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

A Flow-Based Node Dominance Centrality Measure for Complex Networks

  • Original Research
  • Published:
SN Computer Science Aims and scope Submit manuscript

Abstract

We introduce a new class of power centrality measures, which we refer to as Dominance Centrality, for complex networks which may have weighted, directed edges, as well as weights associated with nodes. These measures are similar in spirit to the power-over-powerless class introduced by Bonacich decades ago in the context of eigenvector centrality, but are quite different in their roots and effect. Our Dominance Centrality measure, referred to as DONEX, is derived as a Pareto optimal solution to collective welfare maximisation problem that allocates values to nodes in the network on the basis of expected interaction strengths captured by edge weights between pairs of nodes, where the expectation is taken over the probability of there being an undirected or directed link between them. We show how this formulation yields a new centrality measure which captures the notion of a weighted degree-differential virtual flow along the edges in the network in such a manner that the sum of such flows at each node itself represents the notion of dominance over the neighbourhood. We develop a greedy propagation algorithm called NetProp which allows us to estimate the reach of dominant nodes, and network boundaries where the dominance-derived influence or attention are maximum and minimum. We also demonstrate how DONEX extends to multi-hop neighbourhoods taking account of both local and non-local effects. The new methods are illustrated extensively with synthetic and real networks.

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
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Similar content being viewed by others

Availability of data and materials

All data and materials related to the research reported in this work can be made available on request.

Code availability

Software codes related to the research reported in this work can be made available on request.

Notes

  1. Even though the Configuration Model, as explained by Newman in [16], was proposed and examined by many authors in the context of network applications in physics, its variants have found wider utility in the context of community detection problems [60, 61] as a means for defining the notion of modularity [4, 5]. The central premise in the Newman model is that the probability with which a node with degree \(\alpha_{i}\) attaches its edge to another node in a fixed graph with n nodes, is proportional to its own degree, \(\alpha_{i}\), normalised to values <  = 1, by dividing by 2 m =|E|, the number of edges in the graph, assuming that the probabilities for the two ends of a single edge are independent of each other. Since the probability that there exists an edge between node i and node j must be symmetric, it must be proportional to the product of their degrees.

  2. This, as is evident, is quite different from deriving directions simply on the basis of degree-differential alone.

  3. Note that the Fiedler vector, obtained from eigenspace of the Laplacian in (16) could also give us a sparse-cut into two clusters of nodes that the connected network could have been split into. But they would have been obtained only on the basis of degree connectivity, and not on the basis of flows which lead to dominance.

References

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

    MATH  Google Scholar 

  2. Bonacich P. Power and centrality: a family of measures. Am J Sociol. 1987;92(5):1170–82.

    Google Scholar 

  3. Bonacich P. Some unique properties of eigenvector centrality. Soc Netw. 2007;29(4):555–64.

    Google Scholar 

  4. Barabasi A-L. Network Science, http://networksciencebook.com/ July 2016,

  5. Newman MEJ. Networks: an introduction. Oxford: Oxford University Press; 2010.

    MATH  Google Scholar 

  6. Page L et al. The PageRank citation ranking: Bringing order to the web, Stanford InfoLab, 1999

  7. Kleinberg JM, et al. The web as a graph: measurements, models, and methods. International Computing and Combinatorics Conference. Springer, Berlin, Heidelberg, 1999.

  8. Xing W, Ghorbani A. Weighted pagerank algorithm. Proceedings. Second Annual Conference on Communication Networks and Services Research, IEEE, 2004.

  9. Langville AN, Meyer CD. Deeper inside pagerank. Internet Math. 2004;1(3):335–80.

    MathSciNet  MATH  Google Scholar 

  10. Grassi R, Stefani S, Torriero A. Some new results on the eigenvector centrality. Math Sociol. 2007;31(3):237–48.

    MATH  Google Scholar 

  11. Benzi M, Estrada E, Klymko C. Ranking hubs and authorities using matrix functions. Linear Algebra Appl. 2013;438(5):2447–74.

    MathSciNet  MATH  Google Scholar 

  12. Benzi M, Klymko C. A matrix analysis of different centrality measures arXiv preprint arXiv:1312.6722, 2014.

  13. DiStefano JJ, Stubberud AR, Williams IJ, Schaum’s outline of feedback and control systems, signal flow graphs chapter, McGraw-Hill, New York, 2014.

  14. Tsugawa S. Empirical analysis of the relation between community structure and cascading retweet diffusion. Proceedings of the International AAAI Conference on Web and Social Media. Vol. 13. 2019.

  15. Subbanarasimha RP, Srinivasa S, Mandyam S. Invisible stories that drive online social cognition. IEEE Transact Comput Soc Syst. 2020. https://doi.org/10.1109/TCSS.2020.3009474.

    Article  Google Scholar 

  16. Newman, Mark EJ. Finding community structure in networks using the eigenvectors of matrices. Phys Rev E 2006;74(3):036104.

  17. Parzy M, Bogucka H. Coopetition methodology for resource sharing in distributed OFDM-based cognitive radio networks. IEEE Trans Commun. 2014;62(5):1518–29. https://doi.org/10.1109/TCOMM.2014.031214.130451.

    Article  Google Scholar 

  18. Abdel-Hadi A, Clancy C. A utility proportional fairness approach for resource allocation in 4G-LTE. 2014 International Conference on Computing, Networking and Communications (ICNC). IEEE, 2014.

  19. Alexandris K, et al. Utility-based resource allocation under multi-connectivity in evolved LTE. 2017 IEEE 86th Vehicular Technology Conference (VTC-Fall). IEEE, 2017.

  20. Pennings JME, Smidts A. The shape of utility functions and organizational behavior.” Manag Sci, 2003;49(9):1251–1263. JSTOR, www.jstor.org/stable/4134038. Accessed 11 Jul 2021.

  21. Mandyam S, Sridhar U. Loan allocation and guarantee structure for group borrower networks in microfinance. Stud Microecon. 2016;4(2):100–14.

    Google Scholar 

  22. Sridhar U, Mandyam S. DON and shapley value for allocation among cooperating agents in a network: conditions for equivalence. Stud Microecon. 2017;5(2):1–19.

    Google Scholar 

  23. Borgatti SP. Centrality and network flow. Soc Netw. 2005;27(1):55–71.

    MathSciNet  Google Scholar 

  24. Bonacich P. A behavioral foundation for a structural theory of power in exchange networks. Soc Psychol Q. 1998;61:185–98.

    Google Scholar 

  25. Hubbell CH. An input–output approach to clique identification. Sociometry. 1965;28:377–99.

    Google Scholar 

  26. Taylor M. Influence structures. Sociometry. 1969;32:490–502.

    Google Scholar 

  27. Coleman JS, Katz E, Menzel H. Medical innovation: a diffusion study. Bobbs-Merrill, Indianapolis. 1966

  28. Freeman LC. Centrality in networks: I. Concept Clarifi Soc Netw. 1979;1:215–39.

    Google Scholar 

  29. Freeman LC, Borgatti SP, White DR. Centrality in valued graphs: a measure of betweenness based on network flow. Soc Netw. 1991;13:141–54.

    MathSciNet  Google Scholar 

  30. Friedkin NE. Theoretical foundations for centrality measures. Am J Sociol. 1991;96:1478–504.

    Google Scholar 

  31. Bonacich P, Lloyd P. Eigenvector-like measures of centrality for asymmetric relations. Soc Netw. 2001;23(3):191–201.

    Google Scholar 

  32. Ghosh R, Lerman K. Parameterized centrality metric for network analysis. Phys Rev E. 2011;83(6): 066118.

    Google Scholar 

  33. Bothner MS, Smith EB, White HC. A model of robust positions in social structure. Am J Sociol. 2010;116:943–92.

    Google Scholar 

  34. Rhoades SA. The herfindahl-hirschman index. Fed Res Bull. 1993;79:188.

    Google Scholar 

  35. Singh A, Singh RR, Iyengar SRS. Node-weighted centrality: a new way of centrality hybridization. Comput Soc Netw. 2020;7(1):1–33.

    Google Scholar 

  36. Yang Y, Xie G, Xie J. Mining important nodes in directed weighted complex networks. Discrete Dyn Nat Soc. 2017;2017:7.

    MathSciNet  MATH  Google Scholar 

  37. Qiao T, Shan W, Zhou C. How to identify the most powerful node in complex networks? A novel entropy centrality approach. Entropy. 2017;19(11):614.

    Google Scholar 

  38. Fei L, Zhang Qi, Deng Y. Identifying influential nodes in complex networks based on the inverse-square law. Physica A. 2018;512:1044–59.

    Google Scholar 

  39. Everett MG, Borgatti SP. The centrality of groups and classes. J Math Soc. 1999;23(3):181–201.

    MATH  Google Scholar 

  40. Bonacich P. Simultaneous group and individual centralities. Soc Netw. 1991;13:155–68.

    Google Scholar 

  41. Borgatti SP, Everett MG. A graph-theoretic perspective on centrality. Soc Netw. 2006;28(4):466–84.

    Google Scholar 

  42. Gilles RP. The cooperative game theory of networks and hierarchies. Vol. 44. Springer Science & Business Media, 2010.

  43. del Pozo M, et al. Centrality in directed social networks. A game theoretic approach. Soc Netw. 2011;33(3):191–200.

    Google Scholar 

  44. Grofman B, Owen G. A game theoretic approach to measuring degree of centrality in social networks. Soc Netw. 1982;4(3):213–24.

    MathSciNet  Google Scholar 

  45. Skibski O, Michalak TP, Rahwan T. Axiomatic characterization of game-theoretic centrality. J Artif Intell Res. 2018;62:33–68.

    MathSciNet  MATH  Google Scholar 

  46. Michalak TP, et al. Efficient computation of the Shapley value for game-theoretic network centrality. J Artif Intell Res. 2013;46:607–50.

    MathSciNet  MATH  Google Scholar 

  47. Khmelnitskaya OS, Talman D. The shapley value for directed graph games. Oper Res Lett. 2016;44:143–7.

    MathSciNet  MATH  Google Scholar 

  48. Myerson R. Graphs and cooperation in games. Math Oper Res. 1977;2(3):225–9.

    MathSciNet  MATH  Google Scholar 

  49. Gomez D, et al. Centrality and power in social networks: a game theoretic approach. Math Soc Sci. 2003;46(1):27–54.

    MathSciNet  MATH  Google Scholar 

  50. Suri N, Narahari Y. Determining the top-k nodes in social networks using the shapley value. In AAMAS ’08: Proceedings of the 7th International Joint Conference on Autonomous Agents and Multi-Agent Systems, pp. 1509–1512 2008.

  51. Michalak T, Aadithya K, Szczepanski P, Ravindran B, Jennings N. Efficient computation of the shapley value for game-theoretic network centrality. J Artif Intell Res. 2013;46:607–50.

    MathSciNet  MATH  Google Scholar 

  52. Kempe, Kleinberg JM, Tardos E. Maximizing the spread of influence through a social network. In KDD, pp 137:146, 2003.

  53. Li CT, Lin SD, Shan MK. Influence propagation and maximization for heterogeneous social networks. In: Proceedings of the 21st International Conference on World Wide Web; 2012 Apr 16–20; Lyon, France; p. 559–60. 2012.

  54. Zhou C, Zhang P, Zang W, Guo L. On the upper bounds of spread for greedy algorithms in social network influence maximization. IEEE Trans Knowl Data Eng. 2015;27(10):2770–83.

    Google Scholar 

  55. Kreps DM, Microeconomic Foundations I. Choice and competitive markets. Princeton: Princeton University Press; 2013.

    Google Scholar 

  56. Newman MEJ, Strogatz SH, Watts DJ. Random graphs with arbitrary degree distributions and their applications. Phys Rev E. 2001;64: 026118.

    Google Scholar 

  57. Chung F, Lu L. Connected components in random graphs with given degree sequences. Ann Comb. 2002;6:125–45.

    MathSciNet  MATH  Google Scholar 

  58. Luczak T. Sparse random graphs with a given degree sequence. In: Frieze AM, Luczak T, editors. Proceedings of the symposium on random graphs, Pozna´n 1989. New York: John Wiley; 1992. p. 165–82.

    Google Scholar 

  59. Molloy M, Reed B. A critical point for random graphs with a given degree sequence. Random Struct Algorithms. 1995;6:161–79.

    MathSciNet  MATH  Google Scholar 

  60. Leicht EA, Newman MEJ. Community structure in directed networks. Phys Rev Lett. 2008;100(11):118703.

    Google Scholar 

  61. Blondel VD, Guillaume J-L, Lambiotte R, Lefebvre E, Fast unfolding of communities in large networks. J Stat Mech 2008(10),P10008(12pp) doi:https://doi.org/10.1088/1742-5468/2008/10/P10008. http://arxiv.org/abs/0803.0476.

  62. Newman MEJ, Girvan M. Finding and evaluating community structure in networks. Phys Rev E. 2004;69: 026113.

    Google Scholar 

  63. Newman MEJ. The structure and function of complex networks. SIAM Rev. 2003;45(2):167–256.

    MathSciNet  MATH  Google Scholar 

  64. Newman MEJ. Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality. Phys Rev E. 2001;64:016132.

    Google Scholar 

  65. Zachary WW. An information flow model for conflict and fission in small groups. J Anthropol Res. 1977;33(4):452–73.

    Google Scholar 

  66. Mcauley J, Leskovec J. Discovering social circles in ego networks. ACM Transact Knowl Discov Data. 2014;8(1):1–28.

    Google Scholar 

  67. Wang J, Hou X, Li K, Ding Y. A novel weight neighborhood centrality algorithm for identifying influential spreaders in complex networks. Phys A. 2017;S0378–4371(17):30121–8.

    Google Scholar 

  68. Zejun S, Bin W, Jinfang S, Yixiang H, Yihan W, Junming S. Identifying influential nodes in complex networks based on weighted formal concept analysis. IEEE Access 2017;5:3777–3789

  69. Newman MEJ. Analysis of weighted networks. Phys Rev E. 2004;389:2134–42.

    Google Scholar 

  70. Barrat A, Barthélemy M, Pastor-Satorras R, Vespignani A. The architecture of complex weighted networks. Proc Natl Acad Sci USA. 2004;101(11):3747–52.

    MATH  Google Scholar 

  71. Li M, Fan Y, Chen J, Gao L, Di Z, Jinshan Wu. Weighted networks of scientific communication: the measurement and topological role of weight. Physica A. 2005;350(2–4):643–65615.

    Google Scholar 

  72. Hougaard JH, Allocation in networks, MIT Press, 2018

  73. Borch K. Economic equilibrium under uncertainty. Int Econ Rev 1968;9(3):339–347

  74. Karlan D, et al. Trust and social collateral. Q J Econ. 2009;124(3):1307–61.

    MATH  Google Scholar 

  75. Mobius M, Quoc-Anh D, Rosenblat TS. Social capital in social networks. Retrieved March 3 (2004):2009.

  76. Michalak TP, et al. A new approach to measure social capital using game-theoretic techniques. ACM SIGecom Exchanges. 2015;14(1):95–100.

    Google Scholar 

  77. Jackson MO. Social and economic networks. NJ: Princeton University Press; 2008.

    MATH  Google Scholar 

  78. Sridhar Mandyam K, Kumar Dasgupta A, Sridhar U, Dasgupta P, Chakrabarti A. Network approaches in anomaly detection for disease conditions. Biomed Signal Process Control. 2021;68:102659. https://doi.org/10.1016/j.bspc.2021.102659.

    Article  Google Scholar 

  79. Negre CFA, et al. Eigenvector centrality for characterization of protein allosteric pathways. Proc Natl Acad Sci. 2018;115(52):E12201–8.

    Google Scholar 

  80. Agryzkov T, et al. A centrality measure for urban networks based on the eigenvector centrality concept. Environ Plann B. 2019;46(4):668–89.

    Google Scholar 

  81. Rossi RA, Ahmed NK. The network data repository with interactive graph analytics and visualization. Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence. http://networkrepository.com. Accessed 3 Jan 2021.

Download references

Acknowledgements

The authors thank anonymous reviewers for comments and suggestions to improve the quality of this paper. The first author thanks Prof Srinath Srinivasa, and research students Raksha RP and Jayati Deshmukh of the Web Science Lab, IIIT, Bangalore for helping to explore many applications of the central ideas of dominance centrality set out in this work, including early versions of DON and DONEX measures. Thanks are also due to Prof Anjan Dasgupta, Department of Biochemistry, Calcutta University, for many illuminating discussions and close collaboration in the development of time-series analysis applications of the basic dominance metrics described here.

Funding

This work has the support of Ecometrix Research.

Author information

Authors and Affiliations

  1. Center for Applied Data Science IIIT, Bangalore, India

    Sridhar Mandyam Kannappan

  2. Ecometrix Research, Bangalore, India

    Usha Sridhar

Authors
  1. Sridhar Mandyam Kannappan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Usha Sridhar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sridhar Mandyam Kannappan.

Ethics declarations

Conflict of interest

There are no conflicts or competing interests in the publication of this research.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file1 (PDF 904 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mandyam Kannappan, S., Sridhar, U. A Flow-Based Node Dominance Centrality Measure for Complex Networks. SN COMPUT. SCI. 3, 379 (2022). https://doi.org/10.1007/s42979-022-01270-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s42979-022-01270-2

Keywords

Search

Navigation