EURO Journal on Computational Optimization

, Volume 5, Issue 4, pp 467–498 | Cite as

Evaluating balancing on social networks through the efficient solution of correlation clustering problems

  • Mario Levorato
  • Rosa FigueiredoEmail author
  • Yuri Frota
  • Lúcia Drummond
Original Paper


One challenge for social network researchers is to evaluate balance in a social network. The degree of balance in a social group can be used as a tool to study whether and how this group evolves to a possible balanced state. The solution of clustering problems defined on signed graphs can be used as a criterion to measure the degree of balance in social networks and this measure can be obtained with the optimal solution of the correlation clustering problem, as well as a variation of it, the relaxed correlation clustering problem. However, solving these problems is no easy task, especially when large network instances need to be analyzed. In this work, we contribute to the efficient solution of both problems by developing sequential and parallel ILS metaheuristics. Then, by using our algorithms, we solve the problem of measuring the structural balance on large real-world social networks.


Correlation clustering Social network Structural balance ILS VND 

Mathematics Subject Classification

90C35 05C22 91D30 90C59 



The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.

Supplementary material

13675_2017_82_MOESM1_ESM.pdf (219 kb)
Supplementary material 1 (pdf 218 KB)


  1. Abell P, Ludwig M (2009) Structural balance: a dynamic perspective. J Math Sociol 33:129–155CrossRefGoogle Scholar
  2. Aiex RM, Resende MGC, Ribeiro CC (2007) TTT plots: a perl program to create time-to-target plots. Optim Lett 1(4):355–366CrossRefGoogle Scholar
  3. Ailon N, Charikar M, Newman A (2008) Aggregating inconsistent information: ranking and clustering. J ACM 55(5):23CrossRefGoogle Scholar
  4. Alba E (2005) Parallel metaheuristics: a new class of algorithms, vol 47. Wiley, New YorkCrossRefGoogle Scholar
  5. Allison GT (1969) Conceptual models and the cuban missile crisis. Am Polit Sci Rev 63(03):689–718CrossRefGoogle Scholar
  6. Bache K, Lichman M (2013) UCI machine learning repository.
  7. Bansal N, Blum A, Chawla S (2002) Correlation clustering. In: Proceedings of the 43rd annual IEEE symposium of foundations of computer science. Vancouver, Canada, pp 238–250Google Scholar
  8. Bhattacharya A, De RK (2008) Divisive correlation clustering algorithm (DCCA) for grouping of genes: detecting varying patterns in expression profiles. Bioinformatics 24(11):1359–1366CrossRefGoogle Scholar
  9. Bonchi F, Gionis A, Ukkonen A (2011) Overlapping correlation clustering. 2011 IEEE 11th international conference on data mining (ICDM). IEEE, pp 51–60Google Scholar
  10. Brandes U, Delling D, Gaertler M, Gorke R, Hoefer M, Nikoloski Z, Wagner D (2008) On modularity clustering. IEEE Trans Knowl Data Eng 20(2):172–188CrossRefGoogle Scholar
  11. Brusco M (2003) An enhanced branch-and-bound algorithm for a partitioning problem. Br J Math Stat Psychol 56:83–92CrossRefGoogle Scholar
  12. Brusco M, Doreian P, Mrvar A, Steinly D (2011) Two algorithms for relaxed structural balance partitioning: linking theory, models and data to understand social network phenomena. Sociol Methods Res 40:57–87CrossRefGoogle Scholar
  13. Brusco MJ, Köhn H-F (2009) Clustering qualitative data based on binary equivalence relations: neighborhood search heuristics for the clique partitioning problem. Psychometrika 74(4):685–703CrossRefGoogle Scholar
  14. Cartwright D, Harary F (1956) Structural balance: a generalization of Heider’s theory. Psychol Rev 63:277–293CrossRefGoogle Scholar
  15. Charikara M, Guruswamib V, Wirtha A (2005) Clustering with qualitative information. J Comput Syst Sci 71:360–383CrossRefGoogle Scholar
  16. Chiang K-Y, Hsieh C-J, Natarajan N, Tewari A, Inderjit SD (2013) Prediction and clustering in signed networks. A local to global perspective. arXiv:1302.5145
  17. Crainic TG, Toulouse M (2010) Parallel meta-heuristics. In: Handbook of metaheuristics. Springer, US, pp 497–541Google Scholar
  18. DasGupta B, Encisob GA, Sontag E, Zhanga Y (2007) Algorithmic and complexity results for decompositions of biological networks into monotone subsystems. BioSystems 90:161–178CrossRefGoogle Scholar
  19. Davis JA (1967) Clustering and structural balance in graphs. Hum Relat 20:181–187CrossRefGoogle Scholar
  20. De Nooy W, Mrvar A, Vladimir B (2011) Exploratory social network analysis with Pajek: revised and expanded, vol 27, 2nd edn. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  21. Demaine ED, Emanuel D, Fiat A, Immorlica N (2006) Correlation clustering in general weighted graphs. Theoret Comput Sci 361:172–187CrossRefGoogle Scholar
  22. Den Besten M, Stützle T, Dorigo M (2001) Design of iterated local search algorithms. In: Workshops on applications of evolutionary computation. Springer Berlin, Heidelberg, pp 441–451Google Scholar
  23. Doreian P, Mrvar A (1996a) A partitioning approach to structural balance. Soc Netw 18:149–168CrossRefGoogle Scholar
  24. Doreian P, Mrvar A (2009) Partitioning signed social networks. Soc Netw 31:1–11CrossRefGoogle Scholar
  25. Doreian P, Krackhardt D (2001) Pre-transitive balance mechanisms for signed networks*. J Math Sociol 25(1):43–67CrossRefGoogle Scholar
  26. Doreian P, Mrvar A (1996b) Structural balance and partitioning signed graphs. Developments in data analysis, pp 195–208Google Scholar
  27. Dowdall AT (2009) The birth and death of a tar baby: Henry kissinger and southern africa. Ph.D. thesis, University of Missouri–ColumbiaGoogle Scholar
  28. Drummond L, Figueiredo R, Frota Y, Levorato M (2013) Efficient solution of the correlation clustering problem: an application to structural balance. In: YanTang D, Herv P (eds) OTM 2013 Workshops, LNCS, vol 8186. Springer, Berlin, pp 674–683Google Scholar
  29. Duch J, Arenas A (2005) Community detection in complex networks using extremal optimization. Phys Rev E 72(2):027104CrossRefGoogle Scholar
  30. Ekşioglu SD, Pardalos PM, Resende MGC (2002) Parallel metaheuristics for combinatorial optimization. In: Models for parallel and distributed computation. Springer, pp 179–206Google Scholar
  31. Elsner M, Schudy W (2009) Bounding and comparing methods for correlation clustering beyond ILP. In: ILP’09 proceedings of the workshop on integer linear programming for natural language processing, pp 19–27Google Scholar
  32. Epinions (1999) Website. Accessed on March 2015
  33. Esmailian P, Abtahi SE, Jalili M (2014) Mesoscopic analysis of online social networks: the role of negative ties. Phys Rev E 90(4):042817CrossRefGoogle Scholar
  34. Facchetti G, Iacono G, Altafini C (2011) Computing global structural balance in large-scale signed social networks. Proc Natl Acad Sci USA 108:20953–20958CrossRefGoogle Scholar
  35. Feo TA, Resende MGC (1995) Greedy randomized adaptive search procedures. J Glob Optim 6(2):109–133CrossRefGoogle Scholar
  36. Figueiredo R, Frota Y (2014) The maximum balanced subgraph of a signed graph: applications and solution approaches. Eur J Oper Res 236(2):473–487CrossRefGoogle Scholar
  37. Figueiredo R, Moura G (2013) Mixed integer programming formulations for clustering problems related to structural balance. Soc Netw 35(4):639–651CrossRefGoogle Scholar
  38. Gendreau M, Potvin JY (2010) Handbook of metaheuristics. International series in operations research and management science. Springer, BerlinGoogle Scholar
  39. Giotis I, Guruswami V (2006) Correlation clustering with a fixed number of clusters. In: Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm. ACM, pp 1167–1176Google Scholar
  40. Golan G (2010) Yom Kippur and after: The Soviet Union and the Middle East Crisis, vol 19. Cambridge University Press, CambridgeGoogle Scholar
  41. Golani M (1995) The historical place of the czech-egyptian arms deal, fall 1955. Middle Eastern Stud 31(4):803–827CrossRefGoogle Scholar
  42. Gülpinar N, Gutin G, Mitra G, Zverovitch A (2004) Extracting pure network submatrices in linear programs using signed graphs. Discrete Appl Math 137:359–372CrossRefGoogle Scholar
  43. Harary F, Lim M, Wunsch DC (2003) Signed graphs for portfolio analysis in risk management. IMA J Manag Math 13:1–10Google Scholar
  44. Heider F (1946) Attitudes and cognitive organization. J Psychol 21:107–112CrossRefGoogle Scholar
  45. Huffner F, Betzler N, Niedermeier R (2010) Separator-based data reduction for signed graph balancing. J Combin Optim 20:335–360CrossRefGoogle Scholar
  46. Inohara T (1998) On conditions for a meeting not to reach a deadlock. Appl Math Comput 90:1–9CrossRefGoogle Scholar
  47. Kim S, Yoo CD, Nowozin S, Kohli P (2014) Image segmentation usinghigher-order correlation clustering. IEEE Trans Pattern Anal Mach Intell 36(9):1761–1774CrossRefGoogle Scholar
  48. Kreps SE (2007) The 2006 Lebanon war: lessons learned. Parameters 37(1):72Google Scholar
  49. Kunegis J, Lommatzsch A, Bauckhage C (2009) The slashdot zoo: mining a social network with negative edges. In: WWW’09 Proceedings of the 18th international conference on World wide web, pp 741–750Google Scholar
  50. Kunegis J, Schmidt S, Lommatzsch A, Lerner J, De Luca EW, Albayrak S (2010) Spectral analysis of signed graphs for clustering, prediction and visualization. SDM, vol 10. SIAM, pp 559–559Google Scholar
  51. Leskovec J, Huttenlocher D, Kleinberg J (2010) Signed networks in social media. In: CHI’10 Proceedings of the SIGCHI conference on human factors in computing systems, pp 1361–1370Google Scholar
  52. Leskovec J, Krevl A (2014) SNAP datasets: Stanford large network dataset collection.
  53. Levorato M, Drummond L, Frota Y, Figueiredo R (2015) An ILS algorithm to evaluate structural balance in signed social networks. In: Symposium on applied computing, SAC 2015, Salamanca, Spain—April 13–17, pp 1117–1122Google Scholar
  54. Lourenço HR, Martin OC, Stützle T (2003) Iterated local search. Springer, BerlinCrossRefGoogle Scholar
  55. Macon KT, Mucha PJ, Porter MA (2012) Community structure in the united nations general assembly. Phys A 391:343–361CrossRefGoogle Scholar
  56. McGreal C (2006) Brothers in arms-Israel’s secret pact with pretoria. Guardian 7. Accessed 23 Jan 2017
  57. Mearsheimer JJ, Walt SM (2006) The Israel lobby and us foreign policy. Middle East Policy 13(3):29–87CrossRefGoogle Scholar
  58. Mehrotra A, Trick MA (1998) Cliques and clustering: a combinatorial approach. Oper Res Lett 22(1):1–12CrossRefGoogle Scholar
  59. Mladenović N, Hansen P (1997) Variable neighborhood search. Comput Oper Res 24(11):1097–1100CrossRefGoogle Scholar
  60. Munem BA (2008) Canada and peace in the middle east. Accessed on Jan 2015
  61. Nascimento MC, Pitsoulis L (2013) Community detection by modularity maximization using GRASP with path relinking. Comput Oper Res 40(12):3121–3131Google Scholar
  62. Nesbitt FN (2004) Race for sanctions: African Americans against apartheid, 1946–1994. Indiana University Press, BloomingtonGoogle Scholar
  63. Newman MEJ (2006) Modularity and community structure in networks. Proc Natl Acad Sci USA 103(23):8577–8582CrossRefGoogle Scholar
  64. Pérez-Stable M (1993) The Cuban revolution: origins, course, and legacy. Oxford University Press, New YorkGoogle Scholar
  65. Ruiz R, Stützle T (2007) A simple and effective iterated greedy algorithm for the permutation flowshop scheduling problem. Eur J Oper Res 177(3):2033–2049CrossRefGoogle Scholar
  66. Slashdot Website (1997) Accessed on March 2015
  67. Smith CD (2010) Palestine and the Arab-Israeli conflict:[a history with documents]. Bedford/St. Martin’s,Google Scholar
  68. Srinivasan A (2011) Local balancing influences global structure in social networks. Proc Natl Acad Sci USA 108:1751–1752CrossRefGoogle Scholar
  69. Stinnett DM, Tir J, Diehl PF, Schafer P, Gochman C (2002) The correlates of war (cow) project direct contiguity data, version 3.0. Confl Manag Peace Sci 19:59–67CrossRefGoogle Scholar
  70. Swamy Chaitanya (2004) Correlation clustering: maximizing agreements via semidefinite programming. In: Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms. Society for Industrial and Applied Mathematics, pp 526–527Google Scholar
  71. Traag VA, Bruggeman J (2009) Community detection in networks with positive and negative links. Phys Rev E 80:036115CrossRefGoogle Scholar
  72. Wang Ning, Li Jie (2013) Restoring: A greedy heuristic approach based on neighborhood for correlation clustering. In: Advanced data mining and applications. Springer, pp 348–359Google Scholar
  73. Yang B, Cheung WK, Liu J (2007) Community mining from signed social networks. IEEE Trans Knowl Data Eng 19:1333–1348CrossRefGoogle Scholar
  74. Zhang S, Wang R-S, Zhang X-S (2007) Identification of overlapping community structure in complex networks using fuzzy c-means clustering. Phys A 374(1):483–490CrossRefGoogle Scholar
  75. Zhang Z, Cheng H, Chen W, Zhang S, Fang Q (2008) Correlation clustering based on genetic algorithm for documents clustering. IEEE congress on evolutionary computation, pp 3193–3198Google Scholar

Copyright information

© EURO - The Association of European Operational Research Societies 2017

Authors and Affiliations

  • Mario Levorato
    • 1
  • Rosa Figueiredo
    • 2
    Email author
  • Yuri Frota
    • 1
  • Lúcia Drummond
    • 1
  1. 1.Department of Computer ScienceFluminense Federal UniversityNiteróiBrazil
  2. 2.Laboratoire d’Informatique d’AvignonUniversity of AvignonAvignonFrance

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