithme additif a la programmation en nombres entiers et a la programmation non lineaire. C R Acad Sci Paris 258:5136–5139Google Scholar
  3. Borgatti S, Everett M (1993) Two algorithms for computing regular equivalence. Soc Netw 15(4):361–376CrossRefGoogle Scholar
  4. Borgatti S, Everett M (1992) Regular blockmodels of multiway, multimode matrices. Soc Netw 14:91–120CrossRefGoogle Scholar
  5. Borgatti SP, Everett MG, Freeman LC (2002) Ucinet for Windows: software for social network analysis. Analytic Technologies, HarvardGoogle Scholar
  6. Brusco MJ, Steinley D (2009) Integer programs for one-and two-mode blockmodeling based on prespecified image matrices for structural and regular equivalence. J Math Psychol 53(6):577–585CrossRefGoogle Scholar
  7. De Nooy W, Mrvar A, Batagelj V (2005) Exploratory social network analysis with Pajek. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  8. Ferreira C, Martin A, de Souza C, Weismantel R, Wolsey L (1996) Formulations and valid inequalities for the node capacitated graph partitioning problem. Math Prog 74(3):247–266CrossRefGoogle Scholar
  9. Frieze A, Yadegar J (1983) On the quadratic assignment problem. Discrete Appl Math 5(1):89–98CrossRefGoogle Scholar
  10. Grötschel M, Wakabayashi Y (1989) A cutting plane algorithm for a clustering problem. Math Prog 45(1–3):59–96CrossRefGoogle Scholar
  11. Johnson EL, Mehrotra A, Nemhauser GL (1993) Min-cut clustering. Math Prog 62(1–3):133–151CrossRefGoogle Scholar
  12. Kaibel V (1997) Polyhedral combinatorics of the quadratic assignment problem. Ph.D. thesis, Köln UniversityGoogle Scholar
  13. Kernighan BW, Lin S (1970) An efficient heuristic procedure for partitioning graphs. Bell Sys Tech J 49(1):291–307CrossRefGoogle Scholar
  14. Liberti L (2007) Compact linearization for binary quadratic problems. 4OR 5(3):231–245CrossRefGoogle Scholar
  15. Luczkovic JJ, Borgatti SP, Johnson JC, Everett MG (2003) Defining and measuring trophic role similarity in food webs using regular equivalence. J Theor Biol 220:303–321CrossRefGoogle Scholar
  16. Melancon G (2006) Just how dense are dense graphs in the real world? In: Proceedings of the 2006 AVI workshop on beyond time and errors, pp 1–7Google Scholar
  17. Newman MEJ, Girvan M (2004) Finding and evaluating community structure in networks. Phys Rev E 69(2):026113CrossRefGoogle Scholar
  18. Nordlund C (2007) Identifying regular blocks in valued networks: a heuristic applied to the St. Marks carbon flow data, and international trade in cereal products. Soc Netw 29:59–69CrossRefGoogle Scholar
  19. Padberg M (1989) The boolean quadric polytope: some characteristics, facets and relatives. Math Prog 45:139–172CrossRefGoogle Scholar
  20. Paige R, Tarjan R (1987) Three partition refinement algorithms. SIAM J Comput 16(6):973–989CrossRefGoogle Scholar
  21. Rašković M, Žnidaršič A, Udovič B (2011) Application of weighted blockmodeling in the analysis of small EU states’ export patterns. Manuskript, Ljubljana UniversityGoogle Scholar
  22. Roberts FS, Sheng L (2001) How hard is it to determine if a graph has a 2-role assignment? Networks 37(2):67–73CrossRefGoogle Scholar
  23. Smith DA, White DR (1992) Structure and dynamics of the global economy: network analysis of international trade 1965–1980. Soc Forces 70(4):857–893CrossRefGoogle Scholar
  24. Wasserman S, Faust K (1994) Social network analysis: methods and applications, vol 8. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  25. Watts D, Strogatz S (1998) Collective dynamics of small-world networks. Nature 393(6684):440–442CrossRefGoogle Scholar
  26. White DR, Reitz KP (1983) Graph and semigroup homomorphisms on networks of relations. Soc Netw 5(2):193–234CrossRefGoogle Scholar
  27. Wiesberg S, Reinelt G (2013) Relaxations in Practical Clustering and Blockmodeling. In: Proceedings of the 16th international information society multiconference, part A, Ljubljana, pp 607–610Google Scholar

Copyright information

© EURO - The Association of European Operational Research Societies 2015

Authors and Affiliations

  1. 1.Institut für InformatikUniversität HeidelbergHeidelbergGermany

Personalised recommendations