8. Dimension Measure for Complex Networks

  1. Matthias Dehmer2,
  2. Abbe Mowshowitz3 and
  3. Frank Emmert-Streib3
  1. O. Shanker

Published Online: 12 JUL 2013

DOI: 10.1002/9783527670468.ch08

Advances in Network Complexity

Advances in Network Complexity

How to Cite

Shanker, O. (2013) Dimension Measure for Complex Networks, in Advances in Network Complexity (eds M. Dehmer, A. Mowshowitz and F. Emmert-Streib), Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany. doi: 10.1002/9783527670468.ch08

Editor Information

  1. 2

    UMIT, Institut für Bioinformatik und, Translationale Forschung, Eduard-Wallnöfer-Zentrum 1, 6060 Hall in Tyrol, Austria

  2. 3

    The City College of New York, Department of Computer Science, 138th Street at Convent Avenue, New York, NY 10031, USA

Author Information

  1. Shutterfly Inc., 2800 Bridge Pkwy, Redwood City, CA, 94065, USA

Publication History

  1. Published Online: 12 JUL 2013
  2. Published Print: 10 JUL 2013

ISBN Information

Print ISBN: 9783527332915

Online ISBN: 9783527670468

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Keywords:

  • complex network dimension;
  • complex network Zeta Function;
  • Kolmogorov complexity;
  • linguistic text analysis;
  • phase transitions

Summary

In this chapter, the authors survey one of the complexity measures, the measure based on graph dimension. The first section of this chapter reviews the volume definition of dimension for complex networks. The second section defines the dimension using the complex network zeta function and relates it to the Kolmogorov complexity. The third section reviews the interesting analogies between the complex network dimension and the properties of complexity classes in theoretical computer science. The fourth section shows how we can avoid taking averages over the number of nodes by using the concept of lim sup. The fifth section presents an application to linguistic text analysis. The sixth section also presents an application to phase transitions. The seventh section explains the complex network zeta function for some complex networks. The eighth section reviews other definitions of complexity useful for networks. The conclusions are given in the final section.