Positional Dominance: Concepts and Algorithms

  • Ulrik Brandes
  • Moritz Heine
  • Julian Müller
  • Mark Ortmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10156)

Abstract

Centrality indices assign values to the vertices of a graph such that vertices with higher values are considered more central. Triggered by a recent result on the preservation of the vicinal preorder in rankings obtained from common centrality indices, we review and extend notions of domination among vertices. These may serve as building blocks for new concepts of centrality that extend more directly, and more coherently, to more general types of data such as multilayer networks. We also give efficient algorithms to construct the associated partial rankings.

References

  1. 1.
    Brandes, U.: Network positions. Methodol. Innov. 9, 2059799116630650 (2016)Google Scholar
  2. 2.
    Chiba, N., Nishizeki, T.: Arboricity and subgraph listing algorithms. SIAM J. Comput. 14(1), 210–223 (1985)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    van Emde Boas, P.: Preserving order in a forest in less than logarithmic time and linear space. Inf. Process. Lett. 6(3), 80–82 (1977)CrossRefMATHGoogle Scholar
  4. 4.
    Eppstein, D., Spiro, E.S.: The h-index of a graph and its application to dynamic subgraph statistics. J. Graph Algorithms Appl. 16(2), 543–567 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Foldes, S., Hammer, P.L.: The Dilworth number of a graph. Ann. Discret. Math. 2, 211–219 (1978)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Freeman, L.C.: A set of measures of centrality based on betweenness. Sociometry 40(1), 35–41 (1977)CrossRefGoogle Scholar
  7. 7.
    Habib, M., Paul, C.: A survey of the algorithmic aspects of modular decomposition. Comput. Sci. Rev. 4(1), 41–59 (2010)CrossRefMATHGoogle Scholar
  8. 8.
    Heggernes, P., Kratsch, D.: Linear-time certifying recognition algorithms and forbidden induced subgraphs. Nordic J. Comput. 14(1–2), 87–108 (2007)MathSciNetMATHGoogle Scholar
  9. 9.
    Hopcroft, J.E., Karp, R.M.: An \({\rm n}^{{5/2}}\) algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2(4), 225–231 (1973)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Lagraa, S., Seba, H.: An efficient exact algorithm for triangle listing in large graphs. Data Min. Knowl. Disc. 30(5), 1350–1369 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lerner, J.: Role assignments. In: Brandes, U., Erlebach, T. (eds.) Network Analysis. LNCS, vol. 3418, pp. 216–252. Springer, Heidelberg (2005). doi:10.1007/978-3-540-31955-9_9 CrossRefGoogle Scholar
  12. 12.
    Lorrain, F., White, H.C.: Structural equivalence of individuals in social networks. J. Math. Soc. 1(1), 49–80 (1971)CrossRefGoogle Scholar
  13. 13.
    Lü, L., Chen, D., Ren, X.L., Zhang, Q.M., Zhang, Y.C., Zhou, T.: Vital nodes identification in complex networks. Phys. Rep. 650, 1–63 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lueker, G.S.: A data structure for orthogonal range queries. In: 19th Annual Symposium on Foundations of Computer Science, Ann Arbor, Michigan, USA, 16–18 October 1978, pp. 28–34 (1978)Google Scholar
  15. 15.
    Mahadev, N.V., Peled, U.N.: Threshold Graphs and Related Topics, Annals of Discrete Mathematics, vol. 56. Elsevier, Amsterdam (1995)MATHGoogle Scholar
  16. 16.
    Nash-Williams, C.S.J.A.: Decomposition of finite graphs into forests. J. Lond. Math. Soc. 39(1), 12 (1964)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ortmann, M., Brandes, U.: Triangle listing algorithms: back from the diversion. In: Proceedings of the 16th Workshop on Algorithm Engineering and Experiments (ALENEX 2014), pp. 1–8 (2014)Google Scholar
  18. 18.
    Paige, R., Tarjan, R.E.: Three partition refinement algorithms. SIAM J. Comput. 16(6), 973–989 (1987)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Pritchard, P.: On computing the subset graph of a collection of sets. J. Algorithms 33(2), 187–203 (1999)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Pritchard, P.: A simple sub-quadratic algorithm for computing the subset partial order. Inf. Process. Lett. 56(6), 337–341 (1995)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Sabidussi, G.: The centrality index of a graph. Psychometrika 31(4), 581–603 (1966)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Schoch, D., Brandes, U.: Re-conceptualizing centrality in social networks. Eur. J. Appl. Math. 27(6), 971–985 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Yellin, D.M., Jutla, C.S.: Finding extremal sets in less than quadratic time. Inf. Process. Lett. 48(1), 29–34 (1993)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Ulrik Brandes
    • 1
  • Moritz Heine
    • 1
  • Julian Müller
    • 1
  • Mark Ortmann
    • 1
  1. 1.Computer and Information ScienceUniversity of KonstanzKonstanzGermany

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