Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Achterberg T (2009) SCIP: solving constraint integer programs. Math Prog Comp 1(1):1–41
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 258:5136–5139
Borgatti S, Everett M (1993) Two algorithms for computing regular equivalence. Soc Netw 15(4):361–376
Borgatti S, Everett M (1992) Regular blockmodels of multiway, multimode matrices. Soc Netw 14:91–120
Borgatti SP, Everett MG, Freeman LC (2002) Ucinet for Windows: software for social network analysis. Analytic Technologies, Harvard
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):577–585
De Nooy W, Mrvar A, Batagelj V (2005) Exploratory social network analysis with Pajek. Cambridge University Press, Cambridge
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):247–266
Frieze A, Yadegar J (1983) On the quadratic assignment problem. Discrete Appl Math 5(1):89–98
Grötschel M, Wakabayashi Y (1989) A cutting plane algorithm for a clustering problem. Math Prog 45(1–3):59–96
Johnson EL, Mehrotra A, Nemhauser GL (1993) Min-cut clustering. Math Prog 62(1–3):133–151
Kaibel V (1997) Polyhedral combinatorics of the quadratic assignment problem. Ph.D. thesis, Köln University
Kernighan BW, Lin S (1970) An efficient heuristic procedure for partitioning graphs. Bell Sys Tech J 49(1):291–307
Liberti L (2007) Compact linearization for binary quadratic problems. 4OR 5(3):231–245
Luczkovic JJ, Borgatti SP, Johnson JC, Everett MG (2003) Defining and measuring trophic role similarity in food webs using regular equivalence. J Theor Biol 220:303–321
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 1–7
Newman MEJ, Girvan M (2004) Finding and evaluating community structure in networks. Phys Rev E 69(2):026113
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 29:59–69
Padberg M (1989) The boolean quadric polytope: some characteristics, facets and relatives. Math Prog 45:139–172
Paige R, Tarjan R (1987) Three partition refinement algorithms. SIAM J Comput 16(6):973–989
Rašković M, Žnidaršič A, Udovič B (2011) Application of weighted blockmodeling in the analysis of small EU states’ export patterns. Manuskript, Ljubljana University
Roberts FS, Sheng L (2001) How hard is it to determine if a graph has a 2-role assignment? Networks 37(2):67–73
Smith DA, White DR (1992) Structure and dynamics of the global economy: network analysis of international trade 1965–1980. Soc Forces 70(4):857–893
Wasserman S, Faust K (1994) Social network analysis: methods and applications, vol 8. Cambridge University Press, Cambridge
Watts D, Strogatz S (1998) Collective dynamics of small-world networks. Nature 393(6684):440–442
White DR, Reitz KP (1983) Graph and semigroup homomorphisms on networks of relations. Soc Netw 5(2):193–234
Wiesberg S, Reinelt G (2013) Relaxations in Practical Clustering and Blockmodeling. In: Proceedings of the 16th international information society multiconference, part A, Ljubljana, pp 607–610
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wiesberg, S., Reinelt, G. Evaluating the quality of image matrices in blockmodeling. EURO J Comput Optim 3, 111–129 (2015). https://doi.org/10.1007/s13675-015-0034-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13675-015-0034-y