Entropy of Digraphs and Infinite Networks



The information content of a graph G is defined in Mowshowitz (Bull Math Biophys 30:175–204, 1968) as the entropy of a finite probability scheme associated with the vertex partition determined by the automorphism group of G. This provides a quantitative measure of the symmetry structure of a graph that has been applied to problems in such diverse fields as chemistry, biology, sociology, and computer science (Mowshowitz and Mitsou, Entropy, orbits and spectra of graphs, Wiley-VCH, 2009). The measure extends naturally to directed graphs (digraphs) and can be defined for infinite graphs as well (Mowshowitz, Bull Math Biophys 30:225–240, 1968).This chapter focuses on the information content of digraphs and infinite graphs. In particular, the information content of digraph products and recursively defined infinite graphs is examined.


Digraphs Entropy Infinite graphs Information content Networks 



Research was sponsored by the US Army Research Laboratory and the UK Ministry of Defence and was accomplished under Agreement Number W911NF-06-3-0001. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the US Army Research Laboratory, the US Government, the UK Ministry of Defence, or the UK Government. The US and UK Governments are authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation hereon.


  1. 1.
    Balaban, A.T., Balaban, T.S.: New vertex invariants and topological indices of chemical graphs based on information on distances. J. Math. Chem. 8, 383–397 (1991)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bent, G., Dantressangle, P., Vyvyan, D., Mowshowitz, A., Mitsou, V.: A dynamic distributed federated database. Proceedings of the Second Annual Conference of the International Technology Alliance, Imperial College, London, September 2008Google Scholar
  3. 3.
    Bertz, S.H.: The first general index of molecular complexity. J. Am. Chem. Soc. 103, 3241–3243 (1981)CrossRefGoogle Scholar
  4. 4.
    Bloom, G., Kennedy, J.W., Quintas, L.V.: Distance degree regular graphs. In: Chartrand, G. (ed.) The Theory of Applications of Graphs, pp. 95–108. Wiley, New York (1981)Google Scholar
  5. 5.
    Bollobás, B., Riordan, O.: The diameter of a scalefree random graph. Combinatorika 24, 5–34 (2004)MATHCrossRefGoogle Scholar
  6. 6.
    Bonchev, D.: Information Theoretic Indices for Characterization of Chemical Structures. Research Studies Press, Chichester, UK (1983)Google Scholar
  7. 7.
    Bonchev, D.: Complexity in Chemistry, Biology, and Ecology. Mathematical and Computational Chemistry series. Springer, New York (2005)MATHCrossRefGoogle Scholar
  8. 8.
    Bonchev, D., Trinajstic, N.: Information theory, distance matrix and molecular branching. J. Chem. Phys. 67, 4517–4533 (1977)CrossRefGoogle Scholar
  9. 9.
    Dehmer, M., Emmert-Streib, F.: Structural information content of chemical networks. Zeitschrift für Naturforschung A 63a, 155–159 (2008)Google Scholar
  10. 10.
    Dehmer, M., Emmert-Streib, F., Mehler, A. (eds.): Towards an Information Theory of Complex Networks: Statistical Methods and Applications. Springer/Birkhäuser, Berlin (2011)MATHGoogle Scholar
  11. 11.
    Dehmer, M., Mowshowitz, A.: Inequalities for entropy-based measures of network information content. Appl. Math. Comput. 215, 4263–4271 (2010)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Harary, F.: Graph Theory. Addison Wesley, Reading, MA (1969)Google Scholar
  13. 13.
    Hirata, H., Ulanowicz, R.E.: Information theoretical analysis of ecological networks. Int. J. Syst. Sci. 15, 261–270 (1984)MATHCrossRefGoogle Scholar
  14. 14.
    Khinchin, A.I.: Mathematical Foundations of Information Theory. Dover Publications, New York (1957)MATHGoogle Scholar
  15. 15.
    Konstantinova, E.V., Skorobogatov, V.A., Vidyuk, M.V.: Applications of information theory in chemical graph theory. Indian J. Chem. 42, 1227–1240 (2002)Google Scholar
  16. 16.
    Körner, J.: Coding of an information source having ambiguous alphabet and the entropy of graphs. In: Transactions of the 6th Prague Conference on Information Theory, 411–425 (1973)Google Scholar
  17. 17.
    Mowshowitz, A.: Entropy and the complexity of graphs: I. An index of the relative complexity of a graph. Bull. Math. Biophys. 30, 175–204 (1968)MathSciNetMATHGoogle Scholar
  18. 18.
    Mowshowitz, A.: Entropy and the complexity of graphs: II. The information content of digraphs and infinite graphs. Bull. Math. Biophys. 30, 225–240 (1968)MathSciNetMATHGoogle Scholar
  19. 19.
    Mowshowitz, A., Mitsou, V., Bent, G.: Models of network growth by combination. Proceedings of the Second Annual Conference of the International Technology Alliance, Imperial College, London, September 2008Google Scholar
  20. 20.
    Mowshowitz, A., Mitsou, V.: Entropy, orbits and spectra of graphs. In: Dehmer, M. (ed.) Analysis of Complex Networks: From Biology to Linguistics. Wiley-VCH, Weinheim (2009, in press)Google Scholar
  21. 21.
    Rashevsky, N.: Life, information theory, and topology. Bull. Math. Biophys. 17, 229–235 (1955)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Skorobogatov, V.A., Dobrynin, A.A.: Metric analysis of graphs. MATCH 23, 105–151 (1988)MathSciNetMATHGoogle Scholar
  23. 23.
    Sole, R.V., Valverde, S.: Information theory of complex networks: On evolution and architectural constraints. Lect. Notes Phys. 650, 189–207 (2004)MathSciNetGoogle Scholar
  24. 24.
    Wasserman, S., Faust, K.: Social Network Analysis: Methods and Applications, Structural Analysis in the Social Sciences. Cambridge University Press, Cambridge (1994)Google Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Computer ScienceThe City College of New York (CUNY)New YorkUSA

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