Estimating degree rank in complex networks

Original Article
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Abstract

Identifying top-ranked nodes can be performed using different centrality measures, based on their characteristics and influential power. The most basic of all the ranking techniques is based on nodes degree. While finding the degree of a node requires local information, ranking the node based on its degree requires global information, namely the degrees of all the nodes of the network. It is infeasible to collect the global information for some graphs such as (i) the ones emerging from big data, (ii) dynamic networks, and (iii) distributed networks in which the whole graph is not known. In this work, we propose methods to estimate the degree rank of a node, that are faster than the classical method of computing the centrality value of all nodes and then rank a node. The proposed methods are modeled based on the network characteristics and sampling techniques, thus not requiring the entire network. We show that approximately \(1\%\) node samples are adequate to find the rank of a node with high accuracy.

Keywords

Degree centrality Ranking nodes Social network analysis Sampling techniques 

Notes

Acknowledgements

Gera would like to thank the DoD for sponsoring this work. Saxena and Iyengar would like to thank IIT Ropar HPC committee for providing the resources to perform the experiments.

References

  1. Backstrom L, Leskovec J (2011) Supervised random walks: predicting and recommending links in social networks. In: Proceedings of the fourth ACM international conference on Web search and data mining, ACM, pp 635–644Google Scholar
  2. Barabási AL, Albert R (1999) Emergence of scaling in random networks. Science 286(5439):509–512MathSciNetCrossRefMATHGoogle Scholar
  3. Boldi P, Vigna S (2004) The WebGraph framework I: Compression techniques. In: Proc. of the Thirteenth International World Wide Web Conference (WWW 2004), ACM Press, Manhattan, USA, pp 595–601Google Scholar
  4. 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, AustraliaGoogle Scholar
  5. Cem E, Sarac K (2015) Estimating the size and average degree of online social networks at the extreme. In: Communications (ICC), 2015 IEEE International Conference on, IEEE, pp 1268–1273Google Scholar
  6. Cem E, Sarac K (2016a) Average degree estimation under ego-centric sampling design. In: Computer Communications Workshops (INFOCOM WKSHPS), 2016 IEEE Conference on, IEEE, pp 152–157Google Scholar
  7. Cem E, Sarac K (2016b) Estimation of structural properties of online social networks at the extreme. Comput Netw 108:323–344CrossRefGoogle Scholar
  8. Chen D, Lü L, Shang MS, Zhang YC, Zhou T (2012) Identifying influential nodes in complex networks. Physica Stat Mech Appl 391(4):1777–1787CrossRefGoogle Scholar
  9. Chen L, Karbasi A, Crawford FW (2016) Estimating the size of a large network and its communities from a random sample. In: Advances in Neural Information Processing Systems, pp 3072–3080Google Scholar
  10. 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
  11. Cooper C, Radzik T, Siantos Y (2012) A fast algorithm to find all high degree vertices in power law graphs. In: Proceedings of the 21st International Conference on World Wide Web, ACM, pp 1007–1016Google Scholar
  12. Dasgupta A, Kumar R, Sarlos T (2014) On estimating the average degree. In: Proceedings of the 23rd international conference on World wide web, ACM, pp 795–806Google Scholar
  13. Davis B, Gera R, Lazzaro G, Lim BY, Rye EC (2016) The marginal benefit of monitor placement on networks. In: Cherifi H, Gonçalves B, Menezes R, Sinatra R (eds) Complex networks VII, Springer, Cham, pp 93–104CrossRefGoogle Scholar
  14. De Choudhury M, Sundaram H, John A, Seligmann DD (2009) Social synchrony: Predicting mimicry of user actions in online social media. In: Computational Science and Engineering, 2009. CSE’09. International Conference on, IEEE, vol 4, pp 151–158Google Scholar
  15. Eden T, Ron D, Seshadhri C (2016) Sublinear time estimation of degree distribution moments: The arboricity connection. arXiv preprint arXiv:160403661
  16. Erdős P, Rényi A (1960) On the evolution of random graphs. Publ Math Inst Hungar Acad Sci 5:17–61MathSciNetMATHGoogle Scholar
  17. Even S (2011) Graph algorithms. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  18. Fire M, Tenenboim L, Lesser O, Puzis R, Rokach L, Elovici Y (2011) Link prediction in social networks using computationally efficient topological features. In: Privacy, security, risk and trust (PASSAT) and IEEE third international confernece on social computing (SocialCom), IEEE, pp 73–80Google Scholar
  19. Fortunato S, Boguñá M, Flammini A, Menczer F (2006) Approximating pagerank from in-degree. In: Aiello W, Broder A, Janssen J, Milios E (eds) International workshop on algorithms and models for the web-graph. Springer, Berlin, Heidelberg, pp 59–71Google Scholar
  20. Freeman LC (1977) A set of measures of centrality based on betweenness. Sociometry 40(1):35–41CrossRefGoogle Scholar
  21. Ghoshal G, Barabási AL (2011) Ranking stability and super-stable nodes in complex networks. Nat Commun 2:394CrossRefGoogle Scholar
  22. Gjoka M, Kurant M, Butts CT, Markopoulou A (2010) Walking in Facebook: A case study of unbiased sampling of OSNs. In: INFOCOM, 2010 Proceedings IEEE, IEEE, pp 1–9Google Scholar
  23. Goodman LA (1961) Snowball sampling. Ann Math Stat 32(1):148–170MATHGoogle Scholar
  24. Hansen MH, Hurwitz WN (1943) On the theory of sampling from finite populations. Ann Math Stat 14(4):333–362MathSciNetCrossRefMATHGoogle Scholar
  25. Haralabopoulos G, Anagnostopoulos I (2014) Real time enhanced random sampling of online social networks. J Netw Comput Appl 41:126–134CrossRefGoogle Scholar
  26. Hardiman SJ, Katzir L (2013) Estimating clustering coefficients and size of social networks via random walk. In: Proceedings of the 22nd international conference on World Wide Web, International World Wide Web Conferences Steering Committee, pp 539–550Google Scholar
  27. Hogg T, Lerman K (2012) Social dynamics of digg. EPJ Data Sci 1(1):1–26CrossRefGoogle Scholar
  28. Hou B, Yao Y, Liao D (2012) Identifying all-around nodes for spreading dynamics in complex networks. Phys A Stat Mech Appl 391(15):4012–4017CrossRefGoogle Scholar
  29. Katz L (1953) A new status index derived from sociometric analysis. Psychometrika 18(1):39–43CrossRefMATHGoogle Scholar
  30. Konstas I, Stathopoulos V, Jose JM (2009) On social networks and collaborative recommendation. In: Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval, ACM, pp 195–202Google Scholar
  31. Kurant M, Butts CT, Markopoulou A (2012) Graph size estimation. arXiv preprint arXiv:12100460
  32. Leskovec J, Faloutsos C (2006) Sampling from large graphs. In: Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining, ACM, pp 631–636Google Scholar
  33. Leskovec J, Kleinberg J, Faloutsos C (2007) Graph evolution: densification and shrinking diameters. ACM Trans Knowl Discov Data (TKDD) 1(1):2CrossRefGoogle Scholar
  34. Lovász L (1993) Random walks on graphs: A survey. Comb Paul erdos is eighty 2(1):1–46Google Scholar
  35. Lu J, Li D (2012) Sampling online social networks by random walk. In: Proceedings of the First ACM International Workshop on Hot Topics on Interdisciplinary Social Networks Research, ACM, pp 33–40Google Scholar
  36. Lucchese R, Varagnolo D (2015) Networks cardinality estimation using order statistics. In: American Control Conference (ACC), 2015, IEEE, pp 3810–3817Google Scholar
  37. Marchetti-Spaccamela A (1988) On the estimate of the size of a directed graph. In: International Workshop on Graph-Theoretic Concepts in Computer Science, Springer, pp 317–326Google Scholar
  38. McAuley JJ, Leskovec J (2012) Learning to discover social circles in ego networks. NIPS 2012:548–56Google Scholar
  39. Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21(6):1087–1092CrossRefGoogle Scholar
  40. Moré JJ (1978) The levenberg-marquardt algorithm: implementation and theory. In: Watson GA (ed) Numerical analysis. Springer, Berlin, Heidelberg, pp 105–116CrossRefGoogle Scholar
  41. Musco C, Su HH, Lynch N (2016) Ant-inspired density estimation via random walks. In: Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, ACM, pp 469–478Google Scholar
  42. Nazi A, Zhou Z, Thirumuruganathan S, Zhang N, Das G (2015) Walk, not wait: faster sampling over online social networks. Proc VLDB Endow 8(6):678–689CrossRefGoogle Scholar
  43. Ribeiro B, Towsley D (2010) Estimating and sampling graphs with multidimensional random walks. In: Proceedings of the 10th ACM SIGCOMM conference on Internet measurement, ACM, pp 390–403Google Scholar
  44. Ribeiro B, Towsley D (2012) On the estimation accuracy of degree distributions from graph sampling. In: Decision and Control (CDC), 2012 IEEE 51st Annual Conference on, IEEE, pp 5240–5247Google Scholar
  45. Ribeiro B, Wang P, Murai F, Towsley D (2012) Sampling directed graphs with random walks. In: INFOCOM, 2012 Proceedings IEEE, IEEE, pp 1692–1700Google Scholar
  46. Rossi RA, Ahmed NK (2015) The network data repository with interactive graph analytics and visualization. In: Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence, http://networkrepository.com
  47. Sabidussi G (1966) The centrality index of a graph. Psychometrika 31(4):581–603MathSciNetCrossRefMATHGoogle Scholar
  48. Salganik MJ, Heckathorn DD (2004) Sampling and estimation in hidden populations using respondent-driven sampling. Sociol Methodol 34(1):193–240CrossRefGoogle Scholar
  49. Saxena A, Gera R, Iyengar S (2017) Observe locally rank globally. In: Proceedings of the 2017 IEEE/ACM international conference on advances in social networks analysis and mining. ACM, pp 139–144Google Scholar
  50. 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
  51. Stephenson K, Zelen M (1989) Rethinking centrality: methods and examples. Soc Net 11(1):1–37MathSciNetCrossRefGoogle Scholar
  52. Traud AL, Mucha PJ, Porter MA (2012) Social structure of Facebook networks. Phys A 391(16):4165–4180CrossRefGoogle Scholar
  53. Voudigari E, Salamanos N, Papageorgiou T, Yannakoudakis EJ (2016) Rank degree: An efficient algorithm for graph sampling. In: Advances in Social Networks Analysis and Mining (ASONAM), 2016 IEEE/ACM International Conference on, IEEE, pp 120–129Google Scholar
  54. Yang J, Leskovec J (2015) Defining and evaluating network communities based on ground-truth. Knowl Inf Syst 42(1):181–213CrossRefGoogle Scholar
  55. Ye S, Wu SF (2011) Estimating the size of online social networks. Int J Soc Comput Cyber Phys Syst 1(2):160–179CrossRefGoogle Scholar
  56. Yu Y, Fan S (2015) Node importance measurement based on the degree and closeness centrality. J Inf Commput Sci 12(3):1281–1291CrossRefGoogle Scholar
  57. Zafarani R, Liu H (2009) Social computing data repository at ASU. http://socialcomputing.asu.edu. Accessed Jan 2017
  58. Zhou Z, Zhang N, Gong Z, Das G (2016) Faster random walks by rewiring online social networks on-the-fly. ACM Trans Database Syst (TODS) 40(4):26MathSciNetCrossRefGoogle Scholar

Copyright information

© This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply 2018

Authors and Affiliations

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

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