11. Circumscribed Complexity in Ecological Networks

  1. Matthias Dehmer3,
  2. Abbe Mowshowitz4 and
  3. Frank Emmert-Streib4
  1. Robert E. Ulanowicz1,2

Published Online: 12 JUL 2013

DOI: 10.1002/9783527670468.ch11

Advances in Network Complexity

Advances in Network Complexity

How to Cite

Ulanowicz, R. E. (2013) Circumscribed Complexity in Ecological 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.ch11

Editor Information

  1. 3

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

  2. 4

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

Author Information

  1. 1

    University of Florida, Arthur R. Marshall Laboratory, Department of Biology, Gainesville, FL, 32611-8525, USA

  2. 2

    University of Maryland, Chesapeake Biological Laboratory, P.O. Box 38, Solomons, MD, 20688-0038, USA

Publication History

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

ISBN Information

Print ISBN: 9783527332915

Online ISBN: 9783527670468



  • complexity;
  • conditional entropy;
  • constraint;
  • ecosystems;
  • flexibility;
  • link density;
  • mutual information;
  • networks;
  • persistence;
  • statistical entropy;
  • weighted digraphs


Networks, by their very nature, are complex entities, because they are metaphors for the entanglement of constraint and indeterminacy. The apportionment of these polar opposite behaviors can be quantified by two complementary indices borrowed from information theory – average mutual information and conditional “entropy.” The link density of a weighted digraph is a function of its conditional entropy and, according to the Wigner semicircle criterion, is circumscribed in magnitude to be less than about 3.01 links per node. In ecosystems, the mutual information appears to be functionally related to the link density and hence similarly constrained. Although weighted ecological networks may possess hundreds of actual nodes and thousands of real edges, these results indicate that the dynamics of even the most complicated examples should be no more complex than those of a network of at most 13 effective nodes and 40 effective edges.