A heuristic approach to estimate nodes’ closeness rank using the properties of real world networks

  • Akrati Saxena
  • Ralucca Gera
  • S. R. S. IyengarEmail author
Original Article


Centrality measures capture the intuitive notion of the importance of a node in a network. Importance of a node can be a very subjective term and is defined based on the context and the application. Closeness centrality is one of the most popular centrality measures which quantifies how close a node is to every other node in the network. It considers the average distance of a given node to all the other nodes in a network and requires one to know the complete information of the network. To compute the closeness rank of a node, we first need to compute the closeness value of all the nodes, and then compare them to get the rank of the node. In this work, we address the problem of estimating the closeness centrality rank of a node without computing the closeness centrality values of all the nodes in the network. We provide linear time heuristic algorithms which run in O(m), versus the classical algorithm which runs in time \(O(m \cdot n)\), where m is the number of edges and n is the number of nodes in the network. The proposed methods are applied to real-world networks, and their accuracy is measured using absolute and weighted error functions.


Closeness centrality Closeness ranking Social network analysis Heuristic method 



Gera thanks the DoD, in particular, the Asymmetric Warfare Group and the West Point Network Science Center for partially sponsoring this work. Saxena and Iyengar would like to thank IIT Ropar HPC committee for providing the resources to perform the experiments.


  1. (2016) Spanish book network dataset – KONECT. URL
  2. Bader DA, Madduri K (2006) Parallel algorithms for evaluating centrality indices in real-world networks. In: Parallel rocessing, 2006. ICPP 2006. International Conference on, IEEE, pp 539–550Google Scholar
  3. Barabási AL, Albert R (1999) Emergence of scaling in random networks. Science 286(5439):509–512MathSciNetzbMATHCrossRefGoogle Scholar
  4. Barzinpour F, Ali-Ahmadi BH, Alizadeh S, Jalali Naini SG (2014) Clustering networks heterogeneous data in defining a comprehensive closeness centrality index. Math Probl EngGoogle Scholar
  5. Bergamini E, Borassi M, Crescenzi P, Marino A, Meyerhenke H (2016) Computing top-k closeness centrality faster in unweighted graphs. In: 2016 Proceedings of the eighteenth workshop on algorithm engineering and experiments (ALENEX). SIAM, pp 68–80Google Scholar
  6. Bollacker KD, Lawrence S, Giles CL(1998) Citeseer: An autonomous web agent for automatic retrieval and identification of interesting publications. In: Proceedings of the second international conference on autonomous agents, ACM, pp 116–123Google Scholar
  7. Brandes U, Pich C (2007) Centrality estimation in large networks. Int J Bifurcat Chaos 17(07):2303–2318MathSciNetzbMATHCrossRefGoogle Scholar
  8. Brandes U, Borgatti SP, Freeman LC (2016) Maintaining the duality of closeness and betweenness centrality. Soc Netw 44:153–159CrossRefGoogle Scholar
  9. Brin S, Page L (1998) The anatomy of a large-scale hypertextual web search engine In: Seventh international world-wide web conference (www 1998), april 14-18, 1998, brisbane, australia. Brisbane, AustraliaCrossRefGoogle Scholar
  10. Carbaugh J, Debnath J, Fletcher M, Gera R, Lee WC, Nelson R (2017) Extracting information based on partial or complete network data. In: International conference on communication, management and information technologyGoogle Scholar
  11. Chan SY, Leung IXY, Liò P (2009) Fast centrality approximation in modular networks. In: Proceedings of the 1st ACM international workshop on Complex networks meet information & knowledge management, ACM, pp 31–38Google Scholar
  12. Chen D, Lü L, Shang MS, Zhang YC, Zhou T (2012) Identifying influential nodes in complex networks. Phys A Stat Mech Appl 391(4):1777–1787CrossRefGoogle Scholar
  13. Cho E, Myers SA, Leskovec J (2011) Friendship and mobility: user movement in location-based social networks. In: Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining, ACM, pp 1082–1090Google Scholar
  14. Cohen E, Delling D, Pajor T, Werneck R (2014) Computing classic closeness centrality, at scale. In: Proceedings of the second ACM conference on online social networks. ACM, pp 37–50Google Scholar
  15. Cormen TH, Leiserson CE, Rivest RL, Stein C (2001) Introduction to algorithms, vol 6. MIT press, CambridgezbMATHGoogle Scholar
  16. Du Y, Gao C, Chen X, Hu Y, Sadiq R, Deng Y (2015) A new closeness centrality measure via effective distance in complex networks. Chaos: an Interdisciplinary. J. Nonlin. Sci. 25(3):033,112Google Scholar
  17. Eppstein D, Wang J (2004) Fast approximation of centrality. J Graph Algorithms Appl 8:39–45MathSciNetzbMATHCrossRefGoogle Scholar
  18. Freeman LC (1977) A set of measures of centrality based on betweenness. Sociometry 35–41CrossRefGoogle Scholar
  19. Freeman LC (1978) Centrality in social networks conceptual clarification. Soc Netw 1(3):215–239CrossRefGoogle Scholar
  20. Hogg T, Lerman K (2012) Social dynamics of digg. EPJ Data Sci 1(1):1–26CrossRefGoogle Scholar
  21. Jarukasemratana S, Murata T, Liu X (2014) Community detection algorithm based on centrality and node closeness in scale-free networks. Trans Jpn Soc Artif Intell 29(2):234–244CrossRefGoogle Scholar
  22. Jeong H, Mason SP, Barabási AL, Oltvai ZN (2001) Lethality and centrality in protein networks. Nature 411(6833):41–42CrossRefGoogle Scholar
  23. Joshi-Tope G, Gillespie M, Vastrik I, D’Eustachio P, Schmidt E, de Bono B, Jassal B, Gopinath G, Wu G, Matthews L (2005) Reactome: a knowledgebase of biological pathways. Nucl Acids Res 33(1):428–432Google Scholar
  24. Kaiser M, Hilgetag CC (2006) Nonoptimal component placement, but short processing paths, due to long-distance projections in neural systems. PLoS Comput Biol 2(7):e95CrossRefGoogle Scholar
  25. Kas M, Wachs M, Carley KM, Carley LR (2013) Incremental algorithm for updating betweenness centrality in dynamically growing networks. In: Proceedings of the 2013 IEEE/ACM international conference on advances in social networks analysis and mining, ACM, pp 33–40Google Scholar
  26. Katz L (1953) A new status index derived from sociometric analysis. Psychometrika 18(1):39–43MathSciNetzbMATHCrossRefGoogle Scholar
  27. Kendall MG (1945) The treatment of ties in ranking problems. Biometrika 33(3):239–251MathSciNetzbMATHCrossRefGoogle Scholar
  28. Kim J, Ahn H, Park M, Kim S, Kim KP (2016) An estimated closeness centrality ranking algorithm and its performance analysis in Large-Scale workflow-supported social networks. KSII Trans Int Inf Syst 10(3):1454–1466Google Scholar
  29. Klimt B, Yang Y (2004) The enron corpus: a new dataset for email classification research. In: Boulicaut JF, Esposito F, Giannotti F, Pedreschi D (eds) European conference on machine learning (ECML 2004). Springer, Berlin, Heidelberg, pp 217–226CrossRefGoogle Scholar
  30. Ko K, Lee KJ, Park C (2008) Rethinking preferential attachment scheme: degree centrality versus closeness centrality. Connections 28(1):4–15Google Scholar
  31. Lehmann KA, Kaufmann M (2003) Decentralized algorithms for evaluating centrality in complex networksGoogle Scholar
  32. Leskovec J, Kleinberg J, Faloutsos C (2007) Graph evolution: densification and shrinking diameters. ACM Trans Knowl Discov Data 1(1):2CrossRefGoogle Scholar
  33. Leskovec J, Huttenlocher D, Kleinberg J (2010) Signed networks in social media. In: Proceedings of the SIGCHI conference on human factors in computing systems, ACM, pp 1361–1370Google Scholar
  34. Lu F, Osthoff C, Ramos D, Nardes R, et al (2015) Mdaccer: Modified distributed assessment of the closeness centrality ranking in complex networks for massively parallel environments. In: 2015 International symposium on computer architecture and high performance computing workshop (SBAC-PADW), IEEE, pp 43–48Google Scholar
  35. McAuley JJ, Leskovec J (2012) Learning to discover social circles in ego networks. In: NIPS 2012:548–56Google Scholar
  36. Moré JJ (1978) The levenberg-marquardt algorithm: implementation and theory. In: Watson GA (ed) Numerical analysis. Springer, Berlin, Heidelberg, pp 105–116CrossRefGoogle Scholar
  37. Newman ME (2001) Scientific collaboration networks. ii. Shortest paths, weighted networks, and centrality. Phys Rev E 64(1):016,132MathSciNetCrossRefGoogle Scholar
  38. Newman ME (2003) The structure and function of complex networks. SIAM Rev 45(2):167–256MathSciNetzbMATHCrossRefGoogle Scholar
  39. Okamoto K, Chen W, Li XY (2008) Ranking of closeness centrality for large-scale social networks. In: Preparata FP, Wu X, Yin J (eds) International workshop on Frontiers in algorithmics. Springer, Berlin, Heidelberg, pp 186–195CrossRefGoogle Scholar
  40. Olsen PW, Labouseur AG, Hwang JH (2014) Efficient top-k closeness centrality search. In: Data engineering (ICDE), 2014 IEEE 30th international conference on, IEEE, pp 196–207Google Scholar
  41. Opsahl T (2011) Why anchorage is not (that) important: Binary ties and sample selection. online] https://www.toreopsahlcom/2011/08/12/why-anchorage-is-not-that-important-binary-tiesand-sample-selection (accessed Sept 2013)
  42. Park S, Park M, Kim H, Kim H, Yoon W, Yoon TB, Kim KP (2013) A closeness centrality analysis algorithm for workflow-supported social networks. In: Advanced communication technology (ICACT), 2013 15th International conference on, IEEE, pp 158–161Google Scholar
  43. Pfeffer J, Carley KM (2012) k-centralities: local approximations of global measures based on shortest paths. In: Proceedings of the 21st international conference companion on World Wide Web, ACM, pp 1043–1050Google Scholar
  44. Rattigan MJ, Maier M, Jensen D (2006) Using structure indices for efficient approximation of network properties. In: Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining, ACM, pp 357–366Google Scholar
  45. Richardson M, Agrawal R, Domingos P(2003) Trust management for the semantic web. In: International semantic Web conference, Springer, pp 351–368Google Scholar
  46. Rochat Y (2009) Closeness centrality extended to unconnected graphs: The harmonic centrality index. In: ASNA, EPFL-CONF-200525Google Scholar
  47. Ruslan N, Sharif S (2015) Improved closeness centrality using arithmetic mean approach. In: innovation and analytics conference and exhibition(IACE 2015): Proceedings of the 2nd innovation and analytics conference & Exhibition, AIP Publishing, vol 1691, p 050022Google Scholar
  48. Sabidussi G (1966) The centrality index of a graph. Psychometrika 31(4):581–603MathSciNetzbMATHCrossRefGoogle Scholar
  49. Sariyuce AE, Kaya K, Saule E, Catalyurek UV (2013) Incremental algorithms for network management and analysis based on closeness centrality. arXiv preprint arXiv:13030422
  50. Saxena A, Iyengar S (2017) Global rank estimation. arXiv preprint arXiv:171011341
  51. Saxena A, Iyengar S (2018) Estimating shell-index in a graph with local information. arXiv preprint arXiv:180510391
  52. Saxena A, Malik V, Iyengar S (2015a) Estimating the degree centrality ranking of a node. arXiv preprint arXiv:151105732
  53. Saxena A, Malik V, Iyengar S (2015b) Rank me thou shalln’t compare me. arXiv preprint arXiv:151109050
  54. Saxena A, Gera R, Iyengar S (2017a) Degree ranking using local information. arXiv preprint arXiv:170601205
  55. Saxena A, Gera R, Iyengar S (2017b) Fast estimation of closeness centrality ranking. In: Proceedings of the 2017 IEEE/ACM international conference on advances in social networks analysis and mining 2017, ACM, pp 80–85Google Scholar
  56. Saxena A, Gera R, Iyengar S (2017c) A faster method to estimate closeness centrality ranking. arXiv preprint arXiv:170602083
  57. Saxena A, Gera R, Iyengar S (2017d) Observe locally rank globally. In: Proceedings of the 2017 IEEE/ACM international conference on advances in social networks analysis and mining 2017, ACM, pp 139–144Google Scholar
  58. Shaw ME (1954) Some effects of unequal distribution of information upon group performance in various communication nets. J Abnorm Soc Psychol 49(4):547–553CrossRefGoogle Scholar
  59. Sporns O, Honey CJ, Kötter R (2007) Identification and classification of hubs in brain networks. PloS One 2(10):e1049CrossRefGoogle Scholar
  60. Stephenson K, Zelen M (1989) Rethinking centrality: methods and examples. Soc Netw 11(1):1–37MathSciNetCrossRefGoogle Scholar
  61. Sudarshan Iyengar S, Veni Madhavan C, Zweig KA, Natarajan A (2012) Understanding human navigation using network analysis. Top Cogn Sci 4(1):121–134CrossRefGoogle Scholar
  62. Tallberg C (2000) Comparing degree-based and closeness-based centrality measures. Univ, Department of StatisticsGoogle Scholar
  63. Szczepański P, Rahwan T, Michalak TP, Wooldridge M (2016) Closeness centrality for networks with overlapping community structure. In: Thirtieth AAAI conference on artificial intelligenceGoogle Scholar
  64. Ufimtsev V, Bhowmick S (2014) An extremely fast algorithm for identifying high closeness centrality vertices in large-scale networks. In: Proceedings of the fourth workshop on irregular applications: architectures and algorithms, IEEE Press, pp 53–56Google Scholar
  65. Viswanath B, Mislove A, Cha M, Gummadi KP (2009) On the evolution of user interaction in facebook. In: Proceedings of the 2nd ACM workshop on Online social networks, ACM, pp 37–42Google Scholar
  66. Wang W, Tang CY (2015) Distributed estimation of closeness centrality. In: Decision and Control (CDC), 2015 IEEE 54th annual conference on, IEEE, pp 4860–4865Google Scholar
  67. Wehmuth K, Ziviani A (2012) Distributed assessment of the closeness centrality ranking in complex networks. In: Proceedings of the fourth annual workshop on simplifying complex networks for practitioners, ACM, pp 43–48Google Scholar
  68. Yan E, Ding Y (2009) Applying centrality measures to impact analysis: a coauthorship network analysis. J Am Soc Inf Sci Technol 60(10):2107–2118CrossRefGoogle Scholar
  69. Yang J, Leskovec J (2015) Defining and evaluating network communities based on ground-truth. Knowl Inf Syst 42(1):181–213CrossRefGoogle Scholar
  70. Yen CC, Yeh MY, Chen MS (2013) An efficient approach to updating closeness centrality and average path length in dynamic networks. In: Data mining (ICDM), 2013 IEEE 13th international conference on, IEEE, pp 867–876Google Scholar
  71. Zar JH (1972) Significance testing of the spearman rank correlation coefficient. J Am Stat Assoc 67(339):578–580zbMATHCrossRefGoogle Scholar
  72. Zhang B, Liu R, Massey D, Zhang L (2005) Collecting the internet as-level topology. ACM SIGCOMM Comput Commun Rev 35(1):53–61CrossRefGoogle Scholar
  73. Zhang J, Ma X, Liu W, Bai Y (2012) Inferring community members in social networks by closeness centrality examination. In: Web information systems and applications conference (WISA), 2012 Ninth, IEEE, pp 131–134Google Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of CSEIndian Institute of Technology RoparRupnagarIndia
  2. 2.Department of Applied MathematicsNaval Postgraduate SchoolMontereyUSA

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