A Graph-Theoretic Formulation of Exploratory Blockmodeling

Authors Alexander Bille, Niels Grüttemeier , Christian Komusiewicz , Nils Morawietz

Thumbnail PDF


  • Filesize: 1.64 MB
  • 20 pages

Document Identifiers

Author Details

Alexander Bille
  • Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Germany
Niels Grüttemeier
  • System Technologies and Image Exploitation, Fraunhofer IOSB, Lemgo, Fraunhofer Institute of Optronics, Germany
Christian Komusiewicz
  • Institute of Computer Science, Friedrich Schiller Universität Jena, Germany
Nils Morawietz
  • Institute of Computer Science, Friedrich Schiller Universität Jena, Germany


We would like to thank the anonymous reviewers of SEA for their helpful comments which have improved the presentation of our results.

Cite AsGet BibTex

Alexander Bille, Niels Grüttemeier, Christian Komusiewicz, and Nils Morawietz. A Graph-Theoretic Formulation of Exploratory Blockmodeling. In 21st International Symposium on Experimental Algorithms (SEA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 265, pp. 14:1-14:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We present a new simple graph-theoretic formulation of the exploratory blockmodeling problem on undirected and unweighted one-mode networks. Our formulation takes as input the network G and the maximum number t of blocks for the solution model. The task is to find a minimum-size set of edge insertions and deletions that transform the input graph G into a graph G' with at most t neighborhood classes. Herein, a neighborhood class is a maximal set of vertices with the same neighborhood. The neighborhood classes of G' directly give the blocks and block interactions of the computed blockmodel. We analyze the classic and parameterized complexity of the exploratory blockmodeling problem, provide a branch-and-bound algorithm, an ILP formulation and several heuristics. Finally, we compare our exact algorithms to previous ILP-based approaches and show that the new algorithms are faster for t ≥ 4.

Subject Classification

ACM Subject Classification
  • Theory of computation → Social networks
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Branch-and-bound
  • Clustering
  • Exact Algorithms
  • ILP-Formulation
  • Branch-and-Bound
  • Social Networks


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Vladimir Batagelj, Anuška Ferligoj, and Patrick Doreian. Direct and indirect methods for structural equivalence. Social Networks, 14(1):63-90, 1992. Special Issue on Blockmodels. Google Scholar
  2. Stephen P. Borgatti and Martin G. Everett. Models of core/periphery structures. Social Networks, 21(4):375-395, 2000. Google Scholar
  3. Stephen P. Borgatti, Martin G. Everett, and Jeffrey C. Johnson. Analyzing social networks. SAGE, 2013. Google Scholar
  4. Sharon Bruckner, Falk Hüffner, and Christian Komusiewicz. A graph modification approach for finding core-periphery structures in protein interaction networks. Algorithms for Molecular Biology, 10:16, 2015. Google Scholar
  5. Michael J. Brusco and Douglas Steinley. Integer programs for one- and two-mode blockmodeling based on prespecified image matrices for structural and regular equivalence. Journal of Mathematical Psychology, 53(6):577-585, 2009. Google Scholar
  6. Jeffrey Chan, Wei Liu, Andrey Kan, Christopher Leckie, James Bailey, and Kotagiri Ramamohanarao. Discovering latent blockmodels in sparse and noisy graphs using non-negative matrix factorisation. In 22nd ACM International Conference on Information and Knowledge Management (CIKM '13), pages 811-816. ACM, 2013. Google Scholar
  7. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  8. Matthew Dabkowski, Neng Fan, and Ronald L. Breiger. Exploratory blockmodeling for one-mode, unsigned, deterministic networks using integer programming and structural equivalence. Social Networks, 47:93-106, 2016. Google Scholar
  9. Peter Damaschke and Olof Mogren. Editing simple graphs. Journal of Graph Algorithms and Applications, 18(4):557-576, 2014. Google Scholar
  10. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. Google Scholar
  11. Michael R. Fellows, Jiong Guo, Christian Komusiewicz, Rolf Niedermeier, and Johannes Uhlmann. Graph-based data clustering with overlaps. Discrete Optimization, 8(1):2-17, 2011. Google Scholar
  12. Michel Habib and Christophe Paul. A survey of the algorithmic aspects of modular decomposition. Computer Science Review, 4(1):41-59, 2010. Google Scholar
  13. Falk Hüffner, Christian Komusiewicz, and André Nichterlein. Editing graphs into few cliques: Complexity, approximation, and kernelization schemes. In Proceedings of the 14th International Symposium on Algorithms and Data Structures (WADS '15), volume 9214 of Lecture Notes in Computer Science, pages 410-421. Springer, 2015. Google Scholar
  14. Alan Jessop. Blockmodels with maximum concentration. European Journal of Operational Research, 148(1):56-64, 2003. Google Scholar
  15. Ivan Kovác, Ivana Selecéniová, and Monika Steinová. On the clique editing problem. In Proceedings of the 39th International Symposium on Mathematical Foundations of Computer Science (MFCS '14), volume 8635 of Lecture Notes in Computer Science, pages 469-480. Springer, 2014. Google Scholar
  16. Jérôme Kunegis. KONECT: the koblenz network collection. In Proceedings of the 22nd International World Wide Web Conference (WWW '13), pages 1343-1350. International World Wide Web Conferences Steering Committee / ACM, 2013. Google Scholar
  17. Michael Lampis. Algorithmic meta-theorems for restrictions of treewidth. Algorithmica, 64(1):19-37, 2012. Google Scholar
  18. Ross M. McConnell and Jeremy P. Spinrad. Modular decomposition and transitive orientation. Discrete Mathematics, 201(1-3):189-241, 1999. Google Scholar
  19. Les G. Proll. ILP approaches to the blockmodel problem. European Journal of Operational Research, 177(2):840-850, 2007. Google Scholar
  20. K. E. Read. Cultures of the central highlands, new guinea. Southwestern Journal of Anthropology, 10(1):1-43, 1954. Google Scholar
  21. Jörg Reichardt and Douglas R White. Role models for complex networks. The European Physical Journal B, 60(2):217-224, 2007. Google Scholar
  22. Samuel F Sampson. Crisis in a cloister. PhD thesis, Ph. D. Thesis. Cornell University, Ithaca, 1969. Google Scholar
  23. Ron Shamir, Roded Sharan, and Dekel Tsur. Cluster graph modification problems. Discrete Applied Mathematics, 144(1-2):173-182, 2004. Google Scholar
  24. Gerhard G Van de Bunt, Marijtje AJ Van Duijn, and Tom AB Snijders. Friendship networks through time: An actor-oriented dynamic statistical network model. Computational & Mathematical Organization Theory, 5(2):167-192, 1999. Google Scholar
  25. Stanley Wasserman and Katherine Faust. Social Network Analysis: Methods and Applications. Cambridge University Press, 1994. Google Scholar
  26. Wayne W. Zachary. An information flow model for conflict and fission in small groups. Journal of Anthropological Research, 33(4):452-473, 1977. Google Scholar