Advertisement

Minimum Spanning Markovian Trees: Introducing Context-Sensitivity into the Generation of Spanning Trees

  • Alexander Mehler
Chapter

Abstract

This chapter introduces a novel class of graphs: Minimum Spanning Markovian Trees (MSMTs). The idea behind MSMTs is to provide spanning trees that minimize the costs of edge traversals in a Markovian manner, that is, in terms of the path starting with the root of the tree and ending at the vertex under consideration. In a second part, the chapter generalizes this class of spanning trees in order to allow for damped Markovian effects in the course of spanning. These two effects, (1) the sensitivity to the contexts generated by consecutive edges and (2) the decreasing impact of more antecedent (or “weakly remembered”) vertices, are well known in cognitive modeling [6, 10, 21, 23]. In this sense, the chapter can also be read as an effort to introduce a graph model to support the simulation of cognitive systems. Note that MSMTs are not to be confused with branching Markov chains or Markov trees [20] as we focus on generating spanning trees from given weighted undirected networks.

Keywords

Markovian trees Minimum spanning trees Cohesion trees Linguistic networks Semiotic networks 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

Financial support of the German Federal Ministry of Education (BMBF) through the research project Linguistic Networks and of the German Research Foundation (DFG) through the Excellence Cluster 277 Cognitive Interaction Technology (via the Project Knowledge Enhanced Embodied Cognitive Interaction Technologies (KnowCIT)), the SFB 673 Alignment in Communication (via the Project X1 Multimodal Alignment Corpora: Statistical Modeling and Information Management), the Research Group 437 Text Technological Information Modeling (via the Project A4 Induction of Document Grammars for Webgenre Representation), and the LIS-Project Content-Based P2P-Agents for Thematic Structuring and Search Optimization in Digital Libraries at Bielefeld University is gratefully acknowledged.

References

  1. 1.
    Caldarelli G, Vespignani A (eds) (2007) Large scale structure and dynamics of complex networks. World Scientific, Hackensack, NJMATHGoogle Scholar
  2. 2.
    Dehmer M, Mehler A (2007) A new method of measuring the similarity for a special class of directed graphs. Tatra Mt Math Publ 36:39–59MATHMathSciNetGoogle Scholar
  3. 3.
    Dehmer M, Mehler A, Emmert-Streib F (2007) Graph-theoretical characterizations of generalized trees. In: Proceedings of the 2007 international conference on machine learning: models, technologies, and applications (MLMTA’07), Las Vegas, pp 25–28Google Scholar
  4. 4.
    Diestel R (2005) Graph theory. Springer, HeidelbergMATHGoogle Scholar
  5. 5.
    Fensel D, Hendler J, Lieberman H, Wahlster W (2003) Spinning the semantic Web. Bringing the World Wide Web to its full potential. MIT Press, Cambridge, MAGoogle Scholar
  6. 6.
    Gärdenfors P (2000) Conceptual spaces. MIT Press, Cambridge, MAGoogle Scholar
  7. 7.
    Gritzmann P (2007) On the mathematics of semantic spaces. In: Mehler A, Köhler R (eds) Aspects of automatic text analysis, vol 209. Studies in fuzziness and soft computing. Springer, Berlin, pp 95–115CrossRefGoogle Scholar
  8. 8.
    Halliday MAK, Hasan R (1976) Cohesion in english. Longman, LondonGoogle Scholar
  9. 9.
    Jungnickel D (2008) Graphs, networks and algorithms. Springer, BerlinMATHCrossRefGoogle Scholar
  10. 10.
    Kintsch W (1998) Comprehension: a paradigm for cognition. Cambridge University Press, CambridgeGoogle Scholar
  11. 11.
    Kintsch W (2001) Predication. Cognitive Sci 25:173–202CrossRefGoogle Scholar
  12. 12.
    Lin D (1998) Automatic retrieval and clustering of similar words. In: Proceedings of the COLING-ACL ’98, pp 768–774Google Scholar
  13. 13.
    Martin JR (1992) English text: system and structure. John Benjamins, Philadelphia, PAGoogle Scholar
  14. 14.
    Mehler A (2002) Hierarchical analysis of text similarity data. Künstliche Intelligenz (KI) 2: 12–16Google Scholar
  15. 15.
    Mehler A (2002) Hierarchical orderings of textual units. In: Proceedings of the 19th international conference on computational linguistics (COLING ’02), Taipei, San Francisco, pp 646–652Google Scholar
  16. 16.
    Mehler A (2005) Lexical chaining as a source of text chaining. In: Patrick J, Matthiessen C (eds) Proceedings of the 1st computational systemic functional grammar conference, University of Sydney, Australia, pp 12–21Google Scholar
  17. 17.
    Mehler A (2008) Structural similarities of complex networks: a computational model by example of wiki graphs. Appl Artif Intell 22(7&8):619–683CrossRefGoogle Scholar
  18. 18.
    Mehler A (2009) Generalized shortest paths trees: a novel graph class applied to semiotic networks. In: Dehmer M, Emmert-Streib F (eds) Analysis of complex networks: from biology to linguistics. Wiley-VCH, Weinheim, pp 175–220Google Scholar
  19. 19.
    Menczer F (2004) Lexical and semantic clustering by web links. J Am Soc Inf Sci Technol 55(14):1261–1269CrossRefGoogle Scholar
  20. 20.
    Menshikov MV, Volkov SE (1997) Branching Markov chains: qualitative characteristics. Markov Proc Relat Fields 3:1–18MathSciNetGoogle Scholar
  21. 21.
    Murphy GL (2002) The big book of concepts. MIT Press, CambridgeGoogle Scholar
  22. 22.
    Rieger BB (1978) Feasible fuzzy semantics. In: 7th International conference on computational linguistics (COLING-78), pp 41–43Google Scholar
  23. 23.
    Sharkey AJC, Sharkey NE (1992) Weak contextual constraints in text and word priming. J Mem Lang 31(4):543–572CrossRefGoogle Scholar
  24. 24.
    Tarjan RE (1983) Data structures and network algorithms. Society for Industrial and Applied Mathematics, PhiladelphiaGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Goethe-University Frankfurt am MainFrankfurt am MainGermany

Personalised recommendations