Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content

Advertisement

Springer Nature Link
Log in

Estimating degree rank in complex networks

  • Original Article
  • Published:
Social Network Analysis and Mining Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Notes

  1. Here, discrete probability values are considered as continuous probability density function, as this introduces a very small error.

References

  • 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 635–644

  • Barabási AL, Albert R (1999) Emergence of scaling in random networks. Science 286(5439):509–512

    Article  MathSciNet  MATH  Google Scholar 

  • 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 595–601

  • 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, Australia

  • 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 1268–1273

  • 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 152–157

  • Cem E, Sarac K (2016b) Estimation of structural properties of online social networks at the extreme. Comput Netw 108:323–344

    Article  Google Scholar 

  • Chen D, Lü L, Shang MS, Zhang YC, Zhou T (2012) Identifying influential nodes in complex networks. Physica Stat Mech Appl 391(4):1777–1787

    Article  Google Scholar 

  • 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 3072–3080

  • 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 1082–1090

  • 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 1007–1016

  • 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 795–806

  • 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 93–104

    Chapter  Google Scholar 

  • 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 151–158

  • Eden T, Ron D, Seshadhri C (2016) Sublinear time estimation of degree distribution moments: The arboricity connection. arXiv preprint arXiv:160403661

  • Erdős P, Rényi A (1960) On the evolution of random graphs. Publ Math Inst Hungar Acad Sci 5:17–61

    MathSciNet  MATH  Google Scholar 

  • Even S (2011) Graph algorithms. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • 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 73–80

  • 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 59–71

    Google Scholar 

  • Freeman LC (1977) A set of measures of centrality based on betweenness. Sociometry 40(1):35–41

    Article  Google Scholar 

  • Ghoshal G, Barabási AL (2011) Ranking stability and super-stable nodes in complex networks. Nat Commun 2:394

    Article  Google Scholar 

  • 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 1–9

  • Goodman LA (1961) Snowball sampling. Ann Math Stat 32(1):148–170

    MATH  Google Scholar 

  • Hansen MH, Hurwitz WN (1943) On the theory of sampling from finite populations. Ann Math Stat 14(4):333–362

    Article  MathSciNet  MATH  Google Scholar 

  • Haralabopoulos G, Anagnostopoulos I (2014) Real time enhanced random sampling of online social networks. J Netw Comput Appl 41:126–134

    Article  Google Scholar 

  • 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 539–550

  • Hogg T, Lerman K (2012) Social dynamics of digg. EPJ Data Sci 1(1):1–26

    Article  Google Scholar 

  • Hou B, Yao Y, Liao D (2012) Identifying all-around nodes for spreading dynamics in complex networks. Phys A Stat Mech Appl 391(15):4012–4017

    Article  Google Scholar 

  • Katz L (1953) A new status index derived from sociometric analysis. Psychometrika 18(1):39–43

    Article  MathSciNet  MATH  Google Scholar 

  • 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 195–202

  • Kurant M, Butts CT, Markopoulou A (2012) Graph size estimation. arXiv preprint arXiv:12100460

  • 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 631–636

  • Leskovec J, Kleinberg J, Faloutsos C (2007) Graph evolution: densification and shrinking diameters. ACM Trans Knowl Discov Data (TKDD) 1(1):2

    Article  Google Scholar 

  • Lovász L (1993) Random walks on graphs: A survey. Comb Paul erdos is eighty 2(1):1–46

    Google Scholar 

  • 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 33–40

  • Lucchese R, Varagnolo D (2015) Networks cardinality estimation using order statistics. In: American Control Conference (ACC), 2015, IEEE, pp 3810–3817

  • 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 317–326

  • McAuley JJ, Leskovec J (2012) Learning to discover social circles in ego networks. NIPS 2012:548–56

  • Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21(6):1087–1092

    Article  Google Scholar 

  • Moré JJ (1978) The levenberg-marquardt algorithm: implementation and theory. In: Watson GA (ed) Numerical analysis. Springer, Berlin, Heidelberg, pp 105–116

    Chapter  Google Scholar 

  • 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 469–478

  • 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):678–689

    Article  Google Scholar 

  • 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 390–403

  • 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 5240–5247

  • Ribeiro B, Wang P, Murai F, Towsley D (2012) Sampling directed graphs with random walks. In: INFOCOM, 2012 Proceedings IEEE, IEEE, pp 1692–1700

  • 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

  • Sabidussi G (1966) The centrality index of a graph. Psychometrika 31(4):581–603

    Article  MathSciNet  MATH  Google Scholar 

  • Salganik MJ, Heckathorn DD (2004) Sampling and estimation in hidden populations using respondent-driven sampling. Sociol Methodol 34(1):193–240

    Article  Google Scholar 

  • 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 139–144

  • Shaw ME (1954) Some effects of unequal distribution of information upon group performance in various communication nets. J Abnorm Soc Psychol 49(4):547–553

    Article  Google Scholar 

  • Stephenson K, Zelen M (1989) Rethinking centrality: methods and examples. Soc Net 11(1):1–37

    Article  MathSciNet  Google Scholar 

  • Traud AL, Mucha PJ, Porter MA (2012) Social structure of Facebook networks. Phys A 391(16):4165–4180

    Article  Google Scholar 

  • 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 120–129

  • Yang J, Leskovec J (2015) Defining and evaluating network communities based on ground-truth. Knowl Inf Syst 42(1):181–213

    Article  Google Scholar 

  • Ye S, Wu SF (2011) Estimating the size of online social networks. Int J Soc Comput Cyber Phys Syst 1(2):160–179

    Article  Google Scholar 

  • Yu Y, Fan S (2015) Node importance measurement based on the degree and closeness centrality. J Inf Commput Sci 12(3):1281–1291

    Article  Google Scholar 

  • Zafarani R, Liu H (2009) Social computing data repository at ASU. http://socialcomputing.asu.edu. Accessed Jan 2017

  • 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):26

    Article  MathSciNet  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Department of CSE, Indian Institute of Technology Ropar, Rupnagar, India

    Akrati Saxena & S. R. S. Iyengar

  2. Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA, 93943, USA

    Ralucca Gera

Authors
  1. Akrati Saxena
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ralucca Gera
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. R. S. Iyengar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Akrati Saxena.

Appendices

Appendix

Results on real-world scale-free networks

See Table 6.

Table 6 Absolute and weighted error in the estimated degree rank on real-world social networks
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Saxena, A., Gera, R. & Iyengar, S.R.S. Estimating degree rank in complex networks. Soc. Netw. Anal. Min. 8, 42 (2018). https://doi.org/10.1007/s13278-018-0520-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13278-018-0520-3

Keywords

Search

Navigation