Skip to main content
SpringerLink
Log in
Book cover

Business and Consumer Analytics: New Ideas pp 401–433Cite as

Centrality in Networks: Finding the Most Important Nodes

  • Chapter
  • First Online:

Abstract

Real networks are heterogeneous structures, with edges unevenly distributed among nodes, presenting community structure, motifs, transitivity, rich clubs, and other kinds of topological patterns. Consequently, the roles played by nodes in a network can differ greatly. For example, some nodes may be connectors between parts of the network, others may be central or peripheral, etc. The objective of this chapter is to describe how we can find the most important nodes in networks. The idea is to define a centrality measure for each node in the network, sort the nodes according to their centralities, and fix our attention to the first ranked nodes, which can be considered as the most relevant ones with respect to this centrality measure.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   299.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   379.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    In some texts the adjacency matrix is defined in the opposite direction, i.e., a ij is used to encode an edge from node j to node i, for example, in [44]. This is an important issue to care about when dealing with directed networks.

  2. 2.

    Network of Thrones: https://www.macalester.edu/~abeverid/thrones.html.

  3. 3.

    Pajek: http://mrvar.fdv.uni-lj.si/pajek.

  4. 4.

    Gephi: https://gephi.org.

  5. 5.

    Radatools: http://deim.urv.cat/~sergio.gomez/radatools.php.

  6. 6.

    Cytoscape: http://www.cytoscape.org.

  7. 7.

    igraph: http://igraph.org.

  8. 8.

    NetworkX: http://networkx.github.io.

  9. 9.

    SNAP: http://snap.stanford.edu/snap.

  10. 10.

    Visone: https://www.visone.info.

  11. 11.

    MuxViz: http://muxviz.net.

  12. 12.

    graph-tool: https://graph-tool.skewed.de.

References

  1. Ahnert, S., Garlaschelli, D., Fink, T., Caldarelli, G.: Ensemble approach to the analysis of weighted networks. Physical Review E 76(1), 016,101 (2007)

    Article  Google Scholar 

  2. Anthonisse, J.M.: The rush in a directed graph. Stichting Mathematisch Centrum. Mathematische Besliskunde (BN 9/71), 1–10 (1971)

    Google Scholar 

  3. Arenas, A., Fernandez, A., Gomez, S.: Analysis of the structure of complex networks at different resolution levels. New Journal of Physics 10(5), 053,039 (2008)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Barrat, A., Barthelemy, M., Pastor-Satorras, R., Vespignani, A.: The architecture of complex weighted networks. Proceedings of the National Academy of Sciences USA 101(11), 3747–3752 (2004)

    Article  Google Scholar 

  6. Bastian, M., Heymann, S., Jacomy, M., et al.: Gephi: an open source software for exploring and manipulating networks. ICWSM 8, 361–362 (2009)

    Google Scholar 

  7. Batagelj, V., Mrvar, A.: Pajek – program for large network analysis. Connections 21(2), 47–57 (1998)

    MATH  Google Scholar 

  8. Battiston, F., Nicosia, V., Latora, V.: Structural measures for multiplex networks. Physical Review E 89(3), 032,804 (2014)

    Article  Google Scholar 

  9. Bavelas, A.: Communication patterns in task-oriented groups. The Journal of the Acoustical Society of America 22(6), 725–730 (1950)

    Article  Google Scholar 

  10. Beauchamp, M.A.: An improved index of centrality. Behavioral Science 10(2), 161–163 (1965)

    Article  Google Scholar 

  11. Beveridge, A., Shan, J.: Network of thrones. Math Horizons 23(4), 18–22 (2016)

    Article  MathSciNet  Google Scholar 

  12. Boccaletti, S., Bianconi, G., Criado, R., Del Genio, C.I., Gómez-Gardenes, J., Romance, M., Sendina-Nadal, I., Wang, Z., Zanin, M.: The structure and dynamics of multilayer networks. Physics Reports 544(1), 1–122 (2014)

    Article  MathSciNet  Google Scholar 

  13. Bonacich, P.: Factoring and weighting approaches to status scores and clique identification. Journal of Mathematical Sociology 2(1), 113–120 (1972)

    Article  Google Scholar 

  14. Bonacich, P.: Technique for analyzing overlapping memberships. Sociological Methodology 4, 176–185 (1972)

    Article  Google Scholar 

  15. Bonacich, P.: Power and centrality: A family of measures. American Journal of Sociology 92(5), 1170–1182 (1987)

    Article  Google Scholar 

  16. Brandes, U.: A faster algorithm for betweenness centrality. Journal of Mathematical Sociology 25(2), 163–177 (2001)

    Article  Google Scholar 

  17. Brandes, U., Erlebach, T.: Network Analysis: Methodological Foundations, Lecture Notes in Computer Science, vol. 3418. Springer (2005)

    Google Scholar 

  18. Brandes, U., Wagner, D.: Analysis and visualization of social networks. Graph drawing software pp. 321–340 (2004)

    Google Scholar 

  19. Brin, S., Page, L.: The anatomy of a large-scale hypertextual web search engine. Computer Networks and ISDN Systems 30(1), 107–117 (1998)

    Article  Google Scholar 

  20. Csardi, G., Nepusz, T.: The igraph software package for complex network research. InterJournal, Complex Systems 1695(5), 1–9 (2006)

    Google Scholar 

  21. Danon, L., Diaz-Guilera, A., Duch, J., Arenas, A.: Comparing community structure identification. Journal of Statistical Mechanics: Theory and Experiment 2005(09), P09,008 (2005)

    Article  Google Scholar 

  22. De Domenico, M., Porter, M.A., Arenas, A.: Muxviz: a tool for multilayer analysis and visualization of networks. Journal of Complex Networks 3(2), 159 (2015).

    Article  Google Scholar 

  23. De Domenico, M., Solé-Ribalta, A., Cozzo, E., Kivelä, M., Moreno, Y., Porter, M.A., Gómez, S., Arenas, A.: Mathematical formulation of multilayer networks. Physical Review X 3(4), 041,022 (2013)

    Article  Google Scholar 

  24. De Domenico, M., Solé-Ribalta, A., Omodei, E., Gómez, S., Arenas, A.: Ranking in interconnected multilayer networks reveals versatile nodes. Nature Communications 6 (2015)

    Google Scholar 

  25. Dekker, A.: Conceptual distance in social network analysis. Journal of Social Structure (JOSS) 6 (2005)

    Google Scholar 

  26. Duch, J., Arenas, A.: Community detection in complex networks using extremal optimization. Physical review E 72(2), 027,104 (2005)

    Article  Google Scholar 

  27. Floyd, R.W.: Algorithm 97: shortest path. Communications of the ACM 5(6), 345 (1962)

    Article  Google Scholar 

  28. Fortunato, S.: Community detection in graphs. Physics reports 486(3), 75–174 (2010)

    Article  MathSciNet  Google Scholar 

  29. Freeman, L.C.: A set of measures of centrality based on betweenness. Sociometry pp. 35–41 (1977)

    Article  Google Scholar 

  30. Freeman, L.C.: Centrality in social networks conceptual clarification. Social Networks 1(3), 215–239 (1979)

    Article  Google Scholar 

  31. Frobenius, G.: Über matrizen aus nicht negativen elementen. Sitzungsber. Königl. Preuss. Akad. Wiss. pp. 456–477 (1912)

    Google Scholar 

  32. Garrison, W.L.: Connectivity of the interstate highway system. Papers and Proceedings of the Regional Science Association 6, 121–137 (1960)

    Article  Google Scholar 

  33. Gómez, S., Fernández, A.: Radatools software, communities detection in complex networks and other tools (2011)

    Google Scholar 

  34. Guimerà, R., Diaz-Guilera, A., Vega-Redondo, F., Cabrales, A., Arenas, A.: Optimal network topologies for local search with congestion. Physical Review Letters 89(24), 248,701 (2002)

    Article  Google Scholar 

  35. Halu, A., Mondragón, R.J., Panzarasa, P., Bianconi, G.: Multiplex pagerank. PLOS ONE 8(10), e78,293 (2013)

    Article  Google Scholar 

  36. Jacob, R., Koschützki, D., Lehmann, K., Peeters, L., Tenfelde-Podehl, D.: Algorithms for centrality indices. Network Analysis pp. 62–82 (2005)

    Google Scholar 

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

    Article  Google Scholar 

  38. Kivelä, M., Arenas, A., Barthelemy, M., Gleeson, J.P., Moreno, Y., Porter, M.A.: Multilayer networks. Journal of Complex Networks 2(3), 203–271 (2014)

    Article  Google Scholar 

  39. Kleinberg, J.M.: Authoritative sources in a hyperlinked environment. Journal of the ACM (JACM) 46(5), 604–632 (1999)

    Article  MathSciNet  Google Scholar 

  40. Koschützki, D., Lehmann, K., Peeters, L., Richter, S., Tenfelde-Podehl, D., Zlotowski, O.: Centrality indices. Network Analysis pp. 16–61 (2005)

    Google Scholar 

  41. Koschützki, D., Lehmann, K., Tenfelde-Podehl, D., Zlotowski, O.: Advanced centrality concepts. Network Analysis pp. 83–111 (2005)

    Google Scholar 

  42. Leskovec, J., Sosič, R.: Snap: Stanford network analysis platform (2013)

    Google Scholar 

  43. Lovász, L.: Random walks on graphs: A survey. Combinatorics, Paul Erdos is Eighty 2(1), 1–46 (1993)

    Google Scholar 

  44. Newman, M.: Networks: An Introduction. Oxford University Press, Inc., New York, NY, USA (2010)

    Book  Google Scholar 

  45. Newman, M.E.: Fast algorithm for detecting community structure in networks. Physical review E 69(6), 066,133 (2004)

    Article  Google Scholar 

  46. Newman, M.E.: A measure of betweenness centrality based on random walks. Social Networks 27(1), 39–54 (2005)

    Article  Google Scholar 

  47. Newman, M.E., Girvan, M.: Finding and evaluating community structure in networks. Physical review E 69(2), 026,113 (2004)

    Article  Google Scholar 

  48. Nieminen, J.: On the centrality in a directed graph. Social Science Research 2, 371–378 (1973)

    Article  Google Scholar 

  49. Noh, J.D., Rieger, H.: Random walks on complex networks. Physical Review Letters. 92(11), 118,701 (2004)

    Article  Google Scholar 

  50. Peixoto, T.P.: The graph-tool python library. figshare (2014)

    Google Scholar 

  51. Perron, O.: Zur theorie der matrices. Mathematische Annalen 64(2), 248–263 (1907)

    Article  MathSciNet  Google Scholar 

  52. Pitts, F.R.: A graph theoretic approach to historical geography. The Professional Geographer 17, 15–20 (1965)

    Article  Google Scholar 

  53. Rosvall, M., Bergstrom, C.T.: Maps of random walks on complex networks reveal community structure. Proceedings of the National Academy of Sciences 105(4), 1118–1123 (2008)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  55. Schult, D.A., Swart, P.: Exploring network structure, dynamics, and function using networkx. In: Proceedings of the 7th Python in Science Conferences (SciPy 2008), vol. 2008, pp. 11–16 (2008)

    Google Scholar 

  56. Shannon, P., Markiel, A., Ozier, O., Baliga, N.S., Wang, J.T., Ramage, D., Amin, N., Schwikowski, B., Ideker, T.: Cytoscape: a software environment for integrated models of biomolecular interaction networks. Genome research 13(11), 2498–2504 (2003)

    Article  Google Scholar 

  57. Shaw, M.E.: Group structure and the behavior of individuals in small groups. Journal of Psychology 38, 139–149 (1954)

    Article  Google Scholar 

  58. Solá, L., Romance, M., Criado, R., Flores, J., del Amo, A.G., Boccaletti, S.: Eigenvector centrality of nodes in multiplex networks. Chaos 3, 033,131 (2013)

    Article  Google Scholar 

  59. Solé-Ribalta, A., De Domenico, M., Gómez, S., Arenas, A.: Random walk centrality in interconnected multilayer networks. Physica D: Nonlinear Phenomena 323, 73–79 (2016)

    Article  MathSciNet  Google Scholar 

  60. Solé-Ribalta, A., Gómez, S., Arenas, A.: A model to identify urban traffic congestion hotspots in complex networks. Royal Society Open Science 3(10), 160,098 (2016)

    Article  Google Scholar 

  61. Yang, S.J.: Exploring complex networks by walking on them. Physical Review E 71(1), 016,107 (2005)

    Article  MathSciNet  Google Scholar 

  62. Zhang, Z., Julaiti, A., Hou, B., Zhang, H., Chen, G.: Mean first-passage time for random walks on undirected networks. The European Physical Journal B-Condensed Matter and Complex Systems 84(4), 691–697 (2011)

    Article  Google Scholar 

Download references

Acknowledgements

S.G. acknowledges funding from the Spanish Ministerio de Economía y Competitividad (grant number FIS2015-71582-C2-1).

Author information

Authors and Affiliations

  1. Universitat Rovira i Virgili, Tarragona, Spain

    Sergio Gómez

Authors
  1. Sergio Gómez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergio Gómez .

Editor information

Editors and Affiliations

  1. School of Electrical Engineering and Computing, The University of Newcastle, Callaghan, NSW, Australia

    Pablo Moscato

  2. School of Electrical Engineering and Computing, The University of Newcastle, Callaghan, NSW, Australia

    Natalie Jane de Vries

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Gómez, S. (2019). Centrality in Networks: Finding the Most Important Nodes. In: Moscato, P., de Vries, N. (eds) Business and Consumer Analytics: New Ideas. Springer, Cham. https://doi.org/10.1007/978-3-030-06222-4_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-06222-4_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-06221-7

  • Online ISBN: 978-3-030-06222-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation